## Ongoing Version (Draft)

The square root of 2 is irrational.

- A number is irrational if and only if it is not rational.

The square root of 2 is not rational.

- A number is rational if and only if it can be expressed as the ratio of two integers.

The square root of 2 is not a ratio of two integers.

- If we assume something and we get a contradiction, we may conclude that the assumption is false.

Assuming the square root of 2 is a ratio of two integers produces a contradiction. Therefore our original assumption is wrong, and therefore the square root of 2 is not the ratio of two integers.

- A rational number can be expressed as a ratio of two integers, and this ratio can be reduced to lowest terms.

Assuming the square root of 2 is a ratio of two integers produces a contradiction. If the square root of 2 were a ratio of two integers then it would be possible to keep on reducing it to lowest terms, over and over again. But that would mean you *can't* reduce it to lower terms. But you *can* do such a simplification for a ratio of two integers. Therefore our original assumption is wrong, and therefore the square root of 2 is not the ratio of two integers.

## Older Version

The square root of 2 is irrational.

*If you believe…*

The square root of 2 is not rational.^{1}

*If you believe…*

Assuming the square root of 2 is rational produces a contradiction.

*If you believe…*

Assuming the square root of 2 equals *a**/b* for positive integers *a*, *b*, with no common factors, produces a contradiction.^{2}

*If you believe…*

*a* and *b* have at least one common factor.

*If you believe…*

*a* is divisible by 2, and *b* is divisible by 2.

*If you believe…*

*a* = 2*c* and *b* = 2*d* (*c*, *d* are positive integers).

*If you believe…*

*a* = 2*c* and *b* is even.

*If you believe…*

*a* = 2*c* and *b*^{2} is even.^{3}

*If you believe…*

*a* = 2*c* and 2*c*^{2} = *b*^{2}.

*If you believe…*

*a* = 2*c* and 4*c*^{2} = 2*b*^{2}.

*If you believe…*

*a* = 2*c* and (2*c*)^{2} = 2*b*^{2}.

*If you believe…*

*a* = 2*c* and *a*^{2} = 2*b*^{2}.

*If you believe…*

*a* = 2*c* and *a*^{2}*/b*^{2} = 2.

*If you believe…*

*a* = 2*c* and (*a**/b*)^{2} = 2 (which we assume).

*If you believe…*

*a* = 2*c*.

*If you believe…*

*a* is even.

*If you believe…*

*a*^{2} is even.^{4}

*If you believe…*

*a*^{2} = 2*b*^{2} for some positive integer *b*.

*If you believe…*

*a*^{2}*/b*^{2} = 2

*If you believe…*

(*a**/b*)^{2} = 2 (which we assume).

## Advisory

*Please note that I am not a mathematician and so the presentation of proofs that I make may be deeply flawed. I'm using this writing process to figure out what I'm reading. Please consult more authoritative sources as well.*

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