Polar coordinates

Complex numbers: Introduction to Complex numbers

Polar coordinates

Multiplication of complex numbers also has a geometric interpretation. For that we will use the absolute value and argument, which we are going to deal with here.

Except for with Cartesian coordinates, we can also indicate a point in the plane with polar coordinates, which consist of two values, the absolute value and the argument:

the absolute value is the distance from the origin;
the argument is the angle of the vector with the positive $x$ -axis measured in radians.

The absolute value of the complex number $z=a+b\ii$ , with real $a$ and $b$ , is equal to $\sqrt{a^2+b^2}$ and is indicated with $|z|$ .

The argument of a complex number $z$ unequal to $0$ , is only defined up to a multiple of $2\pi$ . If we choose the argument of $z$ greater than $- \pi$ and smaller than or equal to $\pi$ , we speak of the principal value of the argument, which is indicated as $\arg(z)$ .

The absolute value is a function with domain $\mathbb C$ and range $\left\{x\in{\mathbb R}\mid x\ge0\right\}$ , the set of non-negative real numbers. If $z$ is a real number, then to the previously known definition of absolute value coincides with this new definition (recall that $\sqrt{a^2}=|a|$ ).

The absolute value is also known as norm.

Complex argument

For the origin, the argument is not defined. With the requirement $-\pi\lt\arg(z)\le\pi$ , or $\arg(z)\in\ivoc{-\pi}{\pi}$ we achieve that $\arg$ is a function with domain ${\mathbb C}\setminus\{0\}$ and range $\ivoc{-\pi}{\pi}$ .

The principal value of the argument of points on the negative real axis is $\pi$ .

The word argument is also used as an argument of a function. In mathematics, it happens more often that a word has multiple meanings. By continuing to clarify whether the argument is part of a complex number or of a function, confusion can be avoided. Sometimes we also write complex argument instead of argument to make the distinction with the argument of a function.

From polar coordinates to Cartesian coordinates

Let $r$ be a positive number and $\varphi$ an arbitrary real number. Then

$r\cdot\cos(\varphi)+r\cdot\sin(\varphi)\ii$ is the unique complex number with absolute value $r$ and argument $\varphi$ .

It is known that the point on the unit circle in the complex plane with angle $\varphi$ to the positive $x$ axis, has coordinates $\rv{\cos(\varphi),\sin(\varphi)}$ . The corresponding complex number is $\cos(\varphi)+\sin(\varphi)\ii$ . The complex number with absolute value $r$ and argument $\varphi$ can be obtained from $\cos(\varphi)+\sin(\varphi)\ii$ by multiplying with scalar $r$ , because the angle remains the same and the absolute value is multiplied by $r$ .

In terms of the absolute value $|z|$ and the principal value $\arg(z)$ of the argument, the formula can also be written as

$z = |z|\cdot \left(\cos(\arg(z))+\sin(\arg(z))\cdot\ii\right)\tiny.$

Find the polar coordinates of the complex number $-\ii$ .

$\rv{r,\varphi}=$ $\rv{1,-{{\pi}\over{2}}}$

After all, the radius $r$ and argument $\varphi$ of $-\ii$ satisfy

$\begin{array}{rcl}r &=&\sqrt{\left(0\right)^2+\left(-1\right)^2}=1\\ \\ \cos(\varphi) &=&\dfrac{0}{r}= \dfrac{0}{1}=0\\ \sin(\varphi) &=&\dfrac{-1}{r}= \dfrac{-1}{1}=-1 \end {array}$
From the last two equations we deduce that the principal value of the argument is $\varphi=-{{\pi}\over{2}}$ .

See the figure below for a geometric interpretation of the transition from the standard form of a complex number to polar coordinates. The complex number is marked blue. The absolute value $r$ and argument $\varphi$ are marked red.

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