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`sqrt(2) = 1.4142135623730951...`

1.4142135623730951

This constant provides an approximate value for the square root of two (`sqrt(2)`). The `sqrt(2)` is an irrational number and appears very often in many scientific and mathematical applications.

An **irrational number** is any real number that cannot be expressed as a ratio of integers. Irrational numbers are those real numbers that cannot be represented as terminating or repeating decimals. As a consequence of Cantor's proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational.^{[1]}

When the ratio of lengths of two line segments is irrational, the line segments are also described as being *incommensurable*, meaning they share no measure in common.

Numbers which are irrational include the ratio of a circle's circumference to its diameter ?, Euler's number e, the golden ratio ?, and the square root of two,^{[2][3][4]} in fact all square roots of natural numbers not being a perfect square are irrational.

It would seem that taking the square root of a number we have all know since childhood should result in something simple -- just because the number 2 is so close to our daily thoughts. We grab a couple of cookies from the cookie jar, I give Brutus (my golden) a couple of treats when he comes in out of the snow, I am in a relationship which makes me a partner in a "couple". Two is commonplace and I was at an early age capable of doubling just about any number within my grasp. So, it is a little magical and surprising to find that the `sqrt(2)` is irrational.

There is elegance in many simple mathematical concepts and no more elegant proof exists than the proof by contradiction. And no more representative application of such a proof-by-contradiction exists than that which proves the `sqrt(2)` is irrational. John Phillip Jones of MathsEasyAsPie.com gives a thorough and easy-to-follow proof of the irrationality of the `sqrt(2)`. This video is worth watching for all those entering high school and college mathematics where proof-by-contradiction will enter into your mathematical tool box.