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1.5 Complex Numbers (HL)

Tags
Complex number
Modulus
Argument
Imaginary number
Conjugate
De Moivre's theorem
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The number ii, where i2=−1i^2=-1. (1.12)
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Cartesian form z=a+biz = a + bi ; the terms real part, imaginary part, conjugate, modulus and argument.
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The complex plane.
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Modulus–argument (polar) form: z=r(cos⁡θ+isin⁡θ)=rcisθz = r(\cosθ + i\sinθ) = rcisθ (1.13)
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Euler form: z=reiθ z = re^{iθ}
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Sums, products and quotients in Cartesian, polar or Euler forms and their geometric interpretation.
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Complex conjugate roots of quadratic and polynomial equations with real coefficients. (1.14)
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De Moivre’s theorem and its extension to rational exponents.
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Powers and roots of complex numbers.
Terminology
Definition
Imaginary number ii
This is defined as i=−1i=\sqrt{-1} with the property of i2=−1i^2=-1.
Complex number zz
Any number of the form z=a+biz=a+bi, where a,b∈R,z∈Ca, b ∈ ℝ, z∈ℂ. zz is purely imaginary if a≠0a ≠ 0.  We cal; Re(z)=aRe(z)=a as the real part, and Im(z)=biIm(z )=bi as the imaginary part. For mathematical operations, treat ii like a variable.  To achieve real denominator for the form a+bic+di\frac{a+bi}{c+di}, we need to perform rationalization. Multiply the conjugate of the denominator to both the numerator and the denominator.
a+bic+di(c−dic−di)=(a+bi)(c−di)c2+d2\frac{a+bi}{c+di}(\frac{c-di}{c-di})=\frac{(a+bi)(c-di)}{c^2+d^2}
Conjugates
If z=a+biz = a+bi, then z‾=z∗=a−bi\overline{z}=z^*=a-bi is the complex conjugate of zz.
Argand Diagram (HL)
The Argand diagram is composed of the real axis and the imaginary axis. The vector OP→=(xy)\overrightarrow {OP}=\binom xy represents x+yix+yi. In the complex plane, the conjugate of zz is a reflection of zz in the real axis.
Modulus (HL)
The modulus of z=a+biz=a+bi is ∣z∣=a2+b2|z|=\sqrt{a^2+b^2}, i.e. the magnitude of the vector (ab)\binom ab. 1. ∣z‾∣=∣z∣|\overline {z}|=|z| 2. ∣z∣2=zz‾|z|^2=z\overline z 3.∣z1z2∣=∣z1∣∣z2∣ |z_1z_2|=|z_1||z_2| 4. ∣z1z2∣=∣z1∣∣z2∣|\frac{z_1}{z_2}|=\frac{|z_1|}{|z_2|} 5. ∣z1z2...zn∣=∣z1∣∣z2∣...∣zn∣,∣zn∣=∣z∣n|z_1z_2...z_n|=|z_1||z_2|...|z_n|, |z^n| = |z|^n Given z1=OP1→z_1=\overrightarrow{OP_1} and z2=OP2→z_2=\overrightarrow{OP_2}, the distance between P1P_1 and P2P_2 is ∣z1−z2∣|z_1-z_2|.
Argument (HL)
The argument of zz, arg zz is the angle in the interval −π≤θ≤π-π ≤ \theta ≤ π which is measured anti-clockwise between the positive real axis and OP→\overrightarrow {OP}. Real numbers have argument of 0 or ππ. Purely imaginary numbers have argument of π2\frac π2 or −π2-\frac π2.
Polar form (HL)
On an Argand diagram, we can represent zz in terms of trigonometry:  z=rcos⁡θ+irsin⁡θ=r(cos⁡θ+isin⁡θ)=rcisθ=∣z∣cisθz = r\cos\theta+ir\sin\theta=r(\cos\theta+i\sin\theta) = rcis\theta=|z|cis\theta. For z=∣z∣cisθz=|z|cis\theta, ∣z∣ |z| is the modulus while is the argument.  1. cisθcisΨ=cis(θ+Ψ)cis\theta cis\Psi=cis(\theta+\Psi) 2. cisθcisΨ=cis(θ−Ψ)\frac{cis\theta}{cis\Psi}=cis(\theta-\Psi) 3. cis(θ+k2π)=cisθcis(\theta+k2\pi)=cis\theta 4. z‾=∣z∣cis(−θ)\overline z=|z|cis(-\theta) 5. If z=∣z∣cisθz=|z|cis\theta and w=∣w∣cisΨw=|w|cis\Psi, then zw=∣z∣∣w∣cis(θ+Ψ)zw=|z||w|cis(\theta+\Psi).
Euler form (HL)
z=rcos⁡θ+irsin⁡θ=r(cos⁡θ+isin⁡θ)=reiθz=r\cos\theta+ir\sin\theta=r(\cos\theta+i\sin\theta)=re^{i\theta}.
De Moivre's theorem (HL)
(∣z∣cisθ)n=∣z∣n(cisnθ)(|z| cis\theta )^n=|z|^n(cis n\theta) Utilize this to solve the equation zn=cz^n=c, expressing cc as a polar form.  The solutions to this equation will form a regular nn-polygon in the Argand diagram. Proof Preposition: (∣z∣cisθ)n=∣z∣n(cisnθ)(|z| cis\theta )^n=|z|^n(cis n\theta) for n∈Z+ n∈Z^+ Base case: For n=1n=1, (∣z∣cisθ)1=∣z∣cisθ(|z|cis\theta)^1=|z|cis\theta  Hence, n=1n=1 is true. Induction: Suppose n=kn=k is true, then (∣z∣cisθ)k=∣z∣kciskθ(|z|cis\theta)^k=|z|^kcisk\theta. Let n=k+1n=k+1, then (∣z∣cisθ)k+1(|z|cis\theta)^{k+1} =∣z∣kciskθ⋅∣z∣cisθ|z|^kcisk\theta \cdot |z|cis\theta =∣z∣k+1cis(kθ+θ)=∣z∣k+1cis(k+1)θ|z|^{k+1}cis(k\theta+\theta)=|z|^{k+1}cis(k+1)\theta Conclusion: n=1n=1 is true n=k+1n=k+1 is true whenever n=kn=k is true, hence the preposition is true for all n∈Z+ n∈Z^+.
As you delve more into mathematics, the Euler form is much more commonly used than the Polar form.
1.5 Complex Numbers
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1.5 Complex Numbers (HL)