Imaginary number

An imaginary number in mathematics is any number that is a multiple the imaginary unit, defined as, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^{2} = -1}

An imaginary number is of the form, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k i} , where k is a real number.

For example Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{-1}} has imaginary representation of

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{-1}= \pm i} .

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 $x^{2}+1=0$ 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2+1=0} has no real solutions goes like:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x)+(-x)=0}

$((x)+(-x))^{2}=0^{2}=0$

$(x)^{2}+(x)(-x)+(-x)(x)+(-x)(-x)=0^{2}=0$ (From the FOIL law (first, outside, inside, last) of multiplication)

$x^{2}+(-x)(-x)=2x^{2}$

$(-x)(-x)=x^{2}$

Because $x^{2}\geq 0$ for every $x$, $(-x)(-x)=x^{2}\geq 0$ for every $x$. The number for every $-1<0$ is negative, therefore the polynomial $x^{2}=-1$ has no real solutions, and the imaginary solution $i$, where $i^{2}=-1$, is called an imaginary number because no such number solves this equation and does not exist, as just shown.

Complex numbers $a+ib$ mean the number $a$ of real numbers plus the number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} of imaginary numbers. This is like saying if elves existed, then the "elf-number" Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a+\mbox{elf}\times b} means the number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} of real people plus the number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} of elves.