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complex numbers

Cartesian, polar & Euler forms, De Moivre, nth roots, polynomials & the complex exponential.

cartesian form · polar & euler form · products & quotients · de moivre, roots & polynomials · complex exponential

Cartesian form

Definition. \(i^2=-1\); \(z=a+bi\) with \(\Re z=a\), \(\Im z=b\); two complex numbers are equal iff both real and imaginary parts agree. Arithmetic follows the ordinary rules of algebra with \(i^2\) replaced by \(-1\).

Conjugate & modulus. \(\bar z=a-bi\); conjugation respects arithmetic — \(\overline{z+w}=\bar z+\bar w\), \(\overline{zw}=\bar z\,\bar w\) (replace \(i\) by \(-i\) throughout any valid identity). \(|z|=\sqrt{a^2+b^2}\); \(z\bar z=|z|^2\); \(z^{-1}=\bar z/|z|^2\), and \(\dfrac zw=\dfrac{z\bar w}{|w|^2}\) — realise a denominator by multiplying through by its conjugate.

Real & imaginary parts. \(\Re z=\dfrac{z+\bar z}2\), \(\Im z=\dfrac{z-\bar z}{2i}\).

Complex plane. \(z=a+bi\) is the point \((a,b)\) (real & imaginary axes); \(|z|\) is its distance from the origin. Addition is vector addition (parallelogram rule); multiplication by \(w\) scales by \(|w|\) and rotates by \(\arg w\). Loci: \(|z-z_0|=r\) is the circle of radius \(r\) about \(z_0\); conditions on \(\Re z\), \(\Im z\), \(|z|\), \(\arg z\) cut lines, half-planes, disks and sectors.

Polar & Euler form

Argument. \(\arg z=\theta\), fixed with quadrant care; \(a=r\cos\theta\), \(b=r\sin\theta\). As an angle it is only defined up to adding multiples of \(2\pi\).

Forms. \(z=r(\cos\theta+i\sin\theta)=r\cis\theta=re^{i\theta}\).

Euler. \(e^{i\theta}=\cos\theta+i\sin\theta\); in particular \(e^{i\pi}+1=0\).

Products & quotients

Product. \(r_1\cis\theta_1\cdot r_2\cis\theta_2=r_1r_2\cis(\theta_1+\theta_2)\).

Quotient. \(\dfrac{r_1\cis\theta_1}{r_2\cis\theta_2}=\dfrac{r_1}{r_2}\cis(\theta_1-\theta_2)\).

Modulus & argument rules. \(|zw|=|z|\,|w|\), \(\arg(zw)=\arg z+\arg w\) — multiply the moduli, add the phases; powers become easy in polar form when Cartesian expansion would be painful.

Trig identities. Matching real and imaginary parts of \(e^{i\theta}e^{i\varphi}=e^{i(\theta+\varphi)}\) gives the angle-addition formulas; taking arguments in \((a+bi)(c+di)\) gives \(\arctan\dfrac ba+\arctan\dfrac dc=\arctan\dfrac{ad+bc}{ac-bd}\).

De Moivre, roots & polynomials

De Moivre. \((r\cis\theta)^n=r^n\cis(n\theta)\); expanding both sides generates multiple-angle identities.

nth roots. Solve \(z^n=w\) by matching polar forms: \(r^n=|w|\) and \(n\theta=\arg w+2\pi k\) — the \(2\pi k\) ambiguity of the angle is what yields \(n\) distinct roots. The \(n\)th roots of \(z=r\cis\theta\) are \(w_k=r^{1/n}\cis\!\left(\dfrac{\theta+2\pi k}{n}\right)\), \(k=0,1,\ldots,n-1\), equally spaced around the circle of radius \(r^{1/n}\).

Fundamental theorem of algebra. A degree-\(n\) polynomial factors completely over \(\C\): \(x^n+c_{n-1}x^{n-1}+\cdots+c_0=(x-z_1)(x-z_2)\cdots(x-z_n)\) — exactly \(n\) complex roots, counted with multiplicity.

Real coefficients. \(ax^2+bx+c=0\) has \(x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\); when \(b^2-4ac\lt0\) the roots are a complex-conjugate pair (\(\sqrt{-5}=i\sqrt5\)). In general the non-real roots of any real-coefficient polynomial come in conjugate pairs, because conjugation respects arithmetic.

Complex exponential

Definition. \(e^z=\sum_{n=0}^{\infty}\dfrac{z^n}{n!}\), convergent for every \(z\in\C\); the exponent law \(e^{z+w}=e^ze^w\) carries over. For \(z=x+iy\):

Euler from the series. Powers of \(i\) cycle \(1,i,-1,-i\); splitting \(e^{i\theta}=\sum i^n\theta^n/n!\) into real and imaginary parts gives \(\Bigl(1-\dfrac{\theta^2}{2!}+\dfrac{\theta^4}{4!}-\cdots\Bigr)+i\Bigl(\theta-\dfrac{\theta^3}{3!}+\dfrac{\theta^5}{5!}-\cdots\Bigr)=\cos\theta+i\sin\theta\) — Euler's formula, valid for all \(\theta\) since the series converges everywhere.