Imaginary number

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.