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The complex plane

Complex numbers

Japanese school year: Math C

What you learn

In this section, you will learn to represent complex numbers as points on a plane, using polar form and De Moivre's theorem. This approach connects algebra with geometry, allowing rotations and scalings to be computed easily. Basic knowledge of complex arithmetic and trigonometry is essential prior to this topic.

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Key points

This formula expresses a complex number in polar form using distance and direction. Here, r=∣z∣r = |z| is the distance from the origin (modulus), and θ\theta is the angle measured from the positive real axis (argument).

z=r(cos⁡θ+isin⁡θ)(r=∣z∣)z = r(\cos\theta + i\sin\theta) \quad (r = |z|)

This formula, known as de Moivre's theorem, simplifies raising complex numbers to the nn-th power. Multiplying the angle θ\theta by nn directly gives the new angle nθn\theta after taking the power.

(cos⁡θ+isin⁡θ)n=cos⁡nθ+isin⁡nθ(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta

This formula describes the geometric effect of multiplying two complex numbers, z1z_1 and z2z_2. The resulting distance from the origin is the product of their moduli ∣z1∣∣z2∣|z_1||z_2|, while the new angle is the sum of their arguments arg⁡z1+arg⁡z2\arg z_1 + \arg z_2.

∣z1z2∣=∣z1∣∣z2∣,arg⁡(z1z2)=arg⁡z1+arg⁡z2|z_1 z_2| = |z_1||z_2|, \quad \arg(z_1 z_2) = \arg z_1 + \arg z_2

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