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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[edit] Addition

Addition of complex numbers is done by separating the real and imaginary components, and adding them together.

(a + bi) + (c +di) = (a + c) + (bi + di) 
Example:
Add the two complex numbers 3 + 4i and 1 + 2i together
(3 + 4i) + (1 + 2i)
(3 + 1) + (4i + 2i)
4 + 6i

[edit] Multiplication (by Scalar)

When multiplying by a scalar, both the real and imaginary parts get multiplied by that figure 

3(a + bi) = 3a +3bi

[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

Who Added These Notes?

Rjt

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