Kevin7
Lv 7

# are there numbers that repesent rotations?

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This is an excellent question!

Yes, there are many different ways to represent rotations mathematically.

First, complex numbers can be used to represent rotations:

Since complex multiplication when viewed using polar coordinates is nothing more than rotations and expansion, a complex number can be thought of as the corresponding rotation and expansion/contraction. For example, let's look at the simple number i.

i = 1(cos (90) + sin (90))

So i represents a counter-clockwise rotation of 90 degrees.

If we take any complex number z, where

z = r (cos (x) + i sin (x))

and multiply by i, we get

iz = ir (cos(x) + i sin (x))

= (cos(90) + i sin (90))(r)(cos(x) + i sin(x))

= r (cos(x + 90) + i sin (x + 90))

You can confirm the above details with trig identities. So you can see that multiplication by i is a 90 degree rotation counterclockwise.

Similarly, multiplication by any complex number of the form

w = (cos (a) + i sin (a)) is going to give you a rotation by a degrees counterclockwise. Note that a negative value for a will give a clockwise rotation by the absolute value of a.

I'll leave the details for you to work out.

Now, keep in mind that rotations can also be represented by matrices of numbers as well (2x2 in the plane, and larger matrices in larger dimensions).

If we use vector notation, then any matrix of the form

R = [ cos (x) - sin(x) ]

___[ sin( x) cos(x) ]

can be thought of as a rotation when multiplied on the right by the column vector {x y}.

Again I will leave the details for you.

These are just a few of the (simplest) ways in which rotations can be represented. There are more subtle and sophisticated ways in which rotations can be measured as well (e.g. rotation groups, quaternions, etc.) which you can explore online or in good textbooks.