### 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**.

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

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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