Squaring a complex number

More on complex arithmetics for the curious

Although there is enough other information elsewhere on this site to write a Mandelbrot program, you may still be curious about the background, or more specifically the strange formula for multiplying to complex numbers.

Here it is: A complex number is really written as a + bi.

bi is called the "imaginary component", and its origin is in the very reason for using complex numbers, namely in making it possible to calculate the square root of a negative number.

In non-complex mathematics, the square root of a negative number is undefined. The only thing complex arithmetic adds is the symbol i representing the square root of -1, and this enables us to represent the square root of any negative number.

The calculation rules are simple. The only thing worth mentioning is that

i² = -1

Now, to multiply two parenthesized sums (a₁ + b₁) (a₂ + b₂), you just multiply and add together the four possible combinations of elements:

(a₁ + b₁)(a₂ + b₂) = a₁a₂ + a₁b₂ + b₁a₂ + b₁b₂

If we do this to two complex numbers (a₁ + b₁i) and (a₂ + b₂i), we end up with

a₁a₂ + a₁b₂i+ b₁a₂i + b₁b₂i²

But i² is of course -1, so we can simplify this:

a₁a₂ - b₁b₂ + (a₁b₂ + a₂b₁)i

If we do this to the z² calculation we used in the Mandelbrot formula above, writing z as (x,y), we just substitute "x" for "a₁" and "a₂", and "y" for "b₁" and "b₂" to yield

x² - y² + 2xyi

or, written as a complex number

(x² - y², 2xy)

which is equvalent to the formula used in the Mandelbrot code.