“i” represents a phase change. You’re rotating a point in the complex plane counterclockwise around the origin.

(A kite on a windy day.)

1

1 * i = 1i

1i * i = -1

-1 * i = -1i

-1i * i = 1

So, how does a phase change translate to being used as an exponent? That is, are we multiplying a number by itself “theta” degrees times?

x^bi = cos(ln(x^b)) + i*sin(ln(x^b))

Basically, we’re rotating the point, maybe?

What is “ln()”?

A logarithm is a conversion formula where

if y = b^x, then x = logb(y)

As an example,

y = 10^2 = 100

x = log10(100) = 2

Or, the log of a number using a particular base will give you the power using that base that gives you that number.

10 squared is 10*10 = 100. The log of 100, using base 10, is 2.

10 cubed is 10*10*10 = 1000. The log of 1000, using base 10, is 3.

Logarithms are useful because they let you draw huge exponential curves as straight lines on a smaller sheet of paper.

Note that as x goes from 1 to 100, log10(x) goes from 0 to 2.

Because our modern counting system is decimal-based, it’s natural to use log10() in many cases. However, we want ln(), which is called the natural logarithm.

There is a special number, e, that is defined as the sum of 1 over n factorial (the product of multiplying the integers between 1 and n). It was first defined as a constant by Jacob Bernoulli in the late 1600’s when he was studying compound interest.

The natural logarithm is “loge()”, or, “log base e”, written as “ln()”.

e has some special properties of its own, including Euler’s Identity:

Kinda cool, eh? It also means that e^i*pi = -1. Rotation, baby!

Going back to imaginary exponents:

x^bi = cos(ln(x^b)) + i*sin(ln(x^b))

This just means that instead of trying to calculate “x^bi”, we can convert it by taking the natural logarithm of x^b, and then plugging that into the sine and cosine wave formulas.

Oh, wait, I haven’t talked about those yet?

(How triangles relate to circles.)

I mentioned earlier that a number in the complex plane can either be represented as the two legs of a right triangle, or as a vector in the form of the magnitude (hypotenuse of the triangle) and the angle theta. The cosine() of an angle extracts the real part of the magnitude of the vector (“a”), and the sine() of the angle extracts the imaginary part (“b”). That is, these are conversion factors to switch between rectangular and polar (circular) graphs.

(Note: Images takes from wikipedia.)

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