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Polynomials & Complex Numbers

Chapter 2 · Complex Numbers

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Imaginary unit
i=−1,i2=−1i = \sqrt{-1},\quad i^2 = -1
Complex number
z=a+biz = a + bi
real part aa, imaginary part bb.
Conjugate
z‾=a−bi,zz‾=a2+b2\overline{z} = a - bi,\quad z\overline{z} = a^2 + b^2
Polar form
z=r(cos⁡θ+isin⁡θ)=r cis θz = r(\cos\theta + i\sin\theta) = r\,\text{cis}\,\theta
Modulus / argument
r=a2+b2,θ=arg⁡(z)r = \sqrt{a^2+b^2},\quad \theta = \arg(z)
De Moivre
[r cis θ]n=rn cis(nθ)[r\,\text{cis}\,\theta]^n = r^n\,\text{cis}(n\theta)
Forms
z=a+bi=r(cos⁡θ+isin⁡θ)=reiθz = a+bi = r(\cos\theta + i\sin\theta) = re^{i\theta}
r = |z| = √(a²+b²), θ = arg z.
Modulus / conjugate
∣z∣2=zzˉ,    a+bi‾=a−bi|z|^2 = z\bar z,\;\; \overline{a+bi}=a-bi
De Moivre
zn=rn(cos⁡nθ+isin⁡nθ)z^n = r^n(\cos n\theta + i\sin n\theta)
Common trap
arg is measured from +x axis; watch the quadrant (add/subtract π).