Question

Can someone prove to me that multiplication by imaginary numbers is a rotation?

I understand the concept of the rotation where multiplication by `-1` is a `180°` rotation and then squared is `360°` so therefore `-1^(1/2)` is `90°` but I don't see how that is a viable solution to problems since it is not on the number line. For example given the equation `y = x^2 + 1` the zeros are `i and -i`, but since that means a `90°` rotation we are in the `3rd` dimension however our equation is only in `2` dimensions. Maybe I am thinking of this in the wrong way, but someone please try and explain and prove that imaginary numbers just rotate and please explain how our `2` dimensional function can have two zeros in the `3rd` dimension.

Answer 1

A few thoughts...

Explanation:

The Real numbers are usually thought of as constituting a line which we call the Real line.

This is the `x`-axis of the Complex plane, representing an extension of the Real numbers to numbers of the form `a+bi`, where `i` is the imaginary unit (i.e. the point `(0, 1)`).

Addition of complex numbers is two dimensional vector addition, that is:

`(a+bi)+(c+di) = (a+c)+(b+d)i`

Multiplication of complex numbers is defined as:

`(a+bi)(c+di) = (ac-bd)+(ad+bc)i`

This has some interesting properties:

• Real numbers are complex numbers of the form `a+0i`

• Multiplication by Real numbers is scalar multiplication:

  `(a+0i)(c+di) = ac+adi`

• The square of `i` is `-1`:

  `i^2 = (0+1i)(0+1i) = ((0)(0)-(1)(1))+((0)(1)+(1)(0))i = -1`

The modulus of a complex number is its length as a vector, which can be deduce from Pythagoras:

`abs(a+bi) = sqrt(a^2+b^2)`

Hence we find that any Complex number of modulus `1` can be represented in the form:

`cos theta + i sin theta`

Multiplication by such a number is pure rotation by `theta` in an anticlockwise direction:

`(cos theta + i sin theta)(a+bi)`

`= (a cos theta - b sin theta)+(b cos theta + a sin theta)i`

By the time we get to this point, some readers might be thinking "matrices". Indeed, complex numbers have a natural representation in terms of `2xx2` matrices with real coefficients. Then addition and multiplication become matrix addition and multiplication:

`((a, b),(-b, a))+((c,d),(-d,c)) = ((a+c,b+d),(-(b+d),a+c))`

`((a, b),(-b,a))((c,d),(-d,c)) = ((ac-bd,ad+bc),(-(ad+bc),ac-bd))`

Notice that complex numbers of modulus `1` take the form:

`((cos theta, sin theta),(-sin theta, cos theta))`

which is recognisable as the matrix representing a rotation through angle `theta` about the origin.

In general, multiplication by a complex number is a combination of rotation about the origin and scaling.