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fix formatting for complex/imaginary notes
author
Andrew Lorimer
<andrew@lorimer.id.au>
Tue, 12 Feb 2019 21:45:12 +0000
(08:45 +1100)
committer
Andrew Lorimer
<andrew@lorimer.id.au>
Tue, 12 Feb 2019 21:45:12 +0000
(08:45 +1100)
spec/complex.md
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diff --git
a/spec/complex.md
b/spec/complex.md
index 58091702e2e7c0c81da39bdc7a968cd05be29f84..374b793d048fa32c484f2e1c9a314fcd73a14d4d 100755
(executable)
--- a/
spec/complex.md
+++ b/
spec/complex.md
@@
-8,9
+8,8
@@
$\therefore i = \sqrt {-1}$
### Simplifying negative surds
### Simplifying negative surds
-$\sqrt{-2} = \sqrt{-1 \times 2}$
-
- $= \sqrt{2}i$
+$\sqrt{-2} = \sqrt{-1 \times 2}$
+$= \sqrt{2}i$
## Complex numbers
## Complex numbers
@@
-23,21
+22,16
@@
General form: $z=a+bi$
### Addition
### Addition
-If $z_1 = a+bi$ and $z_2=c+di$, then
-
- $z_1+z_2 = (a+c)+(b+d)i$
+If $z_1 = a+bi$ and $z_2=c+di$, then
+$z_1+z_2 = (a+c)+(b+d)i$
### Subtraction
### Subtraction
-If $z_1=a+bi$ and $z_2=c+di$, then
-
- $z_1−z_2=(a−c)+(b−d)i$
+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
### Multiplication by a real constant
-If $z=a+bi$ and $k \in \mathbb{R}$, then
-
- $kz=ka+kbi$
+If $z=a+bi$ and $k \in \mathbb{R}$, then $kz=ka+kbi$
### Powers of $i$
$i^0=1$
### Powers of $i$
$i^0=1$
@@
-48,17
+42,18
@@
$i^4=1$
$\dots$
Therefore..
$\dots$
Therefore..
+
- $i^{4n} = 1$
- $i^{4n+1} = i$
- $i^{4n+2} = -1$
- $i^{4n+3} = -i$
- $i^{4n} = 1$
- $i^{4n+1} = i$
- $i^{4n+2} = -1$
- $i^{4n+3} = -i$
-Divide by 4 and take remainder
+Divide by 4 and take remainder
.
### Multiplying complex expressions
### Multiplying complex expressions
-If $z_1 = a+bi$ and $z_2=c+di$, then
-
$z_1 \times z_2 = (ac-bd)+(ad+bc)i$
+If $z_1 = a+bi$ and $z_2=c+di$, then
+$z_1 \times z_2 = (ac-bd)+(ad+bc)i$
### Conjugates
### Conjugates
@@
-98,8
+93,7
@@
${z_1 \over z_2} = {{(a+bi)(c-di)} \over {c^2+d^2}}$
To solve $z^2+a^2=0$ (sum of two squares):
To solve $z^2+a^2=0$ (sum of two squares):
-$z^2+a^2=z^2-(ai)^2$
- $=(z+ai)(z-ai)$
+$z^2+a^2=z^2-(ai)^2=(z+ai)(z-ai)$
## Polar form
## Polar form