# 8a: How does complex arithmetic work?

## Description

This article is from the Fractal FAQ, by Ermel Stepp stepp@muvms6.mu.wvnet.edu with numerous contributions by
others.

# 8a: How does complex arithmetic work?

It works mostly like regular algebra with a couple additional formulas:

(note: a,b are reals, x,y are complex, i is the square root of -1)

Powers of i: i^2 = -1

Addition: (a+i*b)+(c+i*d) = (a+c)+i*(b+d)

Multiplication: (a+i*b)*(c+i*d) = a*c-b*d + i*(a*d+b*c)

Division: (a+i*b)/(c+i*d) = (a+i*b)*(c-i*d)/(c^2+d^2)

Exponentiation: exp(a+i*b) = exp(a)(cos(b)+i*sin(b))

Sine: sin(x) = (exp(i*x)-exp(-i*x))/(2*i)

Cosine: cos(x) = (exp(i*x)+exp(-i*x))/2

Magnitude: |a+i*b= sqrt(a^2+b^2)

Log: log(a+i*b) = log(|a+i*b|)+i*arctan(b/a) (Note: log is multivalued.)

Log (polar coordinates): log(r*e^(i*theta)) = log(r)+i*theta

Complex powers: x^y = exp(y*log(x))

DeMoivre's theorem: x^a = r^a * [cos(a*theta) + i * sin(a*theta)]

More details can be found in any complex analysis book.

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