--- /dev/null
+- Relative isotopic mass - mass of an isotope (compared to Carbon-12)
+- Atomic number (protons) is constant between isotopes - defines element.
+- Mass number - number of nucleons (defines isotope)
+- Relative atomic mass - average mass: $A_r = {{RIM \times abundance} \over 100}$
+- Relative atomic mass ($M_r$) - sum of relative atomic masses
+- Mole $6.02 \times 10^{23}$ (= no of atoms in Carbon-12) $=N_A$ (Avogadro's constant)
+- Molar mass - physical mass of 1 mol of a substance ($6.02 \times 10^{23}$ atoms)
+ - Unit: grams / mole
+
+$$
+n = {m \over M}
--- /dev/null
+\relax
+\citation{gmi}
+\citation{fig1}
+\citation{fig1}
+\citation{nzmfg}
+\@writefile{toc}{\contentsline {section}{\numberline {1}Invention and Manufacture}{1}}
+\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces amorphous vs crystalline polymers\cite {fig1}}}{1}}
+\newlabel{fig:structure}{{1}{1}}
+\@writefile{toc}{\contentsline {paragraph}{}{1}}
+\@writefile{toc}{\contentsline {paragraph}{}{1}}
+\@writefile{toc}{\contentsline {paragraph}{}{1}}
+\@writefile{toc}{\contentsline {paragraph}{}{1}}
+\@writefile{toc}{\contentsline {paragraph}{}{1}}
+\citation{wiki:xxx}
+\citation{wiki:xxx}
+\@writefile{toc}{\contentsline {paragraph}{}{2}}
+\@writefile{toc}{\contentsline {section}{\numberline {2}Structure}{2}}
+\pgfsyspdfmark {pgfid1}{8455454}{32637561}
+\pgfsyspdfmark {pgfid2}{18932125}{32637561}
+\pgfsyspdfmark {pgfid3}{26320774}{32637561}
+\@writefile{toc}{\contentsline {section}{\numberline {3}Uses}{2}}
+\@writefile{toc}{\contentsline {section}{\numberline {4}Issues}{2}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {4.1}Health}{2}}
+\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces underground installation of agricultural HDPE pipe\cite {wiki:xxx}}}{2}}
+\newlabel{fig:pipe}{{2}{2}}
+\bibstyle{plain}
+\bibdata{hdpe}
+\bibcite{gmi}{1}
+\bibcite{fig1}{2}
+\bibcite{pubchem}{3}
+\bibcite{nzmfg}{4}
+\bibcite{macrog}{5}
+\bibcite{wiki:xxx}{6}
+\citation{*}
+\@writefile{toc}{\contentsline {subsection}{\numberline {4.2}Environmental}{3}}
--- /dev/null
+\begin{thebibliography}{1}
+
+\bibitem{gmi}
+{Global Market Insights}.
+\newblock High density polyethylene (hdpe) market size.
+\newblock
+ \url{https://www.gminsights.com/industry-analysis/high-density-polyethylene-HDPE-market},
+ 2017.
+\newblock [Online; accessed 20-May-2018].
+
+\bibitem{fig1}
+Thorsten Graunke, Katrin Schmitt, and J~Wöllenstein.
+\newblock Organic membranes for selectivity enhancement of metal oxide gas
+ sensors.
+\newblock 2016:1--22, 01 2016.
+
+\bibitem{pubchem}
+{National Center for Biotechnology Information}.
+\newblock Pubchem compound database; cid=6325.
+\newblock \url{https://pubchem.ncbi.nlm.nih.gov/compound/6325}.
+\newblock [Online; accessed 20-May-2018].
+
+\bibitem{nzmfg}
+{New Zealand Institute of Chemistry}.
+\newblock The manufacture of polyethylene.
+\newblock
+ \url{https://web.archive.org/web/20180223141558/http://www.nzic.org.nz/ChemProcesses/polymers/10J.pdf}.
+\newblock [Online; accessed 23-May-2018].
+
+\bibitem{macrog}
+{Polymer Science Learning Centre}.
+\newblock Polyethylene.
+\newblock \url{http://www.pslc.ws/macrog/pe.htm}.
+\newblock [Online; accessed 23-May-2018].
+
+\bibitem{wiki:xxx}
+{Wikipedia contributors}.
+\newblock High-density polyethylene --- {Wikipedia}{,} the free encyclopedia.
+\newblock \url{https://en.wikipedia.org/wiki/High-density_polyethylene}, 2018.
+\newblock [Online; accessed 20-May-2018].
+
+\end{thebibliography}
--- /dev/null
+@misc{ wiki:xxx,
+ author = "{Wikipedia contributors}",
+ title = "High-density polyethylene --- {Wikipedia}{,} The Free Encyclopedia",
+ year = "2018",
+ howpublished = {\url{https://en.wikipedia.org/wiki/High-density_polyethylene}},
+ note = "[Online; accessed 20-May-2018]"
+}
+
+@misc{ gmi,
+ author = "{Global Market Insights}",
+ title = "High Density Polyethylene (HDPE) Market Size",
+ year = "2017",
+ howpublished = {\url{https://www.gminsights.com/industry-analysis/high-density-polyethylene-HDPE-market}},
+ note = "[Online; accessed 20-May-2018]"
+}
+
+@misc{ pubchem,
+ author = "{National Center for Biotechnology Information}",
+ title = "PubChem Compound Database; CID=6325",
+ howpublished = {\url{https://pubchem.ncbi.nlm.nih.gov/compound/6325}},
+ note="[Online; accessed 20-May-2018]"
+}
+
+@misc{ nzmfg,
+ author = "{New Zealand Institute of Chemistry}",
+ title = "The Manufacture of Polyethylene",
+ howpublished = {\url{https://web.archive.org/web/20180223141558/http://www.nzic.org.nz/ChemProcesses/polymers/10J.pdf}},
+ note = "[Online; accessed 23-May-2018]"
+
+}
+
+@misc{ macrog,
+ author = "{Polymer Science Learning Centre}",
+ title = "Polyethylene",
+ howpublished = {\url{http://www.pslc.ws/macrog/pe.htm}},
+ note = "[Online; accessed 23-May-2018]"
+}
+
+@article{fig1,
+author = {Graunke, Thorsten and Schmitt, Katrin and Wöllenstein, J},
+year = {2016},
+month = {01},
+pages = {1-22},
+title = {Organic Membranes for Selectivity Enhancement of Metal Oxide Gas Sensors},
+volume = {2016},
+booktitle = {Journal of Sensors}
+}
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--- /dev/null
+\BOOKMARK [1][-]{section.1}{Invention and Manufacture}{}% 1
+\BOOKMARK [1][-]{section.2}{Structure}{}% 2
+\BOOKMARK [1][-]{section.3}{Uses}{}% 3
+\BOOKMARK [1][-]{section.4}{Health and environmental issues}{}% 4
--- /dev/null
+\documentclass[a4paper]{article}
+\usepackage{mhchem}
+\usepackage{chemfig}
+\usepackage{graphicx}
+\usepackage{wrapfig}
+\usepackage{url}
+\usepackage{float}
+\usepackage[margincaption,outercaption,ragged,wide]{sidecap}
+\begin{document}
+\title{Lifecycle of High-Density Polyethene}
+\author{Andrew Lorimer}
+\date{}
+\renewcommand{\abstractname}{}
+\maketitle
+
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+\begin{abstract}
+High-density polyethene (HDPE) is a practical and economical thermoplastic used for pipes, bottles, containers, construction materials, and several other uses. It is UV stable and chemically quite resistant - it is almost impossible to damage with acids or bases. Further, HDPE is easily recycled, and has a resin identification code of “2”. In 2015, the HDPE market was worth about \$80 billion (AUD).\cite{gmi}. HDPE has enabled streamlining of manufacturing processes and cheap access to durable, safe products throughout the world.
+\end{abstract}
+
+\section{Invention and Manufacture}
+
+\begin{wrapfigure}{r}{0.3\textwidth}
+% \begin{SCfigure}[][htb]
+\includegraphics[height=3cm]{ldpe-hdpe.png}
+\caption{\label{fig:structure}amorphous vs crystalline polymers\cite{fig1}}
+\vspace{-1em}
+% \end{SCfigure}
+\end{wrapfigure}
+
+\paragraph{}
+High-density polyethene (polyethylene) consists of many thousands of ethene monomers \ce{H2CCH2}, forming a long chain structure. Whilst ethene occurs naturally in plants, polyethene must be synthesised. In 1898, a German chemist Hans von Pechmann accidentally created a compound, polymethylene, which was the basis for the invention of polyethylene. In 1933, Fawcett and Gibson synthesised polyethylene using high pressure and a small proportion of oxygen, at the British Imperial Chemical Industries. This is illustrated below\cite{nzmfg}:
+
+\paragraph{}
+
+
+
+
+\ce{(H2CCH2)n ->[{$\approx$500 MPa}][{$\textless$ 10ppm}O2] (CH2CH2)n}
+
+\paragraph{}
+
+
+
+However, this high-pressure method can only be used to synthesise low-density polyethene (LDPE) because the high pressure causes a lot of random chain branching (i.e. the molecules are non-linear). In the case of most low-density polyethene, branches occur every 10-20 carbon atoms. Linear molecules used in HDPE are stronger due to their larger intermolecular forces resulting from their surface area, giving them a more amorphous (shapeless) structure \textendash\ see fig. 1.
+
+
+\paragraph{}
+
+During the 1940s-50s, two chemists independently discovered methods of synthesising HDPE using organometallic (carbon-metal) catalysts to break the \ce{C=C} double bond in ethene. Karl Zielger from Germany used titanium halides \ce{TiCl_n}, and Phillips Petroleum in America used chromium trioxide \ce{CrO3}:
+
+\paragraph{}
+
+\ce{(H2CCH2)n ->[{catalyst}][{$\approx$5 MPa}] (CH2CH2)n}
+
+\paragraph{}
+
+Note that this method only requires 5 megapascal of pressure, whilst the first method used to synthesise LDPE required 500 megapascals. This makes the manufacture of HDPE much more energy efficient and practical, allowing mass production.
+
+Soon after the discovery of this method, HDPE was produced industrially using both Zielger's and Phillips' methods, primarily by DuPont. Early HDPE was used for internal shielding in cables, infrared lenses, and generally replaced PVC in many applications. This low-pressure synthesis is currently the only method of HDPE mass production, but is efficient and cheap. The only major change in HDPE synthesis has been the addition of a magnesium-based catalyst.
+
+\section{Structure}
+\chemfig{[:60]H-C(-[:120]H)=[0]C(-[:-60]H)-[:60]H}
+(ethene)
+\hspace{5em}
+\chemfig{\vphantom{CH_@}-[@{op,.75}]CH_2-CH_2-[@{cl,0.25}]}
+\makebraces[5pt,5pt]{\!\!n}{op}{cl}
+(polyethene)
+\vspace*{1em}
+
+Thanks to the low-pressure method of synthesis discovered by Ziegler and Phillips, each molecule of HDPE is many thousands of monomers long, without significant branching which would compromise its strength (see fig. 1). It has a density of about 0.97 g / cm$^3$, versus LDPE's density of 0.91 g / cm$^3$. Despite this minimal increase in density, HDPE has vastly greater strength and hardness. This is because of its low level of chain branching. Other results of the low chain branching are greater opacity and higher temperature resistance.
+
+\section{Uses}
+
+
+
+High-density polyethene is used for rigid plastic packaging (milk jugs, food containers, water bottles), water pipes (see fig. 2), coaxial cables, vehicle fuel tanks, wood-plastic composites (decks etc), and 3D printer filament. Most of these applications require little effort to transform the HDPE into its resulting product, due to its low melting point (about 100 C) and its ease of extrusion from beads.
+
+\begin{wrapfigure}{r}{0.3\textwidth}
+\vspace*{-3.5em}
+\includegraphics[height=3cm]{hdpe_pipe.jpg}
+\vspace*{-2em}
+
+\caption{\label{fig:pipe}underground installation of agricultural HDPE pipe\cite{wiki:xxx}}
+
+\end{wrapfigure}
+
+
+\section{Issues}
+
+\subsection{Health}
+
+HDPE has virtually no known health risks in normal use. Its production is a safe process, and whilst toxic chemicals are used as catalysts, this process is heavily automated and restricted to licensed manufacturers. If HDPE is burnt, it releases noxious hydrocarbons, however this is true of all thermoplastics.
+
+HDPE does not degrade as traditional steel or lead pipes do, so it offers significant advantages in safety and cost for potable water piping.
+
+\subsection{Environmental}
+
+Polyethene's monomer, ethene, is usually obtained from drilling rigs in the ocean. These rigs represent a non-renewable form of ethene, whereas less developed methods of ethene synthesis are more sustainable.
+
+Polyethylene does not biodegrade easily, so it constitutes a significant volume of landfill. Some bacteria can accelerate the biodegredation process, but the specific species are rare.
+
+However, HDPE can be recycled indefinitely and it is very easy to do so. Once the molecules have been synthesised, the material can be melted and formed into a product multiple times. For this reason, recycling of HDPE is commonly encouraged, and most curbside recycling collections accept HDPE, denoted by the number 2. However, it is important to note that recycled HDPE is generally unable to be used for food packaging according to regulations, but with more research this should be possible.
+
+Common uses of recycled HDPE include timber-plastic composites (decking, outdoor chairs, signs etc), containers, wheelie bins, furniture etc.
+
+\bibliographystyle{plain}
+\bibliography{hdpe}
+\nocite{*}
+
+\end{document}
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+\multlinetaggap=\skip54
+\mathdisplay@stack=\toks18
+LaTeX Info: Redefining \[ on input line 2817.
+LaTeX Info: Redefining \] on input line 2818.
+)
+(/usr/share/texmf-dist/tex/latex/chemgreek/chemgreek.sty
+(/usr/share/texmf-dist/tex/latex/l3packages/xparse/xparse.sty
+Package: xparse 2018/02/21 L3 Experimental document command parser
+\l__xparse_current_arg_int=\count181
+\g__xparse_grabber_int=\count182
+\l__xparse_m_args_int=\count183
+\l__xparse_mandatory_args_int=\count184
+\l__xparse_v_nesting_int=\count185
+)
+Package: chemgreek 2016/12/20 v1.1 interfaceforuprightgreeklettersforuseinchemi
+stry (CN)
+\l__chemgreek_tmpa_int=\count186
+\g__chemgreek_tmpa_int=\count187
+.................................................
+. LaTeX info: "xparse/define-command"
+.
+. Defining command \newchemgreekmapping with sig. 'O{}mm' on line 336.
+.................................................
+.................................................
+. LaTeX info: "xparse/define-command"
+.
+. Defining command \renewchemgreekmapping with sig. 'O{}mm' on line 339.
+.................................................
+.................................................
+. LaTeX info: "xparse/define-command"
+.
+. Defining command \declarechemgreekmapping with sig. 'O{}mm' on line 342.
+.................................................
+.................................................
+. LaTeX info: "xparse/define-command"
+.
+. Defining command \newchemgreekmappingalias with sig. 'mm' on line 347.
+.................................................
+.................................................
+. LaTeX info: "xparse/define-command"
+.
+. Defining command \renewchemgreekmappingalias with sig. 'mm' on line 350.
+.................................................
+.................................................
+. LaTeX info: "xparse/define-command"
+.
+. Defining command \declarechemgreekmappingalias with sig. 'mm' on line 353.
+.................................................
+.................................................
+. LaTeX info: "xparse/define-command"
+.
+. Defining command \changechemgreeksymbol with sig. 'mmmm' on line 383.
+.................................................
+.................................................
+. LaTeX info: "xparse/define-command"
+.
+. Defining command \chemgreekmappingsymbol with sig. 'mm' on line 477.
+.................................................
+.................................................
+. LaTeX info: "xparse/define-command"
+.
+. Defining command \activatechemgreekmapping with sig. 'sm' on line 486.
+.................................................
+.................................................
+. LaTeX info: "xparse/define-command"
+.
+. Defining command \selectchemgreekmapping with sig. 'm' on line 491.
+.................................................
+)
+(/usr/share/texmf-dist/tex/latex/graphics/graphics.sty
+Package: graphics 2017/06/25 v1.2c Standard LaTeX Graphics (DPC,SPQR)
+
+(/usr/share/texmf-dist/tex/latex/graphics/trig.sty
+Package: trig 2016/01/03 v1.10 sin cos tan (DPC)
+)
+(/usr/share/texmf-dist/tex/latex/graphics-cfg/graphics.cfg
+File: graphics.cfg 2016/06/04 v1.11 sample graphics configuration
+)
+Package graphics Info: Driver file: xetex.def on input line 99.
+
+(/usr/share/texmf-dist/tex/latex/graphics-def/xetex.def
+File: xetex.def 2017/06/24 v5.0h Graphics/color driver for xetex
+))
+\l__mhchem_ce_distanceFromLastComma_int=\count188
+\__mhchem_arrow_extPgfLength_skip=\skip55
+\mhchem@minispace@tmp=\skip56
+\mhchem@minibackspace@tmp=\skip57
+\mhchem@bondwidth=\skip58
+\mhchem@bondheight=\skip59
+\mhchem@smallbondwidth@tmpA=\skip60
+\mhchem@smallbondwidth@tmpB=\skip61
+\mhchem@smallbondwidth=\skip62
+\mhchem@bondtmp@minussidebearingleft=\skip63
+\mhchem@bondtmp@minussidebearingright=\skip64
+\l__mhchem_option_version_int=\count189
+\mhchem@option@minusmathsidebearingleft=\dimen145
+\mhchem@option@minusmathsidebearingright=\dimen146
+\mhchem@option@minustextsidebearingleft=\dimen147
+\mhchem@option@minustextsidebearingright=\dimen148
+
+
+Package mhchem Warning: You did not specify a 'version' option for the mhchem
+(mhchem) package. Please write \usepackage[version=4]{mhchem}
+(mhchem) in your preamble (or any lower number for
+(mhchem) compatibility mode), because you might get slightly
+(mhchem) different output with the same input in future versions
+.
+
+) (./latextest.aux)
+\openout1 = `latextest.aux'.
+
+LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 3.
+LaTeX Font Info: ... okay on input line 3.
+LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 3.
+LaTeX Font Info: ... okay on input line 3.
+LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 3.
+LaTeX Font Info: ... okay on input line 3.
+LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 3.
+LaTeX Font Info: ... okay on input line 3.
+LaTeX Font Info: Checking defaults for TU/lmr/m/n on input line 3.
+LaTeX Font Info: ... okay on input line 3.
+LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 3.
+LaTeX Font Info: ... okay on input line 3.
+LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 3.
+LaTeX Font Info: ... okay on input line 3.
+.................................................
+. chemgreek info: "mapping-activated"
+.
+. Activating mapping `default' on line 3.
+.................................................
+ [1
+
+] (./latextest.aux) )
+Here is how much of TeX's memory you used:
+ 10163 strings out of 493007
+ 203788 string characters out of 6140076
+ 256914 words of memory out of 5000000
+ 14151 multiletter control sequences out of 15000+600000
+ 530750 words of font info for 27 fonts, out of 8000000 for 9000
+ 1348 hyphenation exceptions out of 8191
+ 34i,15n,49p,10367b,368s stack positions out of 5000i,500n,10000p,200000b,80000s
+
+Output written on latextest.pdf (1 page).
--- /dev/null
+\documentclass{article}
+\usepackage{mhchem}
+\begin{document}
+Hello world!
+\end{document}
--- /dev/null
+# Organic Chemistry
+
+Organic compounds > 90% of known chemicals
+Carbon makes living systems possible
+
+Carbon forms 4 covalent bonds (4 valence pairs)
+Strong bonds with other non-metals
+Bonds with itself with single, double or triple
+Can form chains or rings
+
+Large number of compounds due to:
+- 4 valence pairs
+- single / double / triple bonds
+- cyclic (ring) structures
+
+## Hydrocarbons
+
+Hydrocarbons - only carbon and hydrogen
+If all C-C bonds are single, the hydrocarbon is saturated
+
+Hydrocarbons
+- linear (aliphatic)
+- - saturated (all single C-C bonds)
+- - - alkane ($C_n H_{2n+2}$)
+- - unsaturated
+- - - alkene ($C_n H_{2n}$)
+- - - alkyne ($C_n H_{2n-2}$)
+-
+- cyclic (ring)
+- - saturated
+- - - e.g. cyclohexane
+- - unsaturated
+- - - e.g. benzene
+
+Carbon compounds with 4 branches are tetrahedral
+
+Compounds with $C-H$ are insoluble in water
+These compounds are almost non-polar due to similar electronegativities
+
+Only dispersion forces (between electrons on the outside of two molecules) hold each molecule together
+Dispersion forces increase with size (chain length), therefore MP and BP of straight molecules increases with length
+
+Branched molecules have lower MP and BP due to lower density (molecules are further apart due to their shape, so dispersion forces are weaker)
+
+Homologous series - formula for molecules (e.g. alkanes, alkenes, alkynes) where each member differs by some increment
+
+Structural isomers - same molecular formula, different configuration (structural formula)
+
+## Alkanes
+$$C_n H_{2n+2}$$
+- Single bonds between carbon atoms
+- ending in *-ane*
+
+methane $CH_4$
+ethane $C_2H_6$
+propane $C_3H_8$
+butane $C_4H_10$
+pentane $C_5H_12$
+hexane $C_6H_14$
+$\dots$
+
+Alkanes burn in oxygen forming $CO_2 + H_2O$ (oxidation / combustion reaction)
+
+Alkanes react with chlorine or fluorine (substitution reaction) - Cl or F replace H. Result is a haloalkane.
+
+## Alkenes
+
+$$C_nH_{2n}$$
+
+- C-C double bond
+- ending in *-ene*
+
+methene $CH_2$
+ethene $C_2H_4$
+propene $C_3H_6$
+butene $C_4H_8$
+pentene $C_5H_10$
+hexene $C_6H_12$
+$\dots$
+
+Alkenes burn in oxygen forming $H_2O + CO_2$.
+
+Alkenes react with hydrogen or a halogen (addition reaction) - additive breaks the double bond (catalyst is required).
+
+## Alkynes
+
+$$C_nH_{2n-2}$$
+
+- C-C triple bond
+- ending in *-yne*
+
+Alkynes undergo combustion (oxidation) and addition reactions (same as alkenes).
+
+## Rules for naming HCs
+
+1. Find longest chain of C atoms (incl. branches)
+2. Start at end nearest a branch or double/triple bond
+3. Name carbon atoms from end chosen
+4. Branches end with *-yl*
+5. When there are $\ge$ 2 branches, number of C atom is indicated before each branch, and names are in alphabetic order.
+6. When there are $\ge$ 2 identical branches, *di-*, *tri-*, and *tetra-* are used.
+
+## Combustion
+$$C_xH_y+z0_2 \to\ xCO_n+{y \over 2}H_2O$$
+**Complete**
+$z=x+{y \over 4}, n=2$
+
+**Incomplete**
+$z={{2x+y} \over 4}, n=1$
+$z={{x+y} \over 4}, n=0$
+
+## Functional groups
+
+Atom(s), or a bond, which give a properties to an organic molecule.
+
+### Alcohols / hydroxyls
+
+- ---OH functional group
+- ends in *-ol*
+- alcohols have multiple isomers
+- combustible with oxygen
+- BP and MP are affected by H-bonding
+- **H-bonds are represented by dotted lines**
+
+### Carboxylic acids / carboxys
+
+- ---COOH functional group
+- ends in *-oic acid*
+- first portion of name (e.g. **methan**oic acid) refers to **total** number of carbons
+- weak acids (proton from O is donated)
+- high BP
+
+### Esters
+- carboxylic acid reacts with alcohols to form esters (condensation / esterification)
+- catalyst is $H_2SO_4$
+- low boiling points
+- intermolecular bonding is dipole-dipole and dispersion rather than H-bonding
+- ends in *-oate*
--- /dev/null
+# Chemistry - Semester Revision
--- /dev/null
+OFL FAQ - Frequently Asked Questions about the SIL Open Font License (OFL)
+Version 1.1 - 26 February 2007
+(See http://scripts.sil.org/OFL for updates)
+
+
+1 ABOUT USING AND DISTRIBUTING FONTS LICENSED UNDER THE OFL
+
+1.1 Can I use the fonts in any publication, even embedded in the file?
+Yes. You may use them like most other fonts, but unlike some fonts you may include an embedded subset of the fonts in your document. Such use does not require you to include this license or other files (listed in OFL condition 2), nor does it require any type of acknowledgement within the publication. Some mention of the font name within the publication information (such as in a colophon) is usually appreciated. If you wish to include the complete font as a separate file, you should distribute the full font package, including all existing acknowledgements, and comply with the OFL conditions. Of course, referencing or embedding an OFL font in any document does not change the license of the document itself. The requirement for fonts to remain under the OFL does not apply to any document created using the fonts and their derivatives. Similarly, creating any kind of graphic using a font under OFL does not make the resulting artwork subject to the OFL.
+
+1.2 Can I make web pages using these fonts?
+Yes! Go ahead! Using CSS (Cascading Style Sheets) is recommended.
+
+1.3 Can I make the fonts available to others from my web site?
+Yes, as long as you meet the conditions of the license (do not sell by itself, include the necessary files, rename Modified Versions, do not abuse the Author(s)' name(s) and do not sublicense).
+
+1.4 Can the fonts be included with Free/Libre and Open Source Software collections such as GNU/Linux and BSD distributions?
+Yes! Fonts licensed under the OFL can be freely aggregated with software under FLOSS (Free/Libre and Open Source Software) licenses. Since fonts are much more useful aggregated to than merged with existing software, possible incompatibility with existing software licenses is not a problem. You can also repackage the fonts and the accompanying components in a .rpm or .deb package and include them in distro CD/DVDs and online repositories.
+
+1.5 I want to distribute the fonts with my program. Does this mean my program also has to be free and open source software?
+No. Only the portions based on the font software are required to be released under the OFL. The intent of the license is to allow aggregation or bundling with software under restricted licensing as well.
+
+1.6 Can I include the fonts on a CD of freeware or commercial fonts?
+Yes, as long some other font or software is also on the disk, so the OFL font is not sold by itself.
+
+1.7 Can I sell a software package that includes these fonts?
+Yes, you can do this with both the Original Version and a Modified Version. Examples of bundling made possible by the OFL would include: word processors, design and publishing applications, training and educational software, edutainment software, etc.
+
+1.8 Why won't the OFL let me sell the fonts alone?
+The intent is to keep people from making money by simply redistributing the fonts. The only people who ought to profit directly from the fonts should be the original authors, and those authors have kindly given up potential direct income to distribute their fonts under the OFL. Please honor and respect their contribution!
+
+1.9 I've come across a font released under the OFL. How can I easily get more information about the Original Version? How can I know where it stands compared to the Original Version or other Modified Versions?
+Consult the copyright statement(s) in the license for ways to contact the original authors. Consult the FONTLOG for information on how the font differs from the Original Version, and get in touch with the various contributors via the information in the acknowledgment section. Please consider using the Original Versions of the fonts whenever possible.
+
+1.10 What do you mean in condition 4? Can you provide examples of abusive promotion / endorsement / advertisement vs. normal acknowledgement?
+The intent is that the goodwill and reputation of the author(s) should not be used in a way that makes it sound like the original author(s) endorse or approve of a specific Modified Version or software bundle. For example, it would not be right to advertise a word processor by naming the author(s) in a listing of software features, or to promote a Modified Version on a web site by saying "designed by ...". However, it would be appropriate to acknowledge the author(s) if your software package has a list of people who deserve thanks. We realize that this can seem to be a gray area, but the standard used to judge an acknowledgement is that if the acknowledgement benefits the author(s) it is allowed, but if it primarily benefits other parties, or could reflect poorly on the author(s), then it is not.
+
+
+2 ABOUT MODIFYING OFL LICENSED FONTS
+
+2.1 Can I change the fonts? Are there any limitations to what things I can and cannot change?
+You are allowed to change anything, as long as such changes do not violate the terms of the license. In other words, you are not allowed to remove the copyright statement(s) from the font, but you could add additional information into it that covers your contribution.
+
+2.2 I have a font that needs a few extra glyphs - can I take them from an OFL licensed font and copy them into mine?
+Yes, but if you distribute that font to others it must be under the OFL, and include the information mentioned in condition 2 of the license.
+
+2.3 Can I charge people for my additional work? In other words, if I add a bunch of special glyphs and/or OpenType/Graphite code, can I sell the enhanced font?
+Not by itself. Derivative fonts must be released under the OFL and cannot be sold by themselves. It is permitted, however, to include them in a larger software package (such as text editors, office suites or operating systems), even if the larger package is sold. In that case, you are strongly encouraged, but not required, to also make that derived font easily and freely available outside of the larger package.
+
+2.4 Can I pay someone to enhance the fonts for my use and distribution?
+Yes. This is a good way to fund the further development of the fonts. Keep in mind, however, that if the font is distributed to others it must be under the OFL. You won't be able to recover your investment by exclusively selling the font, but you will be making a valuable contribution to the community. Please remember how you have benefitted from the contributions of others.
+
+2.5 I need to make substantial revisions to the font to make it work with my program. It will be a lot of work, and a big investment, and I want to be sure that it can only be distributed with my program. Can I restrict its use?
+No. If you redistribute a Modified Version of the font it must be under the OFL. You may not restrict it in any way. This is intended to ensure that all released improvements to the fonts become available to everyone. But you will likely get an edge over competitors by being the first to distribute a bundle with the enhancements. Again, please remember how you have benefitted from the contributions of others.
+
+2.6 Do I have to make any derivative fonts (including source files, build scripts, documentation, etc.) publicly available?
+No, but please do share your improvements with others. You may find that you receive more than what you gave in return.
+
+2.7 Why can't I use the Reserved Font Name(s) in my derivative font names? I'd like people to know where the design came from.
+The best way to acknowledge the source of the design is to thank the original authors and any other contributors in the files that are distributed with your revised font (although no acknowledgement is required). The FONTLOG is a natural place to do this. Reserved Font Name(s) ensure that the only fonts that have the original names are the unmodified Original Versions. This allows designers to maintain artistic integrity while allowing collaboration to happen. It eliminates potential confusion and name conflicts. When choosing a name be creative and avoid names that reuse almost all the same letters in the same order or sound like the original. Keep in mind that the Copyright Holder(s) can allow a specific trusted partner to use Reserved Font Name(s) through a separate written agreement.
+
+2.8 What do you mean by "primary name as presented to the user"? Are you referring to the font menu name?
+Yes, the requirement to change the visible name used to differentiate the font from others applies to the font menu name and other mechanisms to specify a font in a document. It would be fine, for example, to keep a text reference to the original fonts in the description field, in your modified source file or in documentation provided alongside your derivative as long as no one could be confused that your modified source is the original. But you cannot use the Reserved Font Names in any way to identify the font to the user (unless the Copyright Holder(s) allow(s) it through a separate agreement; see section 2.7). Users who install derivatives ("Modified Versions") on their systems should not see any of the original names ("Reserved Font Names") in their font menus, for example. Again, this is to ensure that users are not confused and do not mistake a font for another and so expect features only another derivative or the Original Version can actually offer. Ultimately, creating name conflicts will cause many problems for the users as well as for the designer of both the Original and Modified versions, so please think ahead and find a good name for your own derivative. Font substitution systems like fontconfig, or application-level font fallback configuration within OpenOffice.org or Scribus, will also get very confused if the name of the font they are configured to substitute to actually refers to another physical font on the user's hard drive. It will help everyone if Original Versions and Modified Versions can easily be distinguished from one another and from other derivatives. The substitution mechanism itself is outside the scope of the license. Users can always manually change a font reference in a document or set up some kind of substitution at a higher level but at the lower level the fonts themselves have to respect the Reserved Font Name(s) requirement to prevent ambiguity. If a substitution is currently active the user should be aware of it.
+
+2.9 Am I not allowed to use any part of the Reserved Font Names?
+You may not use the words of the font names, but you would be allowed to use parts of words, as long as you do not use any word from the Reserved Font Names entirely. We do not recommend using parts of words because of potential confusion, but it is allowed. For example, if "Foobar" was a Reserved Font Name, you would be allowed to use "Foo" or "bar", although we would not recommend it. Such an unfortunate choice would confuse the users of your fonts as well as make it harder for other designers to contribute.
+
+2.10 So what should I, as an author, identify as Reserved Font Names?
+Original authors are encouraged to name their fonts using clear, distinct names, and only declare the unique parts of the name as Reserved Font Names. For example, the author of a font called "Foobar Sans" would declare "Foobar" as a Reserved Font Name, but not "Sans", as that is a common typographical term, and may be a useful word to use in a derivative font name. Reserved Font Names should also be single words. A font called "Flowing River" should have Reserved Font Names "Flowing" and "River", not "Flowing River".
+
+2.11 Do I, as an author, have to identify any Reserved Font Names?
+No, but we strongly encourage you to do so. This is to avoid confusion between your work and Modified versions. You may, however, give certain trusted parties the right to use any of your Reserved Font Names through separate written agreements. For example, even if "Foobar" is a RFN, you could write up an agreement to give company "XYZ" the right to distribute a modified version with a name that includes "Foobar". This allows for freedom without confusion.
+
+2.12 Are any names (such as the main font name) reserved by default?
+No. That is a change to the license as of version 1.1. If you want any names to be Reserved Font Names, they must be specified after the copyright statement(s).
+
+2.13 What is this FONTLOG thing exactly?
+It has three purposes: 1) to provide basic information on the font to users and other developers, 2) to document changes that have been made to the font or accompanying files, either by the original authors or others, and 3) to provide a place to acknowledge the authors and other contributors. Please use it! See below for details on how changes should be noted.
+
+2.14 Am I required to update the FONTLOG?
+No, but users, designers and other developers might get very frustrated at you if you don't! People need to know how derivative fonts differ from the original, and how to take advantage of the changes, or build on them.
+
+
+3 ABOUT THE FONTLOG
+
+The FONTLOG can take a variety of formats, but should include these four sections:
+
+3.1 FONTLOG for <FontFamilyName>
+This file provides detailed information on the <FontFamilyName> font software. This information should be distributed along with the <FontFamilyName> fonts and any derivative works.
+
+3.2 Basic Font Information
+(Here is where you would describe the purpose and brief specifications for the font project, and where users can find more detailed documentation. It can also include references to how changes can be contributed back to the Original Version. You may also wish to include a short guide to the design, or a reference to such a document.)
+
+3.3 ChangeLog
+(This should list both major and minor changes, most recent first. Here are some examples:)
+
+7 February 2007 (Pat Johnson) <NewFontFamilyName> Version 1.3
+- Added Greek and Cyrillic glyphs
+- Released as "<NewFontFamilyName>"
+
+7 March 2006 (Fred Foobar) <NewFontFamilyName> Version 1.2
+- Tweaked contextual behaviours
+- Released as "<NewFontFamilyName>"
+
+1 Feb 2005 (Jane Doe) <NewFontFamilyName> Version 1.1
+- Improved build script performance and verbosity
+- Extended the smart code documentation
+- Corrected minor typos in the documentation
+- Fixed position of combining inverted breve below (U+032F)
+- Added OpenType/Graphite smart code for Armenian
+- Added Armenian glyphs (U+0531 -> U+0587)
+- Released as "<NewFontFamilyName>"
+
+1 Jan 2005 (Joe Smith) <FontFamilyName> Version 1.0
+- Initial release of font "<FontFamilyName>"
+
+3.4 Acknowledgements
+(Here is where contributors can be acknowledged.
+
+If you make modifications be sure to add your name (N), email (E), web-address (W) and description (D). This list is sorted by last name in alphabetical order.)
+
+N: Jane Doe
+E: jane@university.edu
+W: http://art.university.edu/projects/fonts
+D: Contributor - Armenian glyphs and code
+
+N: Fred Foobar
+E: fred@foobar.org
+W: http://foobar.org
+D: Contributor - misc Graphite fixes
+
+N: Pat Johnson
+E: pat@fontstudio.org
+W: http://pat.fontstudio.org
+D: Designer - Greek & Cyrillic glyphs based on Roman design
+
+N: Tom Parker
+E: tom@company.com
+W: http://www.company.com/tom/projects/fonts
+D: Engineer - original smart font code
+
+N: Joe Smith
+E: joe@fontstudio.org
+W: http://joe.fontstudio.org
+D: Designer - original Roman glyphs
+
+(Original authors can also include information here about their organization.)
+
+
+4 ABOUT MAKING CONTRIBUTIONS
+
+4.1 Why should I contribute my changes back to the original authors?
+It would benefit many people if you contributed back to what you've received. Providing your contributions and improvements to the fonts and other components (data files, source code, build scripts, documentation, etc.) could be a tremendous help and would encourage others to contribute as well and 'give back', which means you will have an opportunity to benefit from other people's contributions as well. Sometimes maintaining your own separate version takes more effort than merging back with the original. Be aware that any contributions, however, must be either your own original creation or work that you own, and you may be asked to affirm that clearly when you contribute.
+
+4.2 I've made some very nice improvements to the font, will you consider adopting them and putting them into future Original Versions?
+Most authors would be very happy to receive such contributions. Keep in mind that it is unlikely that they would want to incorporate major changes that would require additional work on their end. Any contributions would likely need to be made for all the fonts in a family and match the overall design and style. Authors are encouraged to include a guide to the design with the fonts. It would also help to have contributions submitted as patches or clearly marked changes (the use of smart source revision control systems like subversion, svk or bzr is a good idea). Examples of useful contributions are bug fixes, additional glyphs, stylistic alternates (and the smart font code to access them) or improved hinting.
+
+4.3 How can I financially support the development of OFL fonts?
+It is likely that most authors of OFL fonts would accept financial contributions - contact them for instructions on how to do this. Such contributions would support future development. You can also pay for others to enhance the fonts and contribute the results back to the original authors for inclusion in the Original Version.
+
+
+5 ABOUT THE LICENSE
+
+5.1 I see that this is version 1.1 of the license. Will there be later changes?
+Version 1.1 is the first minor revision of the OFL. We are confident that version 1.1 will meet most needs, but are open to future improvements. Any revisions would be for future font releases, and previously existing licenses would remain in effect. No retroactive changes are possible, although the Copyright Holder(s) can re-release the font under a revised OFL. All versions will be available on our web site: http://scripts.sil.org/OFL.
+
+5.2 Can I use the SIL Open Font License for my own fonts?
+Yes! We heartily encourage anyone to use the OFL to distribute their own original fonts. It is a carefully constructed license that allows great freedom along with enough artistic integrity protection for the work of the authors as well as clear rules for other contributors and those who redistribute the fonts. Some additional information about using the OFL is included at the end of this FAQ.
+
+5.3 Does this license restrict the rights of the Copyright Holder(s)?
+No. The Copyright Holder(s) still retain(s) all the rights to their creation; they are only releasing a portion of it for use in a specific way. For example, the Copyright Holder(s) may choose to release a 'basic' version of their font under the OFL, but sell a restricted 'enhanced' version. Only the Copyright Holder(s) can do this.
+
+5.4 Is the OFL a contract or a license?
+The OFL is a license and not a contract and so does not require you to sign it to have legal validity. By using, modifying and redistributing components under the OFL you indicate that you accept the license.
+
+5.5 How about translating the license and the FAQ into other languages?
+SIL certainly recognises the need for people who are not familiar with English to be able to understand the OFL and this FAQ better in their own language. Making the license very clear and readable is a key goal of the OFL.
+
+If you are an experienced translator, you are very welcome to help by translating the OFL and its FAQ so that designers and users in your language community can understand the license better. But only the original English version of the license has legal value and has been approved by the community. Translations do not count as legal substitutes and should only serve as a way to explain the original license. SIL - as the author and steward of the license for the community at large - does not approve any translation of the OFL as legally valid because even small translation ambiguities could be abused and create problems.
+
+We give permission to publish unofficial translations into other languages provided that they comply with the following guidelines:
+
+- put the following disclaimer in both English and the target language stating clearly that the translation is unofficial:
+
+"This is an unofficial translation of the SIL Open Font License into $language. It was not published by SIL International, and does not legally state the distribution terms for fonts that use the OFL. A release under the OFL is only valid when using the original English text.
+
+However, we recognize that this unofficial translation will help users and designers not familiar with English to understand the SIL OFL better and make it easier to use and release font families under this collaborative font design model. We encourage designers who consider releasing their creation under the OFL to read the FAQ in their own language if it is available.
+
+Please go to http://scripts.sil.org/OFL for the official version of the license and the accompanying FAQ."
+
+- keep your unofficial translation current and update it at our request if needed, for example if there is any ambiguity which could lead to confusion.
+
+If you start such a unofficial translation effort of the OFL and its accompanying FAQ please let us know, thank you.
+
+
+6 ABOUT SIL INTERNATIONAL
+
+6.1 Who is SIL International and what does it do?
+SIL International is a worldwide faith-based education and development organization (NGO) that studies, documents, and assists in developing the world's lesser-known languages through literacy, linguistics, translation, and other academic disciplines. SIL makes its services available to all without regard to religious belief, political ideology, gender, race, or ethnic background. SIL's members and volunteers share a Christian commitment.
+
+6.2 What does this have to do with font licensing?
+The ability to read, write, type and publish in one's own language is one of the most critical needs for millions of people around the world. This requires fonts that are widely available and support lesser-known languages. SIL develops - and encourages others to develop - a complete stack of writing systems implementation components available under open licenses. This open stack includes input methods, smart fonts, smart rendering libraries and smart applications. There has been a need for a common open license that is specifically applicable to fonts and related software (a crucial component of this stack) so SIL developed the SIL Open Font License with the help of the FLOSS community.
+
+6.3 How can I contact SIL?
+Our main web site is: http://www.sil.org/
+Our site about complex scripts is: http://scripts.sil.org/
+Information about this license (including contact email information) is at: http://scripts.sil.org/OFL
+
+
+7 ABOUT USING THE OFL FOR YOUR ORIGINAL FONTS
+
+If you want to release your fonts under the OFL, you only need to do the following:
+
+7.1 Put your copyright and reserved font names information in the beginning of the main OFL file.
+7.2 Put your copyright and the OFL references in your various font files (such as in the copyright, license and description fields) and in your other components (build scripts, glyph databases, documentation, rendering samples, etc).
+7.3 Write an initial FONTLOG for your font and include it in the release package.
+7.4 Include the OFL in your release package.
+7.5 We also highly recommend you include the relevant practical documentation on the license by putting the OFL-FAQ in your package.
+7.6 If you wish, you can use the OFL Graphics on your web page.
+
+
+
+That's all. If you have any more questions please get in touch with us.
+
+
--- /dev/null
+Copyright (C) Authors of original metafont fonts:
+Donald Ervin Knuth (cm, concrete fonts)
+1995, 1996, 1997 J"org Knappen, 1990, 1992 Norbert Schwarz (ec fonts)
+1992-2006 A.Khodulev, O.Lapko, A.Berdnikov, V.Volovich (lh fonts)
+1997-2005 Claudio Beccari (cb greek fonts)
+2002 FUKUI Rei (tipa fonts)
+2003-2005 Han The Thanh (Vietnamese fonts)
+1996-2005 Walter Schmidt (cmbright fonts)
+
+Copyright (C) 2003-2009, Andrey V. Panov (panov@canopus.iacp.dvo.ru),
+with Reserved Font Family Name "Computer Modern Unicode fonts".
+
+This Font Software is licensed under the SIL Open Font License, Version 1.1.
+This license is copied below, and is also available with a FAQ at:
+http://scripts.sil.org/OFL
+
+
+-----------------------------------------------------------
+SIL OPEN FONT LICENSE Version 1.1 - 26 February 2007
+-----------------------------------------------------------
+
+PREAMBLE
+The goals of the Open Font License (OFL) are to stimulate worldwide
+development of collaborative font projects, to support the font creation
+efforts of academic and linguistic communities, and to provide a free and
+open framework in which fonts may be shared and improved in partnership
+with others.
+
+The OFL allows the licensed fonts to be used, studied, modified and
+redistributed freely as long as they are not sold by themselves. The
+fonts, including any derivative works, can be bundled, embedded,
+redistributed and/or sold with any software provided that any reserved
+names are not used by derivative works. The fonts and derivatives,
+however, cannot be released under any other type of license. The
+requirement for fonts to remain under this license does not apply
+to any document created using the fonts or their derivatives.
+
+DEFINITIONS
+"Font Software" refers to the set of files released by the Copyright
+Holder(s) under this license and clearly marked as such. This may
+include source files, build scripts and documentation.
+
+"Reserved Font Name" refers to any names specified as such after the
+copyright statement(s).
+
+"Original Version" refers to the collection of Font Software components as
+distributed by the Copyright Holder(s).
+
+"Modified Version" refers to any derivative made by adding to, deleting,
+or substituting -- in part or in whole -- any of the components of the
+Original Version, by changing formats or by porting the Font Software to a
+new environment.
+
+"Author" refers to any designer, engineer, programmer, technical
+writer or other person who contributed to the Font Software.
+
+PERMISSION & CONDITIONS
+Permission is hereby granted, free of charge, to any person obtaining
+a copy of the Font Software, to use, study, copy, merge, embed, modify,
+redistribute, and sell modified and unmodified copies of the Font
+Software, subject to the following conditions:
+
+1) Neither the Font Software nor any of its individual components,
+in Original or Modified Versions, may be sold by itself.
+
+2) Original or Modified Versions of the Font Software may be bundled,
+redistributed and/or sold with any software, provided that each copy
+contains the above copyright notice and this license. These can be
+included either as stand-alone text files, human-readable headers or
+in the appropriate machine-readable metadata fields within text or
+binary files as long as those fields can be easily viewed by the user.
+
+3) No Modified Version of the Font Software may use the Reserved Font
+Name(s) unless explicit written permission is granted by the corresponding
+Copyright Holder. This restriction only applies to the primary font name as
+presented to the users.
+
+4) The name(s) of the Copyright Holder(s) or the Author(s) of the Font
+Software shall not be used to promote, endorse or advertise any
+Modified Version, except to acknowledge the contribution(s) of the
+Copyright Holder(s) and the Author(s) or with their explicit written
+permission.
+
+5) The Font Software, modified or unmodified, in part or in whole,
+must be distributed entirely under this license, and must not be
+distributed under any other license. The requirement for fonts to
+remain under this license does not apply to any document created
+using the Font Software.
+
+TERMINATION
+This license becomes null and void if any of the above conditions are
+not met.
+
+DISCLAIMER
+THE FONT SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
+EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO ANY WARRANTIES OF
+MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT
+OF COPYRIGHT, PATENT, TRADEMARK, OR OTHER RIGHT. IN NO EVENT SHALL THE
+COPYRIGHT HOLDER BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
+INCLUDING ANY GENERAL, SPECIAL, INDIRECT, INCIDENTAL, OR CONSEQUENTIAL
+DAMAGES, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
+FROM, OUT OF THE USE OR INABILITY TO USE THE FONT SOFTWARE OR FROM
+OTHER DEALINGS IN THE FONT SOFTWARE.
--- /dev/null
+This package was compiled by Christian Perfect (http://checkmyworking.com) from the Computer Modern Unicode fonts created by Andrey V. Panov (http://cm-unicode.sourceforge.net/)\r
+\r
+They're released under the SIL Open Font License. See OFL.txt and OFL-FAQ.txt for the terms.\r
+\r
+A demo page for these fonts was at http://www.checkmyworking.com/cm-web-fonts/ when I released them. I can only apologise, citizen of the future, if that address doesn't exist any more.\r
--- /dev/null
+@font-face {\r
+ font-family: 'Computer Modern Serif';\r
+ src: url('cmunrm.eot');\r
+ src: url('cmunrm.eot?#iefix') format('embedded-opentype'),\r
+ url('cmunrm.woff') format('woff'),\r
+ url('cmunrm.ttf') format('truetype'),\r
+ url('cmunrm.svg#cmunrm') format('svg');\r
+ font-weight: normal;\r
+ font-style: normal;\r
+}\r
+\r
+\r
+@font-face {\r
+ font-family: 'Computer Modern Serif';\r
+ src: url('cmunbx.eot');\r
+ src: url('cmunbx.eot?#iefix') format('embedded-opentype'),\r
+ url('cmunbx.woff') format('woff'),\r
+ url('cmunbx.ttf') format('truetype'),\r
+ url('cmunbx.svg#cmunbx') format('svg');\r
+ font-weight: bold;\r
+ font-style: normal;\r
+}\r
+\r
+\r
+@font-face {\r
+ font-family: 'Computer Modern Serif';\r
+ src: url('cmunti.eot');\r
+ src: url('cmunti.eot?#iefix') format('embedded-opentype'),\r
+ url('cmunti.woff') format('woff'),\r
+ url('cmunti.ttf') format('truetype'),\r
+ url('cmunti.svg#cmunti') format('svg');\r
+ font-weight: normal;\r
+ font-style: italic;\r
+}\r
+\r
+\r
+@font-face {\r
+ font-family: 'Computer Modern Serif';\r
+ src: url('cmunbi.eot');\r
+ src: url('cmunbi.eot?#iefix') format('embedded-opentype'),\r
+ url('cmunbi.woff') format('woff'),\r
+ url('cmunbi.ttf') format('truetype'),\r
+ url('cmunbi.svg#cmunbi') format('svg');\r
+ font-weight: bold;\r
+ font-style: italic;\r
+}\r
+\r
+\r
--- /dev/null
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+<defs >
+<font id="cmunbi" horiz-adv-x="510" ><font-face
+ font-family="CMU Serif"
+ units-per-em="1000"
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+<!DOCTYPE html>
+<html xmlns="http://www.w3.org/1999/xhtml" lang="$lang$" xml:lang="$lang$"$if(dir)$ dir="$dir$"$endif$>
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+$if(subtitle)$
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+$endif$
+<p class="navbar">
+<span class="author"><a href="http://lorimer.id.au">Andrew Lorimer</a></span>
+<span class="sitenav"><a href="http://lorimer.id.au">home</a> // <a href="blog.html">blog</a> // <a href="photos.html">photos</a> // <a href="about.html">about</a></span>
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+$endif$
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--- /dev/null
+# Linear and Quadratic Functions
+
+## General binomial expansion
+
+$$(a+b)(c+d) = a(c+d) + b(c+d)$$
+$$= ac+ad+bc+bd$$
+
+## Perfect square expansion
+
+$$(a+b)^2=a^2+2ab+b^2$$
+
+## Difference of perfect squares
+
+$$ a^2-b^2=(a+b)(a-b)$$
+
+## Factorising monic quadratics
+
+$$x^2+bx+c = (x+e)(x-f)$$
+
+where
+$e \times f = c$
+$e+f = b$
+
+## Factorising general quadratics
+
+$ax^2+bx+c=(x+e)(x+f)$
+
+where
+$e \times f = a \times c$
+$e+f = b$
+
+## Completing the square
+
+$ax^2+bx+c=a(x + (b \over 2a))^2+(c-{b^2 \over 2a})^2$
+
+## Axis of symmetry
+
+${vert}_x={-b \over 2a}$
+
+$x={-b \over 2a}$
+(corresponding line)
--- /dev/null
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+</style>
+ </head>
+ <body class='markdown-preview'><h1>Linear and Quadratic Functions</h1>
+<h2>General binomial expansion</h2>
+<span class="math"><script type="math/tex; mode=display">(a+b)(c+d) = a(c+d) + b(c+d)
+</script></span>
+<span class="math"><script type="math/tex; mode=display">= ac+ad+bc+bd
+</script></span>
+<h2>Perfect square expansion</h2>
+<span class="math"><script type="math/tex; mode=display">(a+b)^2=a^2+2ab+b^2
+</script></span>
+<h2>Difference of perfect squares</h2>
+<span class="math"><script type="math/tex; mode=display">a^2-b^2=(a+b)(a-b)
+</script></span>
+<h2>Factorising monic quadratics</h2>
+<span class="math"><script type="math/tex; mode=display">x^2+bx+c = (x+e)(x-f)
+</script></span>
+<p>where<br>
+<span class="math"><script type="math/tex">e \times f = c</script></span><br>
+<span class="math"><script type="math/tex">e+f = b</script></span></p>
+<h2>Factorising general quadratics</h2>
+<span class="math"><script type="math/tex; mode=display">ax^2+bx+c=(x+e)(x+f)
+</script></span>
+<p>where<br>
+<span class="math"><script type="math/tex">e \times f = a \times c</script></span><br>
+<span class="math"><script type="math/tex">e+f = b</script></span></p>
+<h2>Completing the square</h2>
+<span class="math"><script type="math/tex; mode=display">ax^2+bx+c=a(x+{b \over 2a})^2+(c-{b^2 \over 2a})^2
+</script></span>
+<h2>Axis of symmetry</h2>
+<span class="math"><script type="math/tex; mode=display">{vert}_x={-b \over 2a}
+</script></span>
+<span class="math"><script type="math/tex; mode=display">x={-b \over 2a}
+</script></span>
+<p>(corresponding line)</p></body>
+</html>
--- /dev/null
+<h1>Linear and Quadratic Functions</h1>
+<h2>General binomial expansion</h2>
+<span class="math"><script type="math/tex; mode=display">(a+b)(c+d) = a(c+d) + b(c+d)
+</script></span>
+<span class="math"><script type="math/tex; mode=display">= ac+ad+bc+bd
+</script></span>
+<h2>Perfect square expansion</h2>
+<span class="math"><script type="math/tex; mode=display">(a+b)^2=a^2+2ab+b^2
+</script></span>
+<h2>Difference of perfect squares</h2>
+<span class="math"><script type="math/tex; mode=display">a^2-b^2=(a+b)(a-b)
+</script></span>
+<h2>Factorising monic quadratics</h2>
+<span class="math"><script type="math/tex; mode=display">x^2+bx+c = (x+e)(x-f)
+</script></span>
+<p>where<br>
+<span class="math"><script type="math/tex">e \times f = c</script></span><br>
+<span class="math"><script type="math/tex">e+f = b</script></span></p>
+<h2>Factorising general quadratics</h2>
+<p><span class="math"><script type="math/tex">ax^2+bx+c=(x+e)(x+f)</script></span></p>
+<p>where<br>
+<span class="math"><script type="math/tex">e \times f = a \times c</script></span><br>
+<span class="math"><script type="math/tex">e+f = b</script></span></p>
+<h2>Completing the square</h2>
+<p><span class="math"><script type="math/tex">ax^2+bx+c=a(x + (b \over 2a))^2+(c-{b^2 \over 2a})^2</script></span></p>
+<h2>Axis of symmetry</h2>
+<p><span class="math"><script type="math/tex">{vert}_x={-b \over 2a}</script></span></p>
+<p><span class="math"><script type="math/tex">x={-b \over 2a}</script></span><br>
+(corresponding line)</p>
--- /dev/null
+<!DOCTYPE html>
+<html>
+ <head>
+ <meta charset="utf-8" />
+ <title>summary</title><script typem5 fusesdujaycaogg="text/x-mathjax-config">
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+
+</style>
+
+<link rel="stylesheet" href="../fonts/cmun-serif.css"></link>
+
+ </head>
+ <body class="markdown-preview-plus-view">
+ <h1>Methods - Semester 1</h1>
+<h2>Simulatenous equations</h2>
+<p>Methods of solving:</p>
+<ul>
+<li>substitution (state one variable in terms of the other)</li>
+<li>subtraction (subtract one equation from the other, substitute resulting equation into the other)</li>
+</ul>
+<h2>Linear inequatlities</h2>
+<ul>
+<li>Flip operator when multiplying / dividing by <0</li>
+</ul>
+<h2>Coordinate geometry</h2>
+<p>Regarding points <span class="math"><script type="math/tex">(x_1,y_1)(x_2y_2)</script></span>:</p>
+<p><strong>Midpoint:</strong> <span class="math"><script type="math/tex">m=({{\Sigma x}\over 2}, {{\Sigma y}\over 2})</script></span></p>
+<p><strong>Distance:</strong> <span class="math"><script type="math/tex">d=\sqrt{(\Delta x)^2+(\Delta y)^2}</script></span></p>
+<p><strong>Gradient:</strong> <span class="math"><script type="math/tex">m={\operatorname{rise}\over\operatorname{run}}={\Delta y \over \Delta x}</script></span></p>
+<p><strong>Line through points:</strong> <span class="math"><script type="math/tex">y={\Delta y \over \Delta x}(x+x_2)-y_2</script></span></p>
+<ul>
+<li>parallel lines: <span class="math"><script type="math/tex">m_1=m_2</script></span></li>
+<li>perpendicular lines: <span class="math"><script type="math/tex">m_1m_2=-1</script></span></li>
+</ul>
+<h2>Polynomials</h2>
+<p><strong>Binomial expansion:</strong> <span class="math"><script type="math/tex">(a+b)(c+d) = a(c+d) + b(c+d)</script></span><br>
+<strong>Cubic expansion:</strong> <span class="math"><script type="math/tex">(a+b)^3=a^3+3a^2b+3ab^2+b^3</script></span><br>
+<strong>Quartic expansion:</strong> <span class="math"><script type="math/tex">(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4</script></span></p>
+<p><strong>Perfect square expansion:</strong> <span class="math"><script type="math/tex">(a+b)^2=a^2+2ab+b^2</script></span><br>
+<strong>Difference of perfect squares:</strong> <span class="math"><script type="math/tex">a^2-b^2=(a+b)(a-b)</script></span></p>
+<p><strong>Factorising quadratics:</strong> <span class="math"><script type="math/tex">x^2+bx+c = (x+e)(x-f)</script></span><br>
+where <span class="math"><script type="math/tex">e \times f = c, \quad e+f = b</script></span></p>
+<p><strong>Remainder theorem:</strong> if <span class="math"><script type="math/tex">{P({-\alpha \over \beta})}=0,</script></span> then <span class="math"><script type="math/tex">{\beta x+ \alpha}</script></span> is a factor of <span class="math"><script type="math/tex">P(x)</script></span></p>
+<p><strong>Factor theorem:</strong> if <span class="math"><script type="math/tex">P(\alpha)=0,$then</script></span>x-\alpha$ is a factor of $P(x)</p>
+<p><strong>Completing the square (monic):</strong> <span class="math"><script type="math/tex">x^2+bx+c=(x+{b\over2})^2+c-{b^2\over4}</script></span><br>
+<strong>Completing the square (non-monic):</strong> <span class="math"><script type="math/tex">ax^2+bx+c=a(x-{b\over2a})^2+c-{b^2\over4a}</script></span></p>
+<p><strong>Quadratic formula:</strong> <span class="math"><script type="math/tex">x={{-b\pm\sqrt{b^2-4ac}}\over2a}</script></span><br>
+<strong>Discriminant:</strong> <span class="math"><script type="math/tex">\Delta=b^2-4ac</script></span></p>
+<p><strong>Solving</strong> <span class="math"><script type="math/tex">y_1=ax^2+bx+c_1, \quad y_2=mx+c_2</script></span>:</p>
+<ul>
+<li><span class="math"><script type="math/tex">ax^2+(b-m)x+(c_1-c_2)=0</script></span></li>
+<li>Solve for <span class="math"><script type="math/tex">x</script></span>, substitute into <span class="math"><script type="math/tex">y_2</script></span></li>
+</ul>
+<p><strong>Axis of symmetry:</strong> <span class="math"><script type="math/tex">x={-b\over2a}</script></span> (when written as <span class="math"><script type="math/tex">y=ax^2+bx+c</script></span>)</p>
+<p><strong>Determining quadratic rules:</strong><br>
+<span class="math"><script type="math/tex">(1)\quad y=a(x-e)(x-f)</script></span> two <span class="math"><script type="math/tex">x</script></span>-intercepts, one point<br>
+<span class="math"><script type="math/tex">(2)\quad y=a(x-h)^2+k</script></span> turning point, one point<br>
+<span class="math"><script type="math/tex">(3)\quad y=ax^2+bx+c</script></span> three points</p>
+<h2>Graphs</h2>
+<p>In general:</p>
+<ul>
+<li>turning point / centre point at <span class="math"><script type="math/tex">(h,k)</script></span></li>
+<li>asymptotes at <span class="math"><script type="math/tex">x=h</script></span> and <span class="math"><script type="math/tex">y=k</script></span></li>
+<li><span class="math"><script type="math/tex">a+</script></span> dilates graph away from centre point</li>
+<li><span class="math"><script type="math/tex">a<0</script></span> reflects graph across <span class="math"><script type="math/tex">x=0</script></span></li>
+<li><span class="math"><script type="math/tex">h</script></span> is the horizontal (<span class="math"><script type="math/tex">x</script></span>) shift, <span class="math"><script type="math/tex">k</script></span> is the vertical (<span class="math"><script type="math/tex">y</script></span>) shift</li>
+</ul>
+<p><strong>Rectangular hyperbola:</strong> <span class="math"><script type="math/tex">\quad y={a\over x-h}+k</script></span></p>
+<ul>
+<li>as <span class="math"><script type="math/tex">x\rightarrow \pm\infty, \quad y \rightarrow 0^\pm</script></span></li>
+<li>as <span class="math"><script type="math/tex">x \rightarrow 0^{\pm}, \quad y \rightarrow \pm \infty</script></span></li>
+</ul>
+<p><strong>Truncus:</strong><span class="math"><script type="math/tex">\quad y={a\over (x-h)^2}+k</script></span></p>
+<ul>
+<li>as <span class="math"><script type="math/tex">x \rightarrow \pm \infty, \quad y \rightarrow 0^+</script></span></li>
+<li>as <span class="math"><script type="math/tex">x \rightarrow 0^{\pm}, \quad y \rightarrow \infty</script></span></li>
+</ul>
+<p><strong>Square root:</strong><span class="math"><script type="math/tex">\quad y=a\sqrt{x-h}+k</script></span></p>
+<ul>
+<li>parabola rotated 90 degrees</li>
+</ul>
+<p><strong>Square root negative:</strong><span class="math"><script type="math/tex">\quad y=a\sqrt{-(x-h)}+k</script></span></p>
+<ul>
+<li>reflection of <span class="math"><script type="math/tex">y=\sqrt{x}</script></span> across <span class="math"><script type="math/tex">y</script></span>-axis</li>
+</ul>
+<p><strong>Circle:</strong><span class="math"><script type="math/tex">\quad (x-h)^2+(y-x)^2=r^2</script></span></p>
+<ul>
+<li>factorised: <span class="math"><script type="math/tex">x^2+y^2-2hx-2ky+c=0</script></span></li>
+<li>or: <span class="math"><script type="math/tex">y=\pm\sqrt{r^2-x^2}</script></span></li>
+</ul>
+<p><strong>Semicircles</strong> - take +ve or -ve square root</p>
+<ul>
+<li><span class="math"><script type="math/tex">y=\pm\sqrt{r^2-x^2}\quad</script></span>(top or bottom)</li>
+<li><span class="math"><script type="math/tex">y=\pm\sqrt{r^2-y^2}\quad</script></span>(left or right)</li>
+</ul>
+<p><strong>Cubic:</strong> <span class="math"><script type="math/tex">\quad f(x)=a(x-h)^3+k</script></span></p>
+<ul>
+<li>all cubics have >1 root</li>
+</ul>
+<p><strong>Inverse cubic:</strong> <span class="math"><script type="math/tex">\quad f^{-1}(x)=x^{1\over 3}</script></span></p>
+<ul>
+<li>inverse of <span class="math"><script type="math/tex">\quad f(x)=a(x-h)^3+k</script></span></li>
+</ul>
+<p><strong>Quartic:</strong> <span class="math"><script type="math/tex">\quad f(x)=a(x-h)^4+k</script></span></p>
+<h2>Set notation</h2>
+<ul>
+<li>set difference: <span class="math"><script type="math/tex">\quad A \setminus B=\{x:x\in A, x\notin B\}</script></span></li>
+<li>interval notation: <span class="math"><script type="math/tex">(a,b)=\{x : a \lt x \lt b\},\quad</script></span>[a,b]={x : a \lte x \lte b}$</li>
+</ul>
+<h2>Functions</h2>
+<span class="math"><script type="math/tex; mode=display">f:\operatorname{dom}(f) \rightarrow \mathbb{R},\quad f(x)=\dots
+</script></span>
+<ul>
+<li>function - one <span class="math"><script type="math/tex">y</script></span> (image) value per <span class="math"><script type="math/tex">x</script></span> (preimage)</li>
+<li>1:1 function - unique <span class="math"><script type="math/tex">y</script></span> for each <span class="math"><script type="math/tex">x</script></span></li>
+</ul>
+<p><strong>Domain <span class="math"><script type="math/tex">\operatorname{dom}(f)</script></span>:</strong> set of all <span class="math"><script type="math/tex">x</script></span> values in function</p>
+<ul>
+<li>maximal (implied) domain - largest domain for which the rule is defined</li>
+<li>restricted domain: <span class="math"><script type="math/tex">f(x)=\dots,\quad\operatorname{dom}(f)</script></span></li>
+</ul>
+<p><strong>Range <span class="math"><script type="math/tex">\operatorname{ran}(f)</script></span>:</strong> set of all <span class="math"><script type="math/tex">y</script></span> values in function</p>
+<p><strong>Piecewise functions:</strong> each domain has a corresponding equation</p>
+<p><strong>Inverse functions:</strong> <span class="math"><script type="math/tex">f^-1(x)=y</script></span> if <span class="math"><script type="math/tex">f(y)=x,\quad</script></span> for <span class="math"><script type="math/tex">x\in \operatorname{ran}(f), y\in \operatorname{dom}(f)</script></span></p>
+<p><strong>Methods of factorising cubics:</strong></p>
+<ul>
+<li>extract common factor</li>
+<li>factor theorem</li>
+<li>polynomial division or equating coefficients</li>
+<li>sum or difference of two cubes</li>
+<li>quadratic formula</li>
+</ul>
+<p><strong>Polynomials:</strong> <span class="math"><script type="math/tex">P(x)=a_nx^n+a_{n-1}x^{n-1}+\dots a_1x+a_0</script></span></p>
+<ul>
+<li>degree (<span class="math"><script type="math/tex">n</script></span>) of <span class="math"><script type="math/tex">P(x)</script></span> is the highest power of <span class="math"><script type="math/tex">x</script></span></li>
+</ul>
+<h2>Bisection method</h2>
+<p>If <span class="math"><script type="math/tex">f(x)=0</script></span> has a solution <span class="math"><script type="math/tex">\alpha</script></span> between <span class="math"><script type="math/tex">[a_1,b_1]</script></span>:</p>
+<ul>
+<li><span class="math"><script type="math/tex">f(a_1)\lt 0,\quad f(b_1)>0</script></span></li>
+<li>calculate <span class="math"><script type="math/tex">f(c_1)=f({{a_1+b_1}\over 2})</script></span></li>
+<li>if <span class="math"><script type="math/tex">f(c_1)\lt 0</script></span>, a root lies between <span class="math"><script type="math/tex">c_1</script></span> and <span class="math"><script type="math/tex">b_1</script></span></li>
+<li>if <span class="math"><script type="math/tex">f(c_1)\gt 0</script></span>, a root lies between <span class="math"><script type="math/tex">a_1</script></span> and <span class="math"><script type="math/tex">c_1</script></span></li>
+</ul>
+<h2>Matrices</h2>
+<ul>
+<li>addition is only defined when dimensions are equal</li>
+<li>multiplication is only defined when columns in first = rows in second</li>
+<li>identity - equal to one</li>
+<li>inverse: <span class="math"><script type="math/tex">A^{-1}={1\over{ad-bc}}\times A\prime</script></span></li>
+<li>determinant: <span class="math"><script type="math/tex">\det(A)=ad-bc</script></span></li>
+</ul>
+<h2>Probability</h2>
+<ul>
+<li>mutually exclusive: <span class="math"><script type="math/tex">\Pr(A\cap B)=0</script></span></li>
+<li>independent: <span class="math"><script type="math/tex">\Pr(A|B)=\Pr(A)</script></span></li>
+<li>addition rule: <span class="math"><script type="math/tex">\Pr(A\cup B)=\Pr(A)+\Pr(B)-\Pr(A\cap B)</script></span></li>
+<li>multiplication rule: <span class="math"><script type="math/tex">\Pr(A\cap B)=\Pr(A|B)\times \Pr(B)</script></span></li>
+<li>law of total probability: <span class="math"><script type="math/tex">\Pr(A)=\Pr(A|B)\Pr(B)+\Pr(A|B\prime)\Pr(B\prime)</script></span></li>
+<li>conditional probability: <span class="math"><script type="math/tex">\Pr(A|B)={{\Pr(A\cap B)\over\Pr(B)}}</script></span></li>
+</ul>
+<h2>Combinatorics</h2>
+<ul>
+<li>Arrangements of <span class="math"><script type="math/tex">n</script></span> in <span class="math"><script type="math/tex">r</script></span> is given by <span class="math"><script type="math/tex">n!\over{(n-r)!}</script></span></li>
+<li>Combinations of <span class="math"><script type="math/tex">n</script></span> in <span class="math"><script type="math/tex">r</script></span> is given by <span class="math"><script type="math/tex">n! \over{r!(n-r)!}</script></span></li>
+</ul>
+
+ </body>
+</html>
--- /dev/null
+# Methods - Semester 1
+
+## Simulatenous equations
+Methods of solving:
+- substitution (state one variable in terms of the other)
+- subtraction (subtract one equation from the other, substitute resulting equation into the other)
+
+## Linear inequatlities
+
+- Flip operator when multiplying / dividing by <0
+
+## Coordinate geometry
+
+Regarding points $(x_1,y_1)(x_2y_2)$:
+
+**Midpoint:** $m=({{\Sigma x}\over 2}, {{\Sigma y}\over 2})$
+
+**Distance:** $d=\sqrt{(\Delta x)^2+(\Delta y)^2}$
+
+**Gradient:** $m={\operatorname{rise}\over\operatorname{run}}={\Delta y \over \Delta x}$
+
+**Line through points:** $y={\Delta y \over \Delta x}(x+x_2)-y_2$
+
+- parallel lines: $m_1=m_2$
+- perpendicular lines: $m_1m_2=-1$
+
+## Polynomials
+
+**Binomial expansion:** $(a+b)(c+d) = a(c+d) + b(c+d)$
+**Cubic expansion:** $(a+b)^3=a^3+3a^2b+3ab^2+b^3$
+**Quartic expansion:** $(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4$
+
+**Perfect square expansion:** $(a+b)^2=a^2+2ab+b^2$
+**Difference of perfect squares:** $a^2-b^2=(a+b)(a-b)$
+
+**Factorising quadratics:** $x^2+bx+c = (x+e)(x-f)$
+where $e \times f = c, \quad e+f = b$
+
+**Remainder theorem:** if ${P({-\alpha \over \beta})}=0,$ then ${\beta x+ \alpha}$ is a factor of $P(x)$
+
+**Factor theorem:** if $P(\alpha)=0,$ then $x-\alpha$ is a factor of $P(x)$
+
+**Completing the square (monic):** $x^2+bx+c=(x+{b\over2})^2+c-{b^2\over4}$
+**Completing the square (non-monic):** $ax^2+bx+c=a(x-{b\over2a})^2+c-{b^2\over4a}$
+
+**Quadratic formula:** $x={{-b\pm\sqrt{b^2-4ac}}\over2a}$
+**Discriminant:** $\Delta=b^2-4ac$
+
+**Solving** $y_1=ax^2+bx+c_1, \quad y_2=mx+c_2$:
+- $ax^2+(b-m)x+(c_1-c_2)=0$
+- Solve for $x$, substitute into $y_2$
+
+**Axis of symmetry:** $x={-b\over2a}$ (when written as $y=ax^2+bx+c$)
+
+**Determining quadratic rules:**
+$(1)\quad y=a(x-e)(x-f)$ two $x$-intercepts, one point
+$(2)\quad y=a(x-h)^2+k$ turning point, one point
+$(3)\quad y=ax^2+bx+c$ three points
+
+## Graphs
+
+In general:
+- turning point / centre point at $(h,k)$
+- asymptotes at $x=h$ and $y=k$
+- $a+$ dilates graph away from centre point
+- $a<0$ reflects graph across $x=0$
+- $h$ is the horizontal ($x$) shift, $k$ is the vertical ($y$) shift
+
+**Rectangular hyperbola:** $\quad y={a\over x-h}+k$
+- as $x\rightarrow \pm\infty, \quad y \rightarrow 0^\pm$
+- as $x \rightarrow 0^{\pm}, \quad y \rightarrow \pm \infty$
+
+**Truncus:**$\quad y={a\over (x-h)^2}+k$
+- as $x \rightarrow \pm \infty, \quad y \rightarrow 0^+$
+- as $x \rightarrow 0^{\pm}, \quad y \rightarrow \infty$
+
+**Square root:**$\quad y=a\sqrt{x-h}+k$
+- parabola rotated 90 degrees
+
+**Square root negative:**$\quad y=a\sqrt{-(x-h)}+k$
+- reflection of $y=\sqrt{x}$ across $y$-axis
+
+**Circle:**$\quad (x-h)^2+(y-x)^2=r^2$
+- factorised: $x^2+y^2-2hx-2ky+c=0$
+- or: $y=\pm\sqrt{r^2-x^2}$
+
+**Semicircles** - take +ve or -ve square root
+- $y=\pm\sqrt{r^2-x^2}\quad$(top or bottom)
+- $y=\pm\sqrt{r^2-y^2}\quad$(left or right)
+
+**Cubic:** $\quad f(x)=a(x-h)^3+k$
+- all cubics have >1 root
+
+**Inverse cubic:** $\quad f^{-1}(x)=x^{1\over 3}$
+- inverse of $\quad f(x)=a(x-h)^3+k$
+
+**Quartic:** $\quad f(x)=a(x-h)^4+k$
+
+
+## Set notation
+
+- set difference: $\quad A \setminus B=\{x:x\in A, x\notin B\}$
+- interval notation: $(a,b)=\{x : a \lt x \lt b\},\quad$[a,b]=\{x : a \lte x \lte b\}$
+
+## Functions
+$$f:\operatorname{dom}(f) \rightarrow \mathbb{R},\quad f(x)=\dots$$
+- function - one $y$ (image) value per $x$ (preimage)
+- 1:1 function - unique $y$ for each $x$
+
+**Domain $\operatorname{dom}(f)$:** set of all $x$ values in function
+- maximal (implied) domain - largest domain for which the rule is defined
+- restricted domain: $f(x)=\dots,\quad\operatorname{dom}(f)$
+
+**Range $\operatorname{ran}(f)$:** set of all $y$ values in function
+
+**Piecewise functions:** each domain has a corresponding equation
+
+**Inverse functions:** $f^-1(x)=y$ if $f(y)=x,\quad$ for $x\in \operatorname{ran}(f), y\in \operatorname{dom}(f)$
+
+**Methods of factorising cubics:**
+- extract common factor
+- factor theorem
+- polynomial division or equating coefficients
+- sum or difference of two cubes
+- quadratic formula
+
+**Polynomials:** $P(x)=a_nx^n+a_{n-1}x^{n-1}+\dots a_1x+a_0$
+- degree ($n$) of $P(x)$ is the highest power of $x$
+
+## Bisection method
+If $f(x)=0$ has a solution $\alpha$ between $[a_1,b_1]$:
+- $f(a_1)\lt 0,\quad f(b_1)>0$
+- calculate $f(c_1)=f({{a_1+b_1}\over 2})$
+- if $f(c_1)\lt 0$, a root lies between $c_1$ and $b_1$
+- if $f(c_1)\gt 0$, a root lies between $a_1$ and $c_1$
+
+## Matrices
+- addition is only defined when dimensions are equal
+- multiplication is only defined when columns in first = rows in second
+- identity - equal to one
+- inverse: $A^{-1}={1\over{ad-bc}}\times A\prime$
+- determinant: $\det(A)=ad-bc$
+
+## Probability
+- mutually exclusive: $\Pr(A\cap B)=0$
+- independent: $\Pr(A|B)=\Pr(A)$
+- addition rule: $\Pr(A\cup B)=\Pr(A)+\Pr(B)-\Pr(A\cap B)$
+- multiplication rule: $\Pr(A\cap B)=\Pr(A|B)\times \Pr(B)$
+- law of total probability: $\Pr(A)=\Pr(A|B)\Pr(B)+\Pr(A|B\prime)\Pr(B\prime)$
+- conditional probability: $\Pr(A|B)={{\Pr(A\cap B)\over\Pr(B)}}$
+
+## Combinatorics
+
+- Arrangements of $n$ in $r$ is given by $n!\over{(n-r)!}$
+- Combinations of $n$ in $r$ is given by $n! \over{r!(n-r)!}$
--- /dev/null
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+ <h1>Methods - Semester 1</h1>
+<h2>Simulatenous equations</h2>
+<p>Methods of solving:</p>
+<ul>
+<li>substitution (state one variable in terms of the other)</li>
+<li>subtraction (subtract one equation from the other, substitute resulting equation into the other)</li>
+</ul>
+<h2>Linear inequatlities</h2>
+<ul>
+<li>Flip operator when multiplying / dividing by <0</li>
+</ul>
+<h2>Coordinate geometry</h2>
+<p>Regarding points <span class="math"><script type="math/tex">(x_1,y_1)(x_2y_2)</script></span>:</p>
+<p><strong>Midpoint:</strong> <span class="math"><script type="math/tex">m=({{\Sigma x}\over 2}, {{\Sigma y}\over 2})</script></span></p>
+<p><strong>Distance:</strong> <span class="math"><script type="math/tex">d=\sqrt{(\Delta x)^2+(\Delta y)^2}</script></span></p>
+<p><strong>Gradient:</strong> <span class="math"><script type="math/tex">m={\operatorname{rise}\over\operatorname{run}}={\Delta y \over \Delta x}</script></span></p>
+<p><strong>Line through points:</strong> <span class="math"><script type="math/tex">y={\Delta y \over \Delta x}(x+x_2)-y_2</script></span></p>
+<ul>
+<li>parallel lines: <span class="math"><script type="math/tex">m_1=m_2</script></span></li>
+<li>perpendicular lines: <span class="math"><script type="math/tex">m_1m_2=-1</script></span></li>
+</ul>
+<h2>Polynomials</h2>
+<p><strong>Binomial expansion:</strong> <span class="math"><script type="math/tex">(a+b)(c+d) = a(c+d) + b(c+d)</script></span><br>
+<strong>Cubic expansion:</strong> <span class="math"><script type="math/tex">(a+b)^3=a^3+3a^2b+3ab^2+b^3</script></span><br>
+<strong>Quartic expansion:</strong> <span class="math"><script type="math/tex">(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4</script></span></p>
+<p><strong>Perfect square expansion:</strong> <span class="math"><script type="math/tex">(a+b)^2=a^2+2ab+b^2</script></span><br>
+<strong>Difference of perfect squares:</strong> <span class="math"><script type="math/tex">a^2-b^2=(a+b)(a-b)</script></span></p>
+<p><strong>Factorising quadratics:</strong> <span class="math"><script type="math/tex">x^2+bx+c = (x+e)(x-f)</script></span><br>
+where <span class="math"><script type="math/tex">e \times f = c, \quad e+f = b</script></span></p>
+<p><strong>Remainder theorem:</strong> if <span class="math"><script type="math/tex">{P({-\alpha \over \beta})}=0,</script></span> then <span class="math"><script type="math/tex">{\beta x+ \alpha}</script></span> is a factor of <span class="math"><script type="math/tex">P(x)</script></span></p>
+<p><strong>Factor theorem:</strong> if <span class="math"><script type="math/tex">P(\alpha)=0,</script></span> then <span class="math"><script type="math/tex">x-\alpha</script></span> is a factor of <span class="math"><script type="math/tex">P(x)</script></span></p>
+<p><strong>Completing the square (monic):</strong> <span class="math"><script type="math/tex">x^2+bx+c=(x+{b\over2})^2+c-{b^2\over4}</script></span><br>
+<strong>Completing the square (non-monic):</strong> <span class="math"><script type="math/tex">ax^2+bx+c=a(x-{b\over2a})^2+c-{b^2\over4a}</script></span></p>
+<p><strong>Quadratic formula:</strong> <span class="math"><script type="math/tex">x={{-b\pm\sqrt{b^2-4ac}}\over2a}</script></span><br>
+<strong>Discriminant:</strong> <span class="math"><script type="math/tex">\Delta=b^2-4ac</script></span></p>
+<p><strong>Solving</strong> <span class="math"><script type="math/tex">y_1=ax^2+bx+c_1, \quad y_2=mx+c_2</script></span>:</p>
+<ul>
+<li><span class="math"><script type="math/tex">ax^2+(b-m)x+(c_1-c_2)=0</script></span></li>
+<li>Solve for <span class="math"><script type="math/tex">x</script></span>, substitute into <span class="math"><script type="math/tex">y_2</script></span></li>
+</ul>
+<p><strong>Axis of symmetry:</strong> <span class="math"><script type="math/tex">x={-b\over2a}</script></span> (when written as <span class="math"><script type="math/tex">y=ax^2+bx+c</script></span>)</p>
+<p><strong>Determining quadratic rules:</strong><br>
+<span class="math"><script type="math/tex">(1)\quad y=a(x-e)(x-f)</script></span> two <span class="math"><script type="math/tex">x</script></span>-intercepts, one point<br>
+<span class="math"><script type="math/tex">(2)\quad y=a(x-h)^2+k</script></span> turning point, one point<br>
+<span class="math"><script type="math/tex">(3)\quad y=ax^2+bx+c</script></span> three points</p>
+<h2>Graphs</h2>
+<p>In general:</p>
+<ul>
+<li>turning point / centre point at <span class="math"><script type="math/tex">(h,k)</script></span></li>
+<li>asymptotes at <span class="math"><script type="math/tex">x=h</script></span> and <span class="math"><script type="math/tex">y=k</script></span></li>
+<li><span class="math"><script type="math/tex">a+</script></span> dilates graph away from centre point</li>
+<li><span class="math"><script type="math/tex">a<0</script></span> reflects graph across <span class="math"><script type="math/tex">x=0</script></span></li>
+<li><span class="math"><script type="math/tex">h</script></span> is the horizontal (<span class="math"><script type="math/tex">x</script></span>) shift, <span class="math"><script type="math/tex">k</script></span> is the vertical (<span class="math"><script type="math/tex">y</script></span>) shift</li>
+</ul>
+<p><strong>Rectangular hyperbola:</strong> <span class="math"><script type="math/tex">\quad y={a\over x-h}+k</script></span></p>
+<ul>
+<li>as <span class="math"><script type="math/tex">x\rightarrow \pm\infty, \quad y \rightarrow 0^\pm</script></span></li>
+<li>as <span class="math"><script type="math/tex">x \rightarrow 0^{\pm}, \quad y \rightarrow \pm \infty</script></span></li>
+</ul>
+<p><strong>Truncus:</strong><span class="math"><script type="math/tex">\quad y={a\over (x-h)^2}+k</script></span></p>
+<ul>
+<li>as <span class="math"><script type="math/tex">x \rightarrow \pm \infty, \quad y \rightarrow 0^+</script></span></li>
+<li>as <span class="math"><script type="math/tex">x \rightarrow 0^{\pm}, \quad y \rightarrow \infty</script></span></li>
+</ul>
+<p><strong>Square root:</strong><span class="math"><script type="math/tex">\quad y=a\sqrt{x-h}+k</script></span></p>
+<ul>
+<li>parabola rotated 90 degrees</li>
+</ul>
+<p><strong>Square root negative:</strong><span class="math"><script type="math/tex">\quad y=a\sqrt{-(x-h)}+k</script></span></p>
+<ul>
+<li>reflection of <span class="math"><script type="math/tex">y=\sqrt{x}</script></span> across <span class="math"><script type="math/tex">y</script></span>-axis</li>
+</ul>
+<p><strong>Circle:</strong><span class="math"><script type="math/tex">\quad (x-h)^2+(y-x)^2=r^2</script></span></p>
+<ul>
+<li>factorised: <span class="math"><script type="math/tex">x^2+y^2-2hx-2ky+c=0</script></span></li>
+<li>or: <span class="math"><script type="math/tex">y=\pm\sqrt{r^2-x^2}</script></span></li>
+</ul>
+<p><strong>Semicircles</strong> - take +ve or -ve square root</p>
+<ul>
+<li><span class="math"><script type="math/tex">y=\pm\sqrt{r^2-x^2}\quad</script></span>(top or bottom)</li>
+<li><span class="math"><script type="math/tex">y=\pm\sqrt{r^2-y^2}\quad</script></span>(left or right)</li>
+</ul>
+<p><strong>Cubic:</strong> <span class="math"><script type="math/tex">\quad f(x)=a(x-h)^3+k</script></span></p>
+<ul>
+<li>all cubics have >1 root</li>
+</ul>
+<p><strong>Inverse cubic:</strong> <span class="math"><script type="math/tex">\quad f^{-1}(x)=x^{1\over 3}</script></span></p>
+<ul>
+<li>inverse of <span class="math"><script type="math/tex">\quad f(x)=a(x-h)^3+k</script></span></li>
+</ul>
+<p><strong>Quartic:</strong> <span class="math"><script type="math/tex">\quad f(x)=a(x-h)^4+k</script></span></p>
+<h2>Set notation</h2>
+<ul>
+<li>set difference: <span class="math"><script type="math/tex">\quad A \setminus B=\{x:x\in A, x\notin B\}</script></span></li>
+<li>interval notation: <span class="math"><script type="math/tex">(a,b)=\{x : a \lt x \lt b\},\quad</script></span>[a,b]={x : a \lte x \lte b}$</li>
+</ul>
+<h2>Functions</h2>
+<span class="math"><script type="math/tex; mode=display">f:\operatorname{dom}(f) \rightarrow \mathbb{R},\quad f(x)=\dots
+</script></span>
+<ul>
+<li>function - one <span class="math"><script type="math/tex">y</script></span> (image) value per <span class="math"><script type="math/tex">x</script></span> (preimage)</li>
+<li>1:1 function - unique <span class="math"><script type="math/tex">y</script></span> for each <span class="math"><script type="math/tex">x</script></span></li>
+</ul>
+<p><strong>Domain <span class="math"><script type="math/tex">\operatorname{dom}(f)</script></span>:</strong> set of all <span class="math"><script type="math/tex">x</script></span> values in function</p>
+<ul>
+<li>maximal (implied) domain - largest domain for which the rule is defined</li>
+<li>restricted domain: <span class="math"><script type="math/tex">f(x)=\dots,\quad\operatorname{dom}(f)</script></span></li>
+</ul>
+<p><strong>Range <span class="math"><script type="math/tex">\operatorname{ran}(f)</script></span>:</strong> set of all <span class="math"><script type="math/tex">y</script></span> values in function</p>
+<p><strong>Piecewise functions:</strong> each domain has a corresponding equation</p>
+<p><strong>Inverse functions:</strong> <span class="math"><script type="math/tex">f^-1(x)=y</script></span> if <span class="math"><script type="math/tex">f(y)=x,\quad</script></span> for <span class="math"><script type="math/tex">x\in \operatorname{ran}(f), y\in \operatorname{dom}(f)</script></span></p>
+<p><strong>Methods of factorising cubics:</strong></p>
+<ul>
+<li>extract common factor</li>
+<li>factor theorem</li>
+<li>polynomial division or equating coefficients</li>
+<li>sum or difference of two cubes</li>
+<li>quadratic formula</li>
+</ul>
+<p><strong>Polynomials:</strong> <span class="math"><script type="math/tex">P(x)=a_nx^n+a_{n-1}x^{n-1}+\dots a_1x+a_0</script></span></p>
+<ul>
+<li>degree (<span class="math"><script type="math/tex">n</script></span>) of <span class="math"><script type="math/tex">P(x)</script></span> is the highest power of <span class="math"><script type="math/tex">x</script></span></li>
+</ul>
+<h2>Bisection method</h2>
+<p>If <span class="math"><script type="math/tex">f(x)=0</script></span> has a solution <span class="math"><script type="math/tex">\alpha</script></span> between <span class="math"><script type="math/tex">[a_1,b_1]</script></span>:</p>
+<ul>
+<li><span class="math"><script type="math/tex">f(a_1)\lt 0,\quad f(b_1)>0</script></span></li>
+<li>calculate <span class="math"><script type="math/tex">f(c_1)=f({{a_1+b_1}\over 2})</script></span></li>
+<li>if <span class="math"><script type="math/tex">f(c_1)\lt 0</script></span>, a root lies between <span class="math"><script type="math/tex">c_1</script></span> and <span class="math"><script type="math/tex">b_1</script></span></li>
+<li>if <span class="math"><script type="math/tex">f(c_1)\gt 0</script></span>, a root lies between <span class="math"><script type="math/tex">a_1</script></span> and <span class="math"><script type="math/tex">c_1</script></span></li>
+</ul>
+<h2>Matrices</h2>
+<ul>
+<li>addition is only defined when dimensions are equal</li>
+<li>multiplication is only defined when columns in first = rows in second</li>
+<li>identity - equal to one</li>
+<li>inverse: <span class="math"><script type="math/tex">A^{-1}={1\over{ad-bc}}\times A\prime</script></span></li>
+<li>determinant: <span class="math"><script type="math/tex">\det(A)=ad-bc</script></span></li>
+</ul>
+<h2>Probability</h2>
+<ul>
+<li>mutually exclusive: <span class="math"><script type="math/tex">\Pr(A\cap B)=0</script></span></li>
+<li>independent: <span class="math"><script type="math/tex">\Pr(A|B)=\Pr(A)</script></span></li>
+<li>addition rule: <span class="math"><script type="math/tex">\Pr(A\cup B)=\Pr(A)+\Pr(B)-\Pr(A\cap B)</script></span></li>
+<li>multiplication rule: <span class="math"><script type="math/tex">\Pr(A\cap B)=\Pr(A|B)\times \Pr(B)</script></span></li>
+<li>law of total probability: <span class="math"><script type="math/tex">\Pr(A)=\Pr(A|B)\Pr(B)+\Pr(A|B\prime)\Pr(B\prime)</script></span></li>
+<li>conditional probability: <span class="math"><script type="math/tex">\Pr(A|B)={{\Pr(A\cap B)\over\Pr(B)}}</script></span></li>
+</ul>
+<h2>Combinatorics</h2>
+<ul>
+<li>Arrangements of <span class="math"><script type="math/tex">n</script></span> in <span class="math"><script type="math/tex">r</script></span> is given by <span class="math"><script type="math/tex">n!\over{(n-r)!}</script></span></li>
+<li>Combinations of <span class="math"><script type="math/tex">n</script></span> in <span class="math"><script type="math/tex">r</script></span> is given by <span class="math"><script type="math/tex">n! \over{r!(n-r)!}</script></span></li>
+</ul>
+
+ </body>
+</html>
--- /dev/null
+# Time Travel and Quantum Mechanics
+
+## What is time?
+- time is what clocks measure
+ - measure time objectively, therefore does not help our subjective definition of time
+
+- $t$ - just like another quantitative variable (period of time, not absolute)
+
+
+- fourth dimension
+
+- these describe properties of it, not its implications or how it works
+ - implications are generally determined through intuition
+
+
+- sense of time is inherent to all humans - every human has memory and some idea that time is observable through changes in space
+
+ - computers have memory, do they have a concept of time?
+
+Therefore, we have:
+
+**1. Subjective time**
+- presumably everyone experiences this notion of time
+- has the property of lapsing (time approaches $\infty$)
+
+**2. Physical time**
+- represented by $t$
+- objective
+- quantitative (period, not absolute)
+
+> All paradoxes arise from the fact that time lapses / flows - we are basing time on subjective time, not physical time
+
+## Concepts of time in physics
+
+### Thermodynamic time arrow
+- second law of thermodynamics:
+- total entropy cannot decrease over time (randomness of particles)
+- Hence, time has a specific direction (approaching infinite entropy)
+- This can be modelled as $t=e$. If $e$ cannot be subtracted from, $t$ can only go one way.
+- entropy is an irreversible process
+- therefore, time travel backwards is impossible
+
+### Block time
+- time is fourth dimension - all dimensions are related
+- universe exists as a whole, in four dimensions, just like a chair can exist in three dimensions (plus time)
+- you cannot isolate one value in one of the three dimensions and define it by itself - it would be ridiculous to say that a point ($x=2,y=1,z=3$) can exist if y and z are not defined
+- therefore, past and future must also exist as a relation of four values ($x,y,z,t$)
+
+
+2. Are the past and future real?
+By using the present as a proxy for the past or future, we can simulate them, effectively nesting times inside times (a simpler idea of a time machine - normal time machines lose the idea of the present).
+
+3. Do events that occurred in the past / future exist?
+
+> What is the physical prompt to change to a different time?
+
+## Grandfather paradox
+
+- Grandfather is not necessary - could be the suicide paradox
+- Mathematical status - 1 bit (dead or alive):
+ - $x=$ existence at 2 yrs, $y=$ existence at 20 yrs
+ - $0=$ dead, $1=$ alive
+- $x=1$ (alive - assumed because you must exist in the first instance)
+
+
+**(case 1)** if $x=1$, then $y=1$ (assuming you don't die between) $\therefore x = y$
+**(case 2)** if $y=1$, then $x=0$ (because you killed yourself)
+This is not a contradiction - it simply states that $x=0 \lor 1$ (one *or* zero)
+
+**$\therefore$ both situations can exist (superposition) - suggesting multiple universe theory**
+
+- Implies a link to quantum mechanics: dead or alive, or between (superposition):
+- $1 \over 2$ probability of killing grandfather/self (two parallel realities - Murphy's law)
+
+
+### Many worlds interpretation
+
+- different branches of superposition are different universes. Perception jumps between universes.
+- time can be defined only by the physical state of things at the time - this is the only way it can be derived. therefore, by changing the physical state, you are creating a new "time (or timeline)
--- /dev/null
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+ <title>time travel</title>
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+ </head>
+ <body class='markdown-preview'><h1>Time Travel and Quantum Mechanics</h1>
+<h2>What is time?</h2>
+<ul>
+<li>
+<p>time is what clocks measure</p>
+<ul>
+<li>measure time objectively, therefore does not help our subjective definition of time</li>
+</ul>
+</li>
+<li>
+<p><span class="math"><script type="math/tex">t</script></span> - just like another quantitative variable (period of time, not absolute)</p>
+</li>
+</ul>
+<ul>
+<li>
+<p>fourth dimension</p>
+</li>
+<li>
+<p>these describe properties of it, not its implications or how it works</p>
+<ul>
+<li>implications are generally determined through intuition</li>
+</ul>
+</li>
+</ul>
+<ul>
+<li>
+<p>sense of time is inherent to all humans - every human has memory and some idea that time is observable through changes in space</p>
+<ul>
+<li>computers have memory, do they have a concept of time?</li>
+</ul>
+</li>
+</ul>
+<p>Therefore, we have:</p>
+<p><strong>1. Subjective time</strong></p>
+<ul>
+<li>presumably everyone experiences this notion of time</li>
+<li>has the property of lapsing (time approaches <span class="math"><script type="math/tex">\infty</script></span>)</li>
+</ul>
+<p><strong>2. Physical time</strong></p>
+<ul>
+<li>represented by <span class="math"><script type="math/tex">t</script></span></li>
+<li>objective</li>
+<li>quantitative (period, not absolute)</li>
+</ul>
+<blockquote>
+<p>All paradoxes arise from the fact that time lapses / flows - we are basing time on subjective time, not physical time</p>
+</blockquote>
+<h2>Concepts of time in physics</h2>
+<h3>Thermodynamic time arrow</h3>
+<ul>
+<li>second law of thermodynamics:</li>
+<li>total entropy cannot decrease over time (randomness of particles)</li>
+<li>Hence, time has a specific direction (approaching infinite entropy)</li>
+<li>This can be modelled as <span class="math"><script type="math/tex">t=e</script></span>. If <span class="math"><script type="math/tex">e</script></span> cannot be subtracted from, <span class="math"><script type="math/tex">t</script></span> can only go one way.</li>
+<li>entropy is an irreversible process</li>
+<li>therefore, time travel backwards is impossible</li>
+</ul>
+<h3>Block time</h3>
+<ul>
+<li>time is fourth dimension - all dimensions are related</li>
+<li>universe exists as a whole, in four dimensions, just like a chair can exist in three dimensions (plus time)</li>
+<li>you cannot isolate one value in one of the three dimensions and define it by itself - it would be ridiculous to say that a point (<span class="math"><script type="math/tex">x=2,y=1,z=3</script></span>) can exist if y and z are not defined</li>
+<li>therefore, past and future must also exist as a relation of four values (<span class="math"><script type="math/tex">x,y,z,t</script></span>)</li>
+</ul>
+<ol start="2">
+<li>
+<p>Are the past and future real?<br>
+By using the present as a proxy for the past or future, we can simulate them, effectively nesting times inside times (a simpler idea of a time machine - normal time machines lose the idea of the present).</p>
+</li>
+<li>
+<p>Do events that occurred in the past / future exist?</p>
+</li>
+</ol>
+<blockquote>
+<p>What is the physical prompt to change to a different time?</p>
+</blockquote>
+<h2>Grandfather paradox</h2>
+<ul>
+<li>Grandfather is not necessary - could be the suicide paradox</li>
+<li>Mathematical status - 1 bit (dead or alive):
+<ul>
+<li><span class="math"><script type="math/tex">x=</script></span> existence at 2 yrs, <span class="math"><script type="math/tex">y=</script></span> existence at 20 yrs</li>
+<li><span class="math"><script type="math/tex">0=</script></span> dead, <span class="math"><script type="math/tex">1=</script></span> alive</li>
+</ul>
+</li>
+<li><span class="math"><script type="math/tex">x=1</script></span> (alive - assumed because you must exist in the first instance)</li>
+</ul>
+<p><strong>(case 1)</strong> if <span class="math"><script type="math/tex">x=1</script></span>, then <span class="math"><script type="math/tex">y=1</script></span> (assuming you don’t die between) <span class="math"><script type="math/tex">\therefore x = y</script></span><br>
+<strong>(case 2)</strong> if <span class="math"><script type="math/tex">y=1</script></span>, then <span class="math"><script type="math/tex">x=0</script></span> (because you killed yourself)<br>
+This is not a contradiction - it simply states that <span class="math"><script type="math/tex">x=0 \lor 1</script></span> (one <em>or</em> zero)</p>
+<p><strong><span class="math"><script type="math/tex">\therefore</script></span> both situations can exist (superposition) - suggesting multiple universe theory</strong></p>
+<ul>
+<li>Implies a link to quantum mechanics: dead or alive, or between (superposition):</li>
+<li><span class="math"><script type="math/tex">1 \over 2</script></span> probability of killing grandfather/self (two parallel realities - Murphy’s law)</li>
+</ul>
+<h3>Many worlds interpretation</h3>
+<ul>
+<li>different branches of superposition are different universes. Perception jumps between universes.</li>
+<li>time can be defined only by the physical state of things at the time - this is the only way it can be derived. therefore, by changing the physical state, you are creating a new "time (or timeline)</li>
+</ul></body>
+</html>
--- /dev/null
+# Time
+
+## Presentism
+
+- The present time exists exclusively with its objects
+- There are no non-present objects
+- Past & future are human constructs
+- Memories etc are representations of the past in the present
+
+
+<pre>past present future
+---------|----------
+-->-->-->-->-->-->-></pre>
+
+## Eternalism
+
+- Fourth dimension
+- Distant times in past and future exist, like distant places
+- Past, present and future are equally real
+
+## Philosophers' Perspectives
+
+### Aristotle
+- we measure time as the "interval between two nows"
+- time $\propto$ change
+- subjective
+
+### Nietzsche
+- Time repeats infinitely in the exact same sequence
+- "Love your fate"
+
+### Augustine
+- Linking Aristotelian philosophy (infinitesimal yet eternal present) to God (always present)
+- Anything that lasts for a finite time comes from God, yet God is an eternal now
+- Also considered afterlife / multiple phases of time
+- Time tends towards non-existence, because the past and future are the only times that can really exist, and the present is infinitesimal (the only definite time).
+- Yet, time could not exist in infinitesimally small pieces, which go out of existence instantaneously.
+- Hence, time is a function of human perception and memory.
+
+#### Response
+Each premise of the argument seems to appear reasonable, but it is merely meta-analysis - building upon the Aristotelian idea that the present is infinitesimal. Also, there is no proof that time could not exist in infinitesimally small pieces - persistent memory does not disprove this. Events could span multiple "slices" of time. The conclusion that time is an exclusively human construct does not follow from the premise that the present could not exist infinitesimally.
+
+### Borges
+- We think time is a sequence of unique moments, but they can be repeatedf
+- Time = sensations e.g. reading Shakespeare makes you Shakespeare
+- Finite number of variables, therefore time can repeat itself
+
+### Einstein
+- We cannot measure time without being external to it
+- 1 s / s
--- /dev/null
+# Circular forces
+
+## Velocity in a circle
+
+$$v={{2 \pi r} \over T}$$
+
+where $T$ is the period (time for one oscillation)
+
+(derived from $v={d \over t}$ where $d=2 \pi r = \pi D$)
+
+## Frequency and period
+
+$$ f={1 \over T}$$
+$$ t={1 \over f}$$
+
+## Centripetal acceleration & force
+
+$$a={{{v^2} \over r}={{4 \pi^2r}\over T^2}}$$
+
+where
+$a$ is centripetal acceleration
+$v$ is speed
+$r$ is radius
+$T$ is period
+
+and
+
+$v \perp a$
+
+We know that $F=ma$, so
+
+$$F={{mv^2}\over r}={{4\pi^2rm}\over T^2}$$
+
+## Banked track
+
+Forces acting:
+- Weight force $F_g$, vertically down
+- Normal (reaction) force $R$, perpendicular to slope
+
+
+- Net force $\Sigma F$ acts towards centre
+
+### Calculating the angle
+
+$$ tan \theta = {\Sigma F \over F_g} $$
+
+where
+$\Sigma F$ acts towards centre of circle
+$F_g$ is force by gravity on the moving object
+
+In terms of velocity..
+
+$$ \tan \theta = {v^2 \over rg}$$
+
+$$\therefore \theta = \tan^{-1}({v^2 \over rg})$$
+
+$$\therefore v=\sqrt{rg \tan \theta} $$
+
+- If $F_N > F_g$, passenger feels heavier
+- If $F_N < F_g$, passenger feels lighter
+
+### Minimum velocity in circle
+
+$$ v= \sqrt{gr}$$
+
+### Force and acceleration
+
+$$\Sigma F = F_N + mg$$
+
+## Vertical circular motion
+
+## Pulley-mass system
+
+$$a={{m_2g}\over{m_1+m_2}}$$
+
+where $m_2$ is the suspended mass
--- /dev/null
+# Fields
+
+**Non-contact force**
+
+Types of fields:
+- gravitational
+- electric
+- magnetic
+
+Electric and magnetic fields are the two components of the electromagnetic field (effectively the same force). Permeates the universe.
+
+Field diagrams - N to S (current +ve to -ve)
+
+
+Field strength - inverse sq law
+
+## Gravitational fields
+
+### Field strength
+
+$$g={{F_g} \over m}$$
+
+## Electric fields
+
+### Field lines
+- +ve to -ve
+- start and end $\perp$ to surface
+- never cross
+
+### Field strength
+
+$$E = {F \over q}$$
+
+
+
+
+## Transformers
+
+Primary - input
+Secondary - output
+
+As $V \Rightarrow \infty, P_{loss} \Rightarrow 0$
+
+$$P_{loss} = I^2 R$$
+
+where $R$ is determined by resistivity per length of cable
+
+Therefore, power loss is proportional to the square of current.
+
+Electronmagnetism & electromagnetic induction
+
+Changing magnetic field generates current & vice versa
+
+Only for AC, because DC voltage is constant, so does not generate magnetic field
+
+Output is proportional to ratio primary : secondary
+
+
+$${V_{sec} \over V_{pri}} = {turns_{sec} \over turns_{pri}}$$
+
+Iron - good conductor of magnetic fields. Used as a core.
--- /dev/null
+## Gravity
+
+Force that drives the universe - attractive force between masses
+1 of 4 fundamental forces
+
+$$
+{F _{g} }= {{Gm_1m_2}\over{r^2}}
+$$
+
+where
+$G=$ gravitational constant $=6.67 \times10^{-11}Nm^2kg^{-2}$
+$r=$ distance between *centres* of $m_1$ and $m_2$
+
+Area under force-distance graph = work = $\Delta E$
+
+# Newton
--- /dev/null
+# Newtons' Laws
+
+> **Up is always positive**
+
+## First law
+
+Constant velocity if forces are balanced
+- object at rest remains at rest
+- object at motion remains in motion
+**only** if it does not experience an unbalanced force
+
+## Second law
+$$F_{net} = ma$$
+
+## Third law
+
+- Equal and opposite forces
+- Normal force (reaction) is in opposite direction to action force
+
+## Inclined plane
+
+$$F=ma=mg\sin\theta-{\mathit{friction}}$$
--- /dev/null
+# Projectile motion
+
+- Horizontal velocity ($v_x$) remains constant if no drag
+
+- Vertical acceleration ($a_y$) $= -g$
+
+## Calculating velocity from $x$ and $y$ components
+
+${\displaystyle v={\sqrt {v_{x}^{2}+v_{y}^{2}\ }}}$
+
+## Displacement
+
+$x=v_{0}t\cos(\theta)$
+$y=v_{0}t\sin(\theta )-{\frac {1}{2}}gt^{2}$
+
+## Maximum height of an arc
+
+$h={\frac {u^{2}\sin ^{2}(\theta )}{2g}}$
+
+## Time of flight
+$y=u\ t\sin(\theta )-{\frac {1}{2}}gt^{2}$
--- /dev/null
+<!DOCTYPE html>
+<html>
+ <head>
+ <meta charset="utf-8" />
+ <title>relativity</title>
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+ </head>
+ <body class='markdown-preview'><h1>Special Relativity</h1>
+<h2>Postulates of special relativity</h2>
+<ol>
+<li>
+<p>Laws of physics are constant in all inertial reference frames</p>
+</li>
+<li>
+<p>Speed of light <span class="math"><script type="math/tex">c</script></span> is the same to all observers (proved by Michelson-Morley)</p>
+</li>
+</ol>
+<p><span class="math"><script type="math/tex">\therefore</script></span>, <span class="math"><script type="math/tex">t</script></span> must dilate as speed changes.</p>
+<p>Time and space - four dimensional relationship</p>
+<p><strong>Inertial reference frame</strong> - not accelerating</p>
+<p><strong>Proper time / length</strong> - measured by observer in same frame as events</p>
+<h2>Lorentz factor</h2>
+<span class="math"><script type="math/tex; mode=display">\gamma = {1 \over \sqrt{1-{v^2 \over c^2}}}
+</script></span>
+<p>Proper (time|length) denoted by (<span class="math"><script type="math/tex">t|l_0</script></span>) is time observed from moving frame.</p>
+<p>Applied:</p>
+<p><span class="math"><script type="math/tex">t = t_0 \gamma</script></span> (time - longer in moving frame)<br>
+<span class="math"><script type="math/tex">l={l_0 \over \gamma}</script></span> (length - only contracts in direction of motion - shorter in moving frame)<br>
+<span class="math"><script type="math/tex">m=m_0 \gamma</script></span> (mass dilation)</p>
+<span class="math"><script type="math/tex; mode=display">v=c \sqrt{1-{1 \over \gamma ^2}}
+</script></span>
+<p>(<span class="math"><script type="math/tex">c=3 \times 10^8</script></span> m / s)</p>
+<h2>Mass-energy equivalence</h2>
+<p><span class="math"><script type="math/tex">E_0=mc^2</script></span> (rest energy)</p>
+<p><span class="math"><script type="math/tex">E=\gamma mc^2</script></span> (total energy)</p>
+<ul>
+<li>mass is a dense form of energy</li>
+</ul>
+<h3>Relativistic momentum</h3>
+<span class="math"><script type="math/tex; mode=display">\rho = {mv \over {\sqrt{1-{v^2 \over c^2}}}} = \gamma mv=\gamma \rho_0
+</script></span>
+<p>Low speeds - little difference between relativistic & non-relativistic momentum<br>
+<span class="math"><script type="math/tex">\rho</script></span> approaches <span class="math"><script type="math/tex">\infty</script></span> as speed approaches <span class="math"><script type="math/tex">c</script></span><br>
+Impossible to reach speed of light (speed <span class="math"><script type="math/tex">c</script></span> requires <span class="math"><script type="math/tex">\infty</script></span> energy)<br>
+Photon has no mass - zero space or time</p>
+<h3>Total energy of an object</h3>
+<span class="math"><script type="math/tex; mode=display">E_{total}=E_k+E_{rest}=\rho mc^2
+</script></span>
+<h3>Kinetic energy</h3>
+<span class="math"><script type="math/tex; mode=display">E_k=(\gamma - 1)mc^2
+</script></span>
+<p>(takes relativity into account - use for fast objects)</p>
+<h3>Rest energy</h3>
+<span class="math"><script type="math/tex; mode=display">E_{rest}=mc^2
+</script></span>
+<p>(where <span class="math"><script type="math/tex">v<0.1c</script></span>) - this is the innate mass energy (proper energy)</p>
+<h3>Work</h3>
+<span class="math"><script type="math/tex; mode=display">W = \Delta E = \Delta mc^2
+</script></span>
+<h3>Velocity</h3>
+<span class="math"><script type="math/tex; mode=display">v={\rho \over {m \sqrt{1+{p^2 \over {m^2c^2}}}}}
+</script></span>
+<h3>Fusion / Fission equations</h3>
+<p>eV to J: multiply by <span class="math"><script type="math/tex">1.6\times 10^{-19}</script></span> (charge of an electron)</p>
+<h2>High altitude muons</h2>
+<ul>
+<li>Time dilation - more muons reach Earth than expected</li>
+<li>Normal half-life is <span class="math"><script type="math/tex">2.2 \mu s</script></span> in stationary frame of reference</li>
+<li>At close to <span class="math"><script type="math/tex">c</script></span>, muons observed from Earth have lifetime <span class="math"><script type="math/tex">> 2.2 \mu s</script></span></li>
+<li>Slower time - more time to travel, so more muons reach surface</li>
+</ul></body>
+</html>
--- /dev/null
+# Special Relativity
+
+## Postulates of special relativity
+
+1. Laws of physics are constant in all inertial reference frames
+
+2. Speed of light $c$ is the same to all observers (proved by Michelson-Morley)
+
+$\therefore$, $t$ must dilate as speed changes.
+
+Time and space - four dimensional relationship
+
+**Inertial reference frame** - not accelerating
+
+**Proper time / length** - measured by observer in same frame as events
+
+## Lorentz factor
+
+$$\gamma = {1 \over \sqrt{1-{v^2 \over c^2}}}$$
+
+Proper (time|length) denoted by ($t|l_0$) is time observed from moving frame.
+
+Applied:
+
+$t = t_0 \gamma$ (time - longer in moving frame)
+$l={l_0 \over \gamma}$ (length - only contracts in direction of motion - shorter in moving frame)
+$m=m_0 \gamma$ (mass dilation)
+
+$$v=c \sqrt{1-{1 \over \gamma ^2}}$$
+
+($c=3 \times 10^8$ m / s)
+
+## Mass-energy equivalence
+
+$E_0=mc^2$ (rest energy)
+
+$E=\gamma mc^2$ (total energy)
+
+- mass is a dense form of energy
+
+### Relativistic momentum
+$$\rho = {mv \over {\sqrt{1-{v^2 \over c^2}}}} = \gamma mv=\gamma \rho_0$$
+
+Low speeds - little difference between relativistic & non-relativistic momentum
+$\rho$ approaches $\infty$ as speed approaches $c$
+Impossible to reach speed of light (speed $c$ requires $\infty$ energy)
+Photon has no mass - zero space or time
+
+### Total energy of an object
+
+$$E_{total}=E_k+E_{rest}=\rho mc^2$$
+
+### Kinetic energy
+
+$$E_k=(\gamma - 1)mc^2$$
+
+(takes relativity into account - use for fast objects)
+
+### Rest energy
+$$E_{rest}=mc^2$$
+(where $v<0.1c$) - this is the innate mass energy (proper energy)
+
+### Work
+
+$$W = \Delta E = \Delta mc^2$$
+
+### Velocity
+
+$$v={\rho \over {m \sqrt{1+{p^2 \over {m^2c^2}}}}}$$
+
+### Fusion / Fission equations
+
+eV to J: multiply by $1.6\times 10^{-19}$ (charge of an electron)
+
+## High altitude muons
+- Time dilation - more muons reach Earth than expected
+- Normal half-life is $2.2 \mu s$ in stationary frame of reference
+- At close to $c$, muons observed from Earth have lifetime $> 2.2 \mu s$
+- Slower time - more time to travel, so more muons reach surface
--- /dev/null
+<!DOCTYPE html>
+<html>
+ <head>
+ <meta charset="utf-8" />
+ <title>sequences</title><script type="text/x-mathjax-config">
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+.markdown-preview-plus-view:not([data-use-github-style]) h2,
+.markdown-preview-plus-view:not([data-use-github-style]) h3,
+.markdown-preview-plus-view:not([data-use-github-style]) h4,
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+</style>
+
+ </head>
+ <body class="markdown-preview-plus-view">
+ <h1>Complex & Imaginary Numbers</h1>
+<h2>Imaginary numbers</h2>
+<p><span class="math"><script type="math/tex">i^2 = -1</script></span></p>
+<p><span class="math"><script type="math/tex">\therefore i = \sqrt {-1}</script></span></p>
+<h3>Simplifying negative surds</h3>
+<p><span class="math"><script type="math/tex">\sqrt{-2} = \sqrt{-1 \times 2}</script></span></p>
+<p> <span class="math"><script type="math/tex">= \sqrt{2}i</script></span></p>
+<h2>Complex numbers</h2>
+<p><span class="math"><script type="math/tex">\mathbb{C} = \{a+bi : a, b \in \mathbb{R} \}</script></span></p>
+<p>General form: <span class="math"><script type="math/tex">z=a+bi</script></span></p>
+<ul>
+<li><span class="math"><script type="math/tex">\operatorname{Re}(z) = a</script></span></li>
+<li><span class="math"><script type="math/tex">\operatorname{Im}(z) = b</script></span></li>
+</ul>
+<h3>Addition</h3>
+<p>If <span class="math"><script type="math/tex">z_1 = a+bi</script></span> and <span class="math"><script type="math/tex">z_2=c+di</script></span>, then</p>
+<p> <span class="math"><script type="math/tex">z_1+z_2 = (a+c)+(b+d)i</script></span></p>
+<h3>Subtraction</h3>
+<p>If <span class="math"><script type="math/tex">z_1=a+bi</script></span> and <span class="math"><script type="math/tex">z_2=c+di</script></span>, then</p>
+<p> <span class="math"><script type="math/tex">z_1−z_2=(a−c)+(b−d)i</script></span></p>
+<h3>Multiplication by a real constant</h3>
+<p>If <span class="math"><script type="math/tex">z=a+bi</script></span> and <span class="math"><script type="math/tex">k \in \mathbb{R}</script></span>, then</p>
+<p> <span class="math"><script type="math/tex">kz=ka+kbi</script></span></p>
+<h3>Powers of <span class="math"><script type="math/tex">i</script></span></h3>
+<p><span class="math"><script type="math/tex">i^0=1</script></span><br>
+<span class="math"><script type="math/tex">i^1=i</script></span><br>
+<span class="math"><script type="math/tex">i^2=-1</script></span><br>
+<span class="math"><script type="math/tex">i^3=-i</script></span><br>
+<span class="math"><script type="math/tex">i^4=1</script></span><br>
+<span class="math"><script type="math/tex">\dots</script></span></p>
+<p>Therefore…</p>
+<ul>
+<li><span class="math"><script type="math/tex">i^{4n} = 1</script></span></li>
+<li><span class="math"><script type="math/tex">i^{4n+1} = i</script></span></li>
+<li><span class="math"><script type="math/tex">i^{4n+2} = -1</script></span></li>
+<li><span class="math"><script type="math/tex">i^{4n+3} = -i</script></span></li>
+</ul>
+<h3>Multiplying complex expressions</h3>
+<p>If <span class="math"><script type="math/tex">z_1 = a+bi</script></span> and <span class="math"><script type="math/tex">z_2=c+di</script></span>, then<br>
+ <span class="math"><script type="math/tex">z_1 \times z_2 = (ac-bd)+(ad+bc)i</script></span></p>
+<h3>Conjugates</h3>
+<p>If <span class="math"><script type="math/tex">z=a+bi</script></span>, conjugate of <span class="math"><script type="math/tex">z</script></span> is <span class="math"><script type="math/tex">\overline{z} = a-bi</script></span> (flipped operator)</p>
+<p>Also, <span class="math"><script type="math/tex">z \overline{z} = (a+bi)(a-bi) = a^2+b^2</script></span></p>
+<ul>
+<li>Multiplication and addition are associative</li>
+</ul>
+<h3>Modulus</h3>
+<p>Distance from origin.<br>
+<span class="math"><script type="math/tex">|{z}|=\sqrt{a^2+b^2}</script></span></p>
+<p><span class="math"><script type="math/tex">\therefore z \overline{z} = |z|^2</script></span></p>
+<h3>Multiplicative inverse</h3>
+<p><span class="math"><script type="math/tex">z^{-1} = {1 \over z} = {{a-bi} \over {a^2+B^2}} = {\overline{z} \over {|z|^2}}</script></span></p>
+<h3>Dividing complex numbers</h3>
+<p><span class="math"><script type="math/tex">{{z_1} \over {z_2}} = {{z_1\ {z_2}^{-1}}} = {{z_1 \overline{z_2}} \over {{|z_2|}^2}}</script></span></p>
+<p>(using multiplicative inverse)</p>
+<p>In practice, rationalise denominator:<br>
+<span class="math"><script type="math/tex">{z_1 \over z_2} = {{(a+bi)(c-di)} \over {c^2+d^2}}</script></span></p>
+<h2>Argand planes</h2>
+<ul>
+<li>Geometric representation of <span class="math"><script type="math/tex">\mathbb{C}</script></span></li>
+<li>Horizontal <span class="math"><script type="math/tex">= \operatorname{Re}(z)</script></span>; vertical <span class="math"><script type="math/tex">= \operatorname{Im}(z)</script></span></li>
+<li>Multiplication by <span class="math"><script type="math/tex">i</script></span> results in an anticlockwise rotation of <span class="math"><script type="math/tex">\pi \over 2</script></span></li>
+</ul>
+<h2>Solving complex quadratics</h2>
+<p>To solve <span class="math"><script type="math/tex">z^2+a^2=0</script></span> (sum of two squares):</p>
+<p><span class="math"><script type="math/tex">z^2+a^2=z^2-(ai)^2</script></span><br>
+ <span class="math"><script type="math/tex">=(z+ai)(z-ai)</script></span></p>
+<h2>Polar form</h2>
+<p>General form:<br>
+<span class="math"><script type="math/tex">z=r \operatorname{cis} \theta</script></span><br>
+<span class="math"><script type="math/tex">= r\operatorname{cos}\theta+r\operatorname{sin}\theta i</script></span></p>
+<p>where</p>
+<ul>
+<li><span class="math"><script type="math/tex">z=a+bi</script></span></li>
+<li><span class="math"><script type="math/tex">r</script></span> is the distance from origin, given by Pythagoras (<span class="math"><script type="math/tex">r=\sqrt{x^2+y^2}</script></span>)</li>
+<li><span class="math"><script type="math/tex">\theta</script></span> is the argument of <span class="math"><script type="math/tex">z</script></span>, CCW from origin</li>
+</ul>
+<p>Note each complex number has multiple polar representations:<br>
+<span class="math"><script type="math/tex">z=r \operatorname{cis} \theta = r \operatorname{cis} (\theta+2 n\pi</script></span>) where <span class="math"><script type="math/tex">n</script></span> is integer number of revolutions</p>
+<h3>Multiplication and division in polar form</h3>
+<p><span class="math"><script type="math/tex">z_1z_2=r_1r_2\operatorname{cis}(\theta_1+\theta_2)</script></span> (multiply moduli, add angles)</p>
+<p><span class="math"><script type="math/tex">{z_1 \over z_2} = {r_1 \over r_2} \operatorname{cis}(\theta_1-\theta_2)</script></span> (divide moduli, subtract angles)</p>
+<h2>de Moivres’ Theorum</h2>
+<p><span class="math"><script type="math/tex">(r\operatorname{cis}\theta)^n=r^n\operatorname{cis}(n\theta)</script></span></p>
+
+ </body>
+</html>
--- /dev/null
+# Complex & Imaginary Numbers
+
+## Imaginary numbers
+
+$i^2 = -1$
+
+$\therefore i = \sqrt {-1}$
+
+### Simplifying negative surds
+
+$\sqrt{-2} = \sqrt{-1 \times 2}$
+
+ $= \sqrt{2}i$
+
+
+## Complex numbers
+
+$\mathbb{C} = \{a+bi : a, b \in \mathbb{R} \}$
+
+General form: $z=a+bi$
+- $\operatorname{Re}(z) = a$
+- $\operatorname{Im}(z) = b$
+
+### Addition
+
+If $z_1 = a+bi$ and $z_2=c+di$, then
+
+ $z_1+z_2 = (a+c)+(b+d)i$
+
+### Subtraction
+
+If $z_1=a+bi$ and $z_2=c+di$, then
+
+ $z_1−z_2=(a−c)+(b−d)i$
+
+### Multiplication by a real constant
+
+If $z=a+bi$ and $k \in \mathbb{R}$, then
+
+ $kz=ka+kbi$
+
+### Powers of $i$
+$i^0=1$
+$i^1=i$
+$i^2=-1$
+$i^3=-i$
+$i^4=1$
+$\dots$
+
+Therefore..
+- $i^{4n} = 1$
+- $i^{4n+1} = i$
+- $i^{4n+2} = -1$
+- $i^{4n+3} = -i$
+
+### Multiplying complex expressions
+
+If $z_1 = a+bi$ and $z_2=c+di$, then
+ $z_1 \times z_2 = (ac-bd)+(ad+bc)i$
+
+### Conjugates
+
+If $z=a+bi$, conjugate of $z$ is $\overline{z} = a-bi$ (flipped operator)
+
+Also, $z \overline{z} = (a+bi)(a-bi) = a^2+b^2$
+
+- Multiplication and addition are associative
+
+### Modulus
+
+Distance from origin.
+$|{z}|=\sqrt{a^2+b^2}$
+
+$\therefore z \overline{z} = |z|^2$
+
+### Multiplicative inverse
+
+$z^{-1} = {1 \over z} = {{a-bi} \over {a^2+B^2}} = {\overline{z} \over {|z|^2}}$
+
+### Dividing complex numbers
+
+${{z_1} \over {z_2}} = {{z_1\ {z_2}^{-1}}} = {{z_1 \overline{z_2}} \over {{|z_2|}^2}}$
+
+(using multiplicative inverse)
+
+In practice, rationalise denominator:
+${z_1 \over z_2} = {{(a+bi)(c-di)} \over {c^2+d^2}}$
+
+## Argand planes
+
+- Geometric representation of $\mathbb{C}$
+- Horizontal $= \operatorname{Re}(z)$; vertical $= \operatorname{Im}(z)$
+- Multiplication by $i$ results in an anticlockwise rotation of $\pi \over 2$
+
+## Solving complex quadratics
+
+To solve $z^2+a^2=0$ (sum of two squares):
+
+$z^2+a^2=z^2-(ai)^2$
+ $=(z+ai)(z-ai)$
+
+## Polar form
+
+General form:
+$z=r \operatorname{cis} \theta$
+$= r\operatorname{cos}\theta+r\operatorname{sin}\theta i$
+
+where
+- $z=a+bi$
+- $r$ is the distance from origin, given by Pythagoras ($r=\sqrt{x^2+y^2}$)
+- $\theta$ is the argument of $z$, CCW from origin
+
+Note each complex number has multiple polar representations:
+$z=r \operatorname{cis} \theta = r \operatorname{cis} (\theta+2 n\pi$) where $n$ is integer number of revolutions
+
+### Multiplication and division in polar form
+
+$z_1z_2=r_1r_2\operatorname{cis}(\theta_1+\theta_2)$ (multiply moduli, add angles)
+
+${z_1 \over z_2} = {r_1 \over r_2} \operatorname{cis}(\theta_1-\theta_2)$ (divide moduli, subtract angles)
+
+## de Moivres' Theorum
+
+$(r\operatorname{cis}\theta)^n=r^n\operatorname{cis}(n\theta)$
--- /dev/null
+<h1 id="complex-imaginary-numbers">Complex & Imaginary Numbers</h1>
+<h2 id="imaginary-numbers">Imaginary numbers</h2>
+<p><span class="math inline"><em>i</em><sup>2</sup> = −1</span></p>
+<p><span class="math inline">$\therefore i = \sqrt {-1}$</span></p>
+<h3 id="simplifying-negative-surds">Simplifying negative surds</h3>
+<p><span class="math inline">$\sqrt{-2} = \sqrt{-1 \times 2}$</span></p>
+<p> <span class="math inline">$= \sqrt{2}i$</span></p>
+<h2 id="complex-numbers">Complex numbers</h2>
+<p><span class="math inline">ℂ = {<em>a</em> + <em>b</em><em>i</em> : <em>a</em>, <em>b</em> ∈ ℝ}</span></p>
+<p>General form: <span class="math inline"><em>z</em> = <em>a</em> + <em>b</em><em>i</em></span> - <span class="math inline">Re(<em>z</em>)=<em>a</em></span> - <span class="math inline">Im(<em>z</em>)=<em>b</em></span></p>
+<h3 id="addition">Addition</h3>
+<p>If <span class="math inline"><em>z</em><sub>1</sub> = <em>a</em> + <em>b</em><em>i</em></span> and <span class="math inline"><em>z</em><sub>2</sub> = <em>c</em> + <em>d</em><em>i</em></span>, then</p>
+<p> <span class="math inline"><em>z</em><sub>1</sub> + <em>z</em><sub>2</sub> = (<em>a</em> + <em>c</em>)+(<em>b</em> + <em>d</em>)<em>i</em></span></p>
+<h3 id="subtraction">Subtraction</h3>
+<p>If <span class="math inline"><em>z</em><sub>1</sub> = <em>a</em> + <em>b</em><em>i</em></span> and <span class="math inline"><em>z</em><sub>2</sub> = <em>c</em> + <em>d</em><em>i</em></span>, then</p>
+<p> <span class="math inline"><em>z</em><sub>1</sub> − <em>z</em><sub>2</sub> = (<em>a</em> − <em>c</em>)+(<em>b</em> − <em>d</em>)<em>i</em></span></p>
+<h3 id="multiplication-by-a-real-constant">Multiplication by a real constant</h3>
+<p>If <span class="math inline"><em>z</em> = <em>a</em> + <em>b</em><em>i</em></span> and <span class="math inline"><em>k</em> ∈ ℝ</span>, then</p>
+<p> <span class="math inline"><em>k</em><em>z</em> = <em>k</em><em>a</em> + <em>k</em><em>b</em><em>i</em></span></p>
+<h3 id="powers-of-i">Powers of <span class="math inline"><em>i</em></span></h3>
+<p><span class="math inline"><em>i</em><sup>0</sup> = 1</span> <span class="math inline"><em>i</em><sup>1</sup> = <em>i</em></span> <span class="math inline"><em>i</em><sup>2</sup> = −1</span> <span class="math inline"><em>i</em><sup>3</sup> = −<em>i</em></span> <span class="math inline"><em>i</em><sup>4</sup> = 1</span> <span class="math inline">…</span></p>
+<p>Therefore.. - <span class="math inline"><em>i</em><sup>4<em>n</em></sup> = 1</span> - <span class="math inline"><em>i</em><sup>4<em>n</em> + 1</sup> = <em>i</em></span> - <span class="math inline"><em>i</em><sup>4<em>n</em> + 2</sup> = −1</span> - <span class="math inline"><em>i</em><sup>4<em>n</em> + 3</sup> = −<em>i</em></span></p>
+<h3 id="multiplying-complex-expressions">Multiplying complex expressions</h3>
+<p>If <span class="math inline"><em>z</em><sub>1</sub> = <em>a</em> + <em>b</em><em>i</em></span> and <span class="math inline"><em>z</em><sub>2</sub> = <em>c</em> + <em>d</em><em>i</em></span>, then <span class="math inline"><em>z</em><sub>1</sub> × <em>z</em><sub>2</sub> = (<em>a</em><em>c</em> − <em>b</em><em>d</em>)+(<em>a</em><em>d</em> + <em>b</em><em>c</em>)<em>i</em></span></p>
+<h3 id="conjugates">Conjugates</h3>
+<p>If <span class="math inline"><em>z</em> = <em>a</em> + <em>b</em><em>i</em></span>, conjugate of <span class="math inline"><em>z</em></span> is <span class="math inline">$\overline{z} = a-bi$</span> (flipped operator)</p>
+<p>Also, <span class="math inline">$z \overline{z} = (a+bi)(a-bi) = a^2+b^2$</span></p>
+<ul>
+<li>Multiplication and addition are associative</li>
+</ul>
+<h3 id="modulus">Modulus</h3>
+<p>Distance from origin. <span class="math inline">$|{z}|=\sqrt{a^2+b^2}$</span></p>
+<p><span class="math inline">$\therefore z \overline{z} = |z|^2$</span></p>
+<h3 id="multiplicative-inverse">Multiplicative inverse</h3>
+<p><span class="math inline">$z^{-1} = {1 \over z} = {{a-bi} \over {a^2+B^2}} = {\overline{z} \over {|z|^2}}$</span></p>
+<h3 id="dividing-complex-numbers">Dividing complex numbers</h3>
+<p><span class="math inline">${{z_1} \over {z_2}} = {{z_1\ {z_2}^{-1}}} = {{z_1 \overline{z_2}} \over {{|z_2|}^2}}$</span></p>
+<p>(using multiplicative inverse)</p>
+<p>In practice, rationalise denominator: <span class="math inline">${z_1 \over z_2} = {{(a+bi)(c-di)} \over {c^2+d^2}}$</span></p>
+<h2 id="argand-planes">Argand planes</h2>
+<ul>
+<li>Geometric representation of <span class="math inline">ℂ</span></li>
+<li>Horizontal <span class="math inline">=Re(<em>z</em>)</span>; vertical <span class="math inline">=Im(<em>z</em>)</span></li>
+<li>Multiplication by <span class="math inline"><em>i</em></span> results in an anticlockwise rotation of <span class="math inline">$\pi \over 2$</span></li>
+</ul>
+<h2 id="solving-complex-quadratics">Solving complex quadratics</h2>
+<p>To solve <span class="math inline"><em>z</em><sup>2</sup> + <em>a</em><sup>2</sup> = 0</span> (sum of two squares):</p>
+<p><span class="math inline"><em>z</em><sup>2</sup> + <em>a</em><sup>2</sup> = <em>z</em><sup>2</sup> − (<em>a</em><em>i</em>)<sup>2</sup></span> <span class="math inline">=(<em>z</em> + <em>a</em><em>i</em>)(<em>z</em> − <em>a</em><em>i</em>)</span></p>
+<h2 id="polar-form">Polar form</h2>
+<p>General form: <span class="math inline"><em>z</em> = <em>r</em>cis<em>θ</em></span> <span class="math inline">=<em>r</em>cos<em>θ</em> + <em>r</em>sin<em>θ</em><em>i</em></span></p>
+<p>where - <span class="math inline"><em>z</em> = <em>a</em> + <em>b</em><em>i</em></span> - <span class="math inline"><em>r</em></span> is the distance from origin, given by Pythagoras (<span class="math inline">$r=\sqrt{x^2+y^2}$</span>) - <span class="math inline"><em>θ</em></span> is the argument of <span class="math inline"><em>z</em></span>, CCW from origin</p>
+<p>Note each complex number has multiple polar representations: <span class="math inline"><em>z</em> = <em>r</em>cis<em>θ</em> = <em>r</em>cis(<em>θ</em> + 2<em>n</em><em>π</em></span>) where <span class="math inline"><em>n</em></span> is integer number of revolutions</p>
+<h3 id="multiplication-and-division-in-polar-form">Multiplication and division in polar form</h3>
+<p><span class="math inline"><em>z</em><sub>1</sub><em>z</em><sub>2</sub> = <em>r</em><sub>1</sub><em>r</em><sub>2</sub>cis(<em>θ</em><sub>1</sub> + <em>θ</em><sub>2</sub>)</span> (multiply moduli, add angles)</p>
+<p><span class="math inline">${z_1 \over z_2} = {r_1 \over r_2} \operatorname{cis}(\theta_1-\theta_2)$</span> (divide moduli, subtract angles)</p>
+<h2 id="de-moivres-theorum">de Moivres' Theorum</h2>
+<p><span class="math inline">(<em>r</em>cis<em>θ</em>)<sup><em>n</em></sup> = <em>r</em><sup><em>n</em></sup>cis(<em>n</em><em>θ</em>)</span></p>
--- /dev/null
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--- /dev/null
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+ <body class="markdown-preview-plus-view">
+ <h1>Sequences and Series</h1>
+<p><span class="math"><script type="math/tex">\{\ a,\ b,\ c,\ d\ ...\ \}</script></span></p>
+<ul>
+<li><span class="math"><script type="math/tex">n^{th}</script></span> iteration: <span class="math"><script type="math/tex">t_n</script></span></li>
+<li><span class="math"><script type="math/tex">n</script></span> can start at <span class="math"><script type="math/tex">0</script></span> or <span class="math"><script type="math/tex">1</script></span></li>
+</ul>
+<h2>Defining sequences</h2>
+<ul>
+<li>
+<p>sequence rule in terms of <span class="math"><script type="math/tex">n</script></span> e.g. <span class="math"><script type="math/tex">t_n = 2n</script></span></p>
+</li>
+<li>
+<p>recurrence relation - value of term is derived from previous term (recursion)<br>
+e.g. <span class="math"><script type="math/tex">\ t_n=t_{n-1}+7,\ t_1=4</script></span></p>
+<ul>
+<li><span class="math"><script type="math/tex">t_1=4</script></span></li>
+<li><span class="math"><script type="math/tex">t_2=4+7=11</script></span></li>
+<li><span class="math"><script type="math/tex">t_3=11+7=17</script></span></li>
+<li><span class="math"><script type="math/tex">\dots</script></span></li>
+</ul>
+</li>
+</ul>
+<h2>Arithmetic sequences</h2>
+<p><span class="math"><script type="math/tex">t_n = a+(n-1)d</script></span></p>
+<p>where<br>
+<span class="math"><script type="math/tex">a=</script></span> first term<br>
+<span class="math"><script type="math/tex">d=</script></span> common difference</p>
+<h3>Arithmetic mean</h3>
+<p><span class="math"><script type="math/tex">m_a = \frac{a+b}{2}</script></span></p>
+<h3>Arithmetic series</h3>
+<p><span class="math"><script type="math/tex">\sum</script></span> of all terms in an arithmetic sequence.</p>
+<p>e.g. A sequence is defined by <span class="math"><script type="math/tex">t_n = a+(n-1)d</script></span></p>
+<p><span class="math"><script type="math/tex">s_n = {\frac{n}{2}}(2a+(n-1)d)</script></span> <–sum of <span class="math"><script type="math/tex">n</script></span> terms, including <span class="math"><script type="math/tex">t_n</script></span> (works only with <span class="math"><script type="math/tex">t_1</script></span>)<br>
+where</p>
+<ul>
+<li><span class="math"><script type="math/tex">a=t_1</script></span> (first term)</li>
+<li><span class="math"><script type="math/tex">l=t_n</script></span> (last term)</li>
+</ul>
+<h2>Geometric sequences</h2>
+<p><span class="math"><script type="math/tex">t_n=ar^{n-1}</script></span><br>
+where<br>
+<span class="math"><script type="math/tex">a =</script></span> first term<br>
+<span class="math"><script type="math/tex">r =</script></span> common ratio of successive terms (<span class="math"><script type="math/tex">r=\frac{t_k}{t_{k-1}}</script></span>)</p>
+<h3>Geometric mean</h3>
+<p><span class="math"><script type="math/tex">m_g</script></span> of <span class="math"><script type="math/tex">a</script></span> and <span class="math"><script type="math/tex">b</script></span> is <span class="math"><script type="math/tex">\sqrt{ab}</script></span></p>
+<p>If <span class="math"><script type="math/tex">a, c, b</script></span> are positive and consecutive terms in a geometric sequence, then:<br>
+<span class="math"><script type="math/tex">{\frac{c}{a}}={\frac{b}{c}}\therefore c=\sqrt{ab}</script></span>,</p>
+<h3>Geometric series</h3>
+<p><span class="math"><script type="math/tex">\sum</script></span> of all terms in a geometric sequence.</p>
+<p>e.g. <span class="math"><script type="math/tex">s_n = a + ar + ar^2 + ar^2 + \dots + ar^{n-1}</script></span></p>
+<p><span class="math"><script type="math/tex">rs_n = ar+ar^2+ar^3+ar^4+\dots+ar^{n}</script></span></p>
+<p><span class="math"><script type="math/tex">rs_n - s_n = -a + ar^n</script></span></p>
+<hr>
+<p><span class="math"><script type="math/tex">s_n = {\frac{a(r^n-1)}{r-1}}</script></span><br>
+or<br>
+<span class="math"><script type="math/tex">s_n = {\frac{a(r^n-1)}{r-1}}</script></span></p>
+<h2>Infinite series</h2>
+<p>If <span class="math"><script type="math/tex">-1 < r -1</script></span>, the infinite geometric series <span class="math"><script type="math/tex">a+ar+ar^2 \dots</script></span> is convergent.</p>
+<p>Sum to infinity is given by</p>
+<p><span class="math"><script type="math/tex">s_\infty={\frac{a}{1-r}}</script></span></p>
+<p>Tennis ball question - remember down <strong>and</strong> up strokes. Multiply down strokes by 2, subtract 1.</p>
+
+ </body>
+</html>
--- /dev/null
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+entering extended mode
+ restricted \write18 enabled.
+ %&-line parsing enabled.
+**sequences.md
+(./sequences.md
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+l.1 #
+ Sequences and Series
+?
+! Interruption.
+<to be read again>
+
+l.1 #
+ Sequences and Series
+?
+
+! LaTeX Error: Missing \begin{document}.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type H <return> for immediate help.
+ ...
+
+l.1 # S
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+?
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+
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+l.6 -
+ $n^{th}$ iteration: $t_n$
+?
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+ # Defining sequences
+?
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+l.9 ##
+ Defining sequences
+?
+
+! LaTeX Error: Missing \begin{document}.
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+See the LaTeX manual or LaTeX Companion for explanation.
+Type H <return> for immediate help.
+ ...
+
+l.9 ## D
+ efining sequences
+?
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+Type H <return> for immediate help.
+ ...
+
+l.12 -
+ recurrence relation - value of term is derived from previous term (re...
+
+?
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\ No newline at end of file
--- /dev/null
+# Sequences and Series
+
+$\{\ a,\ b,\ c,\ d\ ...\ \}$
+
+
+- $n^{th}$ iteration: $t_n$
+- $n$ can start at $0$ or $1$
+
+## Defining sequences
+- sequence rule in terms of $n$ e.g. $t_n = 2n$
+
+- recurrence relation - value of term is derived from previous term (recursion)
+e.g. $\ t_n=t_{n-1}+7,\ t_1=4$
+ - $t_1=4$
+ - $t_2=4+7=11$
+ - $t_3=11+7=17$
+ - $\dots$
+
+## Arithmetic sequences
+
+$t_n = a+(n-1)d$
+
+where
+$a=$ first term
+$d=$ common difference
+
+### Arithmetic mean
+
+$m_a = \frac{a+b}{2}$
+
+### Arithmetic series
+
+$\sum$ of all terms in an arithmetic sequence.
+
+e.g. A sequence is defined by $t_n = a+(n-1)d$
+
+$s_n = {\frac{n}{2}}(2a+(n-1)d)$ <--sum of $n$ terms, including $t_n$ (works only with $t_1$)
+where
+- $a=t_1$ (first term)
+- $l=t_n$ (last term)
+
+## Geometric sequences
+
+$t_n=ar^{n-1}$
+where
+$a =$ first term
+$r =$ common ratio of successive terms ($r=\frac{t_k}{t_{k-1}}$)
+
+### Geometric mean
+
+$m_g$ of $a$ and $b$ is $\sqrt{ab}$
+
+If $a, c, b$ are positive and consecutive terms in a geometric sequence, then:
+${\frac{c}{a}}={\frac{b}{c}}\therefore c=\sqrt{ab}$,
+
+### Geometric series
+
+$\sum$ of all terms in a geometric sequence.
+
+e.g. $s_n = a + ar + ar^2 + ar^2 + \dots + ar^{n-1}$
+
+$rs_n = ar+ar^2+ar^3+ar^4+\dots+ar^{n}$
+
+$rs_n - s_n = -a + ar^n$
+
+---
+$s_n = {\frac{a(r^n-1)}{r-1}}$
+or
+$s_n = {\frac{a(r^n-1)}{r-1}}$
+
+## Infinite series
+
+If $-1 < r -1$, the infinite geometric series $a+ar+ar^2 \dots$ is convergent.
+
+Sum to infinity is given by
+
+$s_\infty={\frac{a}{1-r}}$
+
+Tennis ball question - remember down **and** up strokes. Multiply down strokes by 2, subtract 1.
--- /dev/null
+# Trigonometry
+
+## Unit circle
+
+For each angle, there is a corresponding point on the unit circle ($r=1$).
+
+- $\cos \theta$ is the $x$-coordinate of this point
+- $\sin \theta$ is the $y$-coordinate
+
+
+
+## Trigonometric ratios
+
+
+## Cosine rule
