Complex numbers

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In mathematics, a complex number is a number of the form

a + bi

where a and b are real numbers, and i is the imaginary unit, with the property i^2 = −1. The real number a is called the real part of the complex number, and the real number b is the imaginary part.

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Contents

[edit] Multiplication

(a + bi)(c + di) = ac + adi + cbi + bdi^2

Note: by definition i^2= -1.

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

[edit] Conjugate

If (a + bi) = {z} Then (a - bi) = <tex>\bar{z} This is known as a conjugate

[edit] Multiplication (by conjugate)

A complex number multiplied by it's conjugate always gives a real number.

({z})(\bar{z}) \in \mathbb{R}

(a + bi)(a - bi) = a^2 - abj + abj + b^2
=a^2+b^2

Which is a real number

[edit] More in this section

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