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The complex numbers form a [[Field_(mathematics)|field]], with the mathematical operations defined as shown below.
The complex numbers form a [[Field_(mathematics)|field]], with the mathematical operations defined as shown below.
==Imaginary numbers==
An '''imaginary number''' in mathematics is any number that is a multiple the imaginary unit, defined as, <math>i^{2} = -1</math>
An imaginary number is of the form, <math>k i</math>, where ''k'' is a [[real number]].
For example <math>\sqrt{-1}</math> has imaginary representation of
:<math>\sqrt{-1}= \pm i</math>.
When a imaginary number is added to real number, they form a [[complex number]]. Imaginary numbers are mathematically useful because they fabricate solutions to every polynomial [[equation]] - for example, the equation <math>x^2+1=0</math> has no real solution.
The analysis of imaginary numbers forms the basis for the field of [[mathematics]] known as
[[complex analysis]].
===Proof that Imaginary Numbers Don't Exist===
The proof that the [[polynomial]] <math>x^2+1=0</math> has no real solutions goes like:
:<math>(x)+(-x)=0</math>
:<math>((x)+(-x))^2=0^2=0</math>
:<math>(x)^2+(x)(-x)+(-x)(x)+(-x)(-x)=0^2=0</math> (From the [[FOIL]] law (first, outside, inside, last) of multiplication)
:<math>x^2+(-x)(-x)=2x^2</math>
:<math>(-x)(-x)=x^2</math>
Because <math>x^2\ge0</math> for every <math>x</math>, <math>(-x)(-x)=x^2\ge0</math> for every <math>x</math>. The number for every <math>-1<0</math> is negative, therefore the [[polynomial]] <math>x^2 = -1</math> has no real solutions, and the ''imaginary'' solution <math>i</math>, where <math>i^2=-1</math>, is called an ''imaginary'' number because no such number solves this equation and does not exist, as just shown.
[[Complex numbers]] <math>a+ib</math> mean the number <math>a</math> of ''real'' numbers plus the number <math>b</math> of ''imaginary'' numbers. This is like saying ''if'' elves existed, then the "elf-number" <math>a+\mbox{elf}\times b</math> means the number <math>a</math> of ''real'' people plus the number <math>b</math> of elves.
==Arithmetic operations==
==Arithmetic operations==