# Scroll thru these 10 frame-by-frame images of my Silly Snake cartoon character. Can you guess what he is doing? He’s rotating like a tire!

## Somebody I stopped following on Quora told people that complex numbers don’t rotate. Correction: Anything can rotate, including numbers!

Rotation is a curvy motion. It technically happens everywhere all the time! Even the planet Earth itself is rotating; it rotates **15 degrees each hour,** bringing everything on its surface with it. Why do you think we have

**day & night?**The entire solar system is rotating, too; the planets revolve around the Sun & even spin on their own axes. (Spinning counts as rotation, too! So does revolution!) As an artist & a mathematician, I always knew that images can rotate. Image-wise, you can rotate anything you examine by changing your point of view, even if the rotation is not physical! Being on the planet Earth, you may not feel like you’re rotating & it mayn’t seem like your house is rotating, but actually, the planet’s forces are rotating you & your house along with it!

It was kind of unimaginative for that guy to conclude that complex numbers don’t rotate. (I won’t use his name publicly!) Who says that complex numbers * can’t* rotate? If complex numbers don’t or can’t rotate, then what do you call

**THIS?**:

All complex numbers have angles in respect to the origin. (The origin is the point marked **(0,0) or 0 + 0 i**) Each complex number is a specific point on the complex plane. The angle of a complex number is measured by starting to the right of the origin on the real number line. Positive angles go counter-clockwise & negative angles go clockwise.

When you multiply or divide complex numbers, you’ll either change the magnitude(also known as absolute value) or the angle of your reference point. ** As long as you changed the angle, you rotated something; angle change is rotation! **For example, if you multiply the complex number

**1 +**by

*i***, then you get**

*i***-1 +**. The angle of

*i***1 +**is

*i***45º;**The angle of

**-1 +**is

*i***135º;**since you multiplied by

**you rotated your reference point**

*i,***90º counter-clockwise!**

*(Division rotates clockwise; multiplication rotates counter-clockwise)*Any complex number in the

**unit circle**causes a rotation without changing the magnitude!

**Pictured below is a graph of 12 complex numbers; these are the same 12 that are tagged in the animation above. (Although, only the names of 3 are printed in it.) Points in the same color have the exact same magnitude, but their angles are different in respect to the origin:**

### Even the calculations themselves count as rotation because you technically change the number(s) of degrees(or radians) of an angle or several angles!

## 45º + 30º = 75º

### You can also multiply an angle by a number of your choice to change the angle! For example:

## 45° × 5 = 225°

### If you convert to radians:

## π/4 r × 5 = 5π/4 r

Initially, the angle was **45 degrees,** but if you add

**30**to it, then it becomes

*degrees***75**. You changed the angle & like I typed just now,

*degrees***Finally, 1 last thing as I conclude this essay: If the number of degrees(or radians) of an angle is**

*angle change is rotation!***, then the rotation is**

*positive**counter-clockwise;*if the number of degrees(or radians) is

**, then the rotation is**

*negative**clockwise.*Mathematicians just decided that it should

*normally*be that way, although you could switch that around if you want to!