Square root of 2 is irrational

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

If you believe…

a = 2c and b is even.

If you believe…

a = 2c and b2 is even.3

If you believe…

a = 2c and 2c2 = b2.

If you believe…

a = 2c and 4c2 = 2b2.

If you believe…

a = 2c and (2c)2 = 2b2.

If you believe…

a = 2c and a2 = 2b2.

If you believe…

a = 2c and a2/b2 = 2.

If you believe…

a = 2c and (a/b)2 = 2 (which we assume).

If you believe…

a = 2c.

If you believe…

a is even.

If you believe…

a2 is even.4

If you believe…

a2 = 2b2 for some positive integer b.

If you believe…

a2/b2 = 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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