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50 <h1>Complex & Imaginary Numbers</h1>
51<h2>Imaginary numbers</h2>
52<p><span class="math"><script type="math/tex">i^2 = -1</script></span></p>
53<p><span class="math"><script type="math/tex">\therefore i = \sqrt {-1}</script></span></p>
54<h3>Simplifying negative surds</h3>
55<p><span class="math"><script type="math/tex">\sqrt{-2} = \sqrt{-1 \times 2}</script></span></p>
56<p> <span class="math"><script type="math/tex">= \sqrt{2}i</script></span></p>
57<h2>Complex numbers</h2>
58<p><span class="math"><script type="math/tex">\mathbb{C} = \{a+bi : a, b \in \mathbb{R} \}</script></span></p>
59<p>General form: <span class="math"><script type="math/tex">z=a+bi</script></span></p>
60<ul>
61<li><span class="math"><script type="math/tex">\operatorname{Re}(z) = a</script></span></li>
62<li><span class="math"><script type="math/tex">\operatorname{Im}(z) = b</script></span></li>
63</ul>
64<h3>Addition</h3>
65<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>
66<p> <span class="math"><script type="math/tex">z_1+z_2 = (a+c)+(b+d)i</script></span></p>
67<h3>Subtraction</h3>
68<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>
69<p> <span class="math"><script type="math/tex">z_1−z_2=(a−c)+(b−d)i</script></span></p>
70<h3>Multiplication by a real constant</h3>
71<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>
72<p> <span class="math"><script type="math/tex">kz=ka+kbi</script></span></p>
73<h3>Powers of <span class="math"><script type="math/tex">i</script></span></h3>
74<p><span class="math"><script type="math/tex">i^0=1</script></span><br>
75<span class="math"><script type="math/tex">i^1=i</script></span><br>
76<span class="math"><script type="math/tex">i^2=-1</script></span><br>
77<span class="math"><script type="math/tex">i^3=-i</script></span><br>
78<span class="math"><script type="math/tex">i^4=1</script></span><br>
79<span class="math"><script type="math/tex">\dots</script></span></p>
80<p>Therefore…</p>
81<ul>
82<li><span class="math"><script type="math/tex">i^{4n} = 1</script></span></li>
83<li><span class="math"><script type="math/tex">i^{4n+1} = i</script></span></li>
84<li><span class="math"><script type="math/tex">i^{4n+2} = -1</script></span></li>
85<li><span class="math"><script type="math/tex">i^{4n+3} = -i</script></span></li>
86</ul>
87<h3>Multiplying complex expressions</h3>
88<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>
89 <span class="math"><script type="math/tex">z_1 \times z_2 = (ac-bd)+(ad+bc)i</script></span></p>
90<h3>Conjugates</h3>
91<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>
92<p>Also, <span class="math"><script type="math/tex">z \overline{z} = (a+bi)(a-bi) = a^2+b^2</script></span></p>
93<ul>
94<li>Multiplication and addition are associative</li>
95</ul>
96<h3>Modulus</h3>
97<p>Distance from origin.<br>
98<span class="math"><script type="math/tex">|{z}|=\sqrt{a^2+b^2}</script></span></p>
99<p><span class="math"><script type="math/tex">\therefore z \overline{z} = |z|^2</script></span></p>
100<h3>Multiplicative inverse</h3>
101<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>
102<h3>Dividing complex numbers</h3>
103<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>
104<p>(using multiplicative inverse)</p>
105<p>In practice, rationalise denominator:<br>
106<span class="math"><script type="math/tex">{z_1 \over z_2} = {{(a+bi)(c-di)} \over {c^2+d^2}}</script></span></p>
107<h2>Argand planes</h2>
108<ul>
109<li>Geometric representation of <span class="math"><script type="math/tex">\mathbb{C}</script></span></li>
110<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>
111<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>
112</ul>
113<h2>Solving complex quadratics</h2>
114<p>To solve <span class="math"><script type="math/tex">z^2+a^2=0</script></span> (sum of two squares):</p>
115<p><span class="math"><script type="math/tex">z^2+a^2=z^2-(ai)^2</script></span><br>
116 <span class="math"><script type="math/tex">=(z+ai)(z-ai)</script></span></p>
117<h2>Polar form</h2>
118<p>General form:<br>
119<span class="math"><script type="math/tex">z=r \operatorname{cis} \theta</script></span><br>
120<span class="math"><script type="math/tex">= r\operatorname{cos}\theta+r\operatorname{sin}\theta i</script></span></p>
121<p>where</p>
122<ul>
123<li><span class="math"><script type="math/tex">z=a+bi</script></span></li>
124<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>
125<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>
126</ul>
127<p>Note each complex number has multiple polar representations:<br>
128<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>
129<h3>Multiplication and division in polar form</h3>
130<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>
131<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>
132<h2>de Moivres’ Theorum</h2>
133<p><span class="math"><script type="math/tex">(r\operatorname{cis}\theta)^n=r^n\operatorname{cis}(n\theta)</script></span></p>
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