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Pure · Complex numbers

Chapter 1 · 3

The idea

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.

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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

Distance and direction instead of across and up

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).

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