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Modulus–argument (polar) form

Writing z = r(cos θ + i sin θ) with r = |z| and θ = arg z (the principal value in (−π, π]), converting between Cartesian a + bi and polar form both ways, and combining numbers in polar form: to MULTIPLY you multiply the moduli and ADD the arguments, to DIVIDE you divide the moduli and SUBTRACT the arguments — then adjust the resulting argument back into the principal range.

Pure · Complex numbers

Modulus–argument (polar) form

Writing z = r(cos θ + i sin θ) with r = |z| and θ = arg z (the principal value in (−π, π]), converting between Cartesian a + bi and polar form both ways, and combining numbers in polar form: to MULTIPLY you multiply the moduli and ADD the arguments, to DIVIDE you divide the moduli and SUBTRACT the arguments — then adjust the resulting argument back into the principal range.

Why it works

On the Argand diagram a complex number zz is fixed by two Cartesian numbers, the real part aa and the imaginary part bb. But the same point is fixed just as well by how far it is from the origin and in what direction — its modulus r=∣z∣r = |z| and its argument θ=arg⁡z\theta = \arg z. Reading the right-angled triangle with hypotenuse rr gives a=rcos⁡θ,b=rsin⁡θ,a = r\cos\theta, \qquad b = r\sin\theta, so z=a+bi=rcos⁡θ+i rsin⁡θ=r(cos⁡θ+isin⁡θ).z = a + bi = r\cos\theta + i\,r\sin\theta = r(\cos\theta + i\sin\theta). This is the modulus–argument form (or polar form). Here r=∣z∣=a2+b2r = |z| = \sqrt{a^2+b^2} is a length, so r≥0r \ge 0, and θ=arg⁡z\theta = \arg z is taken as the principal argument in −π<θ≤π-\pi < \theta \le \pi — you still have to get the quadrant right, exactly as on the Argand diagram. The two pieces play different roles and must not be swapped: rr is the number in front of the bracket, θ\theta is the angle inside the cosine and sine.

Cartesian → polar. Compute r=a2+b2r = \sqrt{a^2+b^2}, then find θ\theta from a sketch: the reference angle is α=arctan⁡∣b∣∣a∣\alpha = \arctan\frac{|b|}{|a|}, and the quadrant of (a,b)(a,b) decides the sign and whether you measure from 00 or from ±π\pm\pi. Then write z=r(cos⁡θ+isin⁡θ)z = r(\cos\theta + i\sin\theta).

Polar → Cartesian. Go the other way: evaluate a=rcos⁡θa = r\cos\theta and b=rsin⁡θb = r\sin\theta and write a+bia + bi. With exact angles this uses the exact trig values, e.g. cos⁡2π3=−12\cos\frac{2\pi}{3} = -\tfrac12.

Multiplication — moduli multiply, arguments add. Take z1=r1(cos⁡θ1+isin⁡θ1)z_1 = r_1(\cos\theta_1 + i\sin\theta_1) and z2=r2(cos⁡θ2+isin⁡θ2)z_2 = r_2(\cos\theta_2 + i\sin\theta_2). Multiplying the brackets and using the addition formulae for cosine and sine, z1z2=r1r2[(cos⁡θ1cos⁡θ2−sin⁡θ1sin⁡θ2)+i(sin⁡θ1cos⁡θ2+cos⁡θ1sin⁡θ2)],z_1 z_2 = r_1 r_2\big[(\cos\theta_1\cos\theta_2 - \sin\theta_1\sin\theta_2) + i(\sin\theta_1\cos\theta_2 + \cos\theta_1\sin\theta_2)\big],  z1z2=r1r2(cos⁡(θ1+θ2)+isin⁡(θ1+θ2)). \boxed{\,z_1 z_2 = r_1 r_2\big(\cos(\theta_1+\theta_2) + i\sin(\theta_1+\theta_2)\big).\,} So ∣z1z2∣=r1r2|z_1 z_2| = r_1 r_2 (the moduli multiply) and arg⁡(z1z2)=θ1+θ2\arg(z_1 z_2) = \theta_1 + \theta_2 (the arguments add). The arguments add because multiplying by z2z_2 rotates by θ2\theta_2 and scales by r2r_2 — a geometric fact you lose completely if you multiply the angles instead.

Division — moduli divide, arguments subtract. The same algebra in reverse gives z1z2=r1r2(cos⁡(θ1−θ2)+isin⁡(θ1−θ2)),\frac{z_1}{z_2} = \frac{r_1}{r_2}\big(\cos(\theta_1-\theta_2) + i\sin(\theta_1-\theta_2)\big), so ∣z1z2∣=r1r2\left|\dfrac{z_1}{z_2}\right| = \dfrac{r_1}{r_2} and arg⁡z1z2=θ1−θ2\arg\dfrac{z_1}{z_2} = \theta_1 - \theta_2.

Bringing the argument back into range. Adding or subtracting two principal arguments can land outside (−π,π](-\pi, \pi]. When it does, add or subtract 2π2\pi once to slide it back in — this changes the label of the angle, not the point. For instance a combined argument of 13π12\frac{13\pi}{12} is bigger than π\pi, so it is not principal; subtract 2π2\pi to get 13π12−2π=−11π12\frac{13\pi}{12} - 2\pi = -\frac{11\pi}{12}, which is in (−π,π](-\pi, \pi] and names the same direction. Always finish a product or quotient by checking the argument is in range.