The number root 2 was discovered by Hippasus who was a Pythagorean, a group of people during Pythagoras’ time that believed that the universe could be explained using rational numbers. Hippasus discovered that this number was irrational, therefore going against the Pythagorean beliefs. When Pythagoras and other Pythagoreans found out what Hippasus had discovered they murdered him, on a boat in the middle of the sea immediately after he told them of his discovery. This information however may not 100% true though because there are very little records dating back to the time of Pythagoras.
This number was discovered by using Pythagorean’s theorem. Pythagorean’s theorem states that A^2 + B^2 = C^2. If you do this with side lengths being 1 your equation would be 1^2 +1^2 = 2, so the length of the hypothesis would be √2. Root 2 calculates to be 1.414213… and so on, making it an irrational number.
Irrational? YES, impractical? NO. Root 2 has very important real world applications. This number can be used to reduce the size of a normal (A size) paper, because of the relationship between the side lengths. It is used to reduce the size of an A size paper without distorting the shape and ratio between the side lengths. If you fold a piece of standard A size paper in half you will get the same ratio between the side lengths, only reduced.
For more information about root 2 (√2) watch: https://youtu.be/5sKah3pJnHIE
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http://cs.fit.edu/~ryan/interdisciplinary_assignments/newton/
http://www.mathblog.dk/project-euler-57-square-root-two/
http://pumas.jpl.nasa.gov/files/10_06_03_1.pdf
http://hsm.stackexchange.com/questions/2/irrationality-of-the-square-root-of-2
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