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The square root of 2 is an irrational number. It can be represented by $ \sqrt{2} $ and has an approximate value of $ 1.41421356 $. The Pythagorean philosopher Hippasus was the first to discover it was irrational. It also the ratio of the length of the hypotenuse to one of the legs of an isosceles right triangle.

This number cannot be written as a fraction.

Proof of the number's irrationality

One proof of the number's irrationality is the following proof by infinite descent. It is also a proof by contradiction, also known as an indirect proof, in that the proposition is proved by assuming that the opposite of the proposition is true and showing that this assumption is false, thereby implying that the proposition must be true.

  1. Assume that 2 is a rational number, meaning that there exists a pair of integers whose ratio is 2.
  2. If the two integers have a common factor, it can be eliminated using the Euclidean algorithm.
  3. Then 2 can be written as an irreducible fraction ab such that a and b are coprime integers (having no common factor).
  4. It follows that a2b2 = 2 and a2 = 2b2.   ( (ab)n = anbn  )
  5. Therefore, a2 is even because it is equal to 2b2. (2b2 is necessarily even because it is 2 times another whole number and multiples of 2 are even.)
  6. It follows that a must be even (as squares of odd integers are never even).
  7. Because a is even, there exists an integer k that fulfills: a = 2k.
  8. Substituting 2k from step 7 for a in the second equation of step 4: 2b2 = (2k)2 is equivalent to 2b2 = 4k2, which is equivalent to b2 = 2k2.
  9. Because 2k2 is divisible by two and therefore even, and because 2k2 = b2, it follows that b2 is also even which means that b is even.
  10. By steps 5 and 8 a and b are both even, which contradicts that ab is irreducible as stated in step 3.
Q.E.D.

Because there is a contradiction, the assumption (1) that 2 is a rational number must be false. This means that 2 is not a rational number; i.e., 2 is irrational.

This proof was hinted at by Aristotle, in his Analytica Priora, §I.23.[1] It appeared first as a full proof in Euclid's Elements, as proposition 117 of Book X. However, since the early 19th century historians have agreed that this proof is an interpolation and not attributable to Euclid.[2]

References

  1. All that Aristotle says, while writing about proofs by contradiction, is that “the diagonal of the square is incommensurate with the side, because odd numbers are equal to evens if it is supposed to be commensurate”.
  2. The edition of the Greek text of the Elements published by E. F. August in Berlin in 1826–1829 already relegates this proof to an Appendix. The same thing occurs with J. L. Heiberg's edition (1883–1888).
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