assume the

contrary:

2

^{.5} =

`p`/

`q`
square both lines.

2 =

`p`^{2}/

`q`^{2}
multiply both sides by

`q`^{2}
2

`q`^{2} =

`p`^{2}
`p`^{2} must be even, for it is equal twice the value of an integer. (

`q`^{2} )

`p` is therefore

even as well, for if the square of a number is even, the number is as well.

because

`p` is even, it can be written as 2m, where m is some other

whole number. placed into the equation, it reads:

2

`q`^{2} = (2

`m`)

^{2} = 4

`m`^{2}
divide both sides by 2.

`q`^{2} = 2

`m`^{2}
with the same arguments used previously, q must also be even, and may be represented as 2n. same steps, resulting in

2

^{.5} =

`p`/

`q` = 2

`m`/2

`n`
lose the 2s.

2

^{.5} =

`m`/

`n`
This

process can continue indefinitely, meaning the fraction is

infinitely reducable. A quality of a fraction is that it may be represented as a ratio between two

relatively prime integers. (like 10/20 can be reduced to 5/10, and finally to 1/2)

yeah...

QED