# Complex Arithmetic

Robert P. Munafo, 2023 Jun 13.

All of the standard arithmetic operations can be performed with complex numbers, and there are a few new operations (notably logarithm and square root of negative numbers) that can be done with complex numbers that cannot be done with ordinary real numbers.

These examples define two complex numbers x and y as:

x = a + b i

y = c + d i

(For clarity, on the right side only the i is in italics)

Addition and subtraction are very simple:

x+y = a+c + (b+d)i

x-y = a-c + (b-d)i

Multiplication is fairly simple too:

xy = ac - bd + (ad + bc)i

Division is a little more elaborate. Provided that c^{2}+d^{2}>0,
the quotient is:

x/y = (ac+bd)/(c^{2}+d^{2}) + (bc-ad)/(c^{2}+d^{2}) i

For those who are curious: starting from (a+bi)/(c+di), multiply
both numerator and denominator by the conjugate c-di; the
denominator is then (c^{2}+cdi-cdi-d^{2}i^{2}) and everything
simplifies to the above.

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revisions: 20100911 oldest version on record; 20230613 more details about division

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.

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