[DEFINITIONS] Polar form

Argument

For $z\ne0$ , an argument of $z$ is a real number $\theta$ satisfying

\cos\theta=\frac{\operatorname{Re}(z)}{|z|}, \qquad \sin\theta=\frac{\operatorname{Im}(z)}{|z|}.

Arguments are determined up to addition of $2\pi k$ , where $k\in\mathbb Z$ . The principal argument $\operatorname{Arg}(z)$ is the unique argument in

-\pi<\operatorname{Arg}(z)\le\pi.

Choose the argument from the point's quadrant, not from $\tan\theta$ alone: tangent has period $\pi$ , so it cannot distinguish opposite quadrants.

The principal argument is one conventional choice; every argument remains valid. The number $z=0$ has no argument, because the origin does not determine a direction.

[1]:Complex number
[2]:Complex modulus
[3]:Polar form
[4]:Principal arguments of products and quotients
[5]:Read loci in the Argand plane

Polar form

For $z\ne0$ , the polar form of $z$ is

z=r(\cos\theta+i\sin\theta)=re^{i\theta}, \qquad r=|z|>0,

where $\theta$ is an argument of $z$ . The expressions $a+bi$ and $re^{i\theta}$ are two writings of the same complex number.

Euler's formula,

e^{i\theta}=\cos\theta+i\sin\theta,

identifies the two expressions in the statement. Replacing $\theta$ by $\theta+2\pi k$ leaves $z$ unchanged.

Euler's formula is taken as given here rather than derived. The Taylor series of the exponential, sine and cosine are the usual route to a proof.

Multiplication by $e^{i\theta}$ rotates a point through angle $\theta$ without changing its modulus; multiplication by $re^{i\theta}$ also scales it by $r$ . In coordinates, for $z=x+iy$ ,

\begin{pmatrix} \operatorname{Re}(e^{i\theta}z)\\ \operatorname{Im}(e^{i\theta}z) \end{pmatrix} = \begin{pmatrix} \cos\theta&-\sin\theta\\ \sin\theta&\cos\theta \end{pmatrix} \begin{pmatrix}x\\y\end{pmatrix}

An nth root of a complex number

For a complex number $z_0$ and a positive integer $n$ , an $n$ th root is a complex number $z$ satisfying

z^n=z_0.

Definitions from Polar form

Argument

Polar form

An nth root of a complex number

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