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)#.

Absolute value 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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