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    • Maths in a minute: The square root of 2 is irrational

      10 November, 2016
      5 comments
      Structure

      The proof of the irrationality of root 2 is often attributed to Hippasus of Metapontum, a member of the Pythagorean cult. He is said to have been murdered for his discovery (though historical evidence is rather murky) as the Pythagoreans didn't like the idea of irrational numbers.

      Here's one of the most elegant proofs in the history of maths. It shows that 2 is an irrational number, in other words, that it cannot be written as a fraction a/b where a and b are whole numbers.  
       
      We start by assuming that 2 can be written as a fraction a/b and that a and b have no common factor — if they did, we could simply cancel it out. In symbols, ab=2. Squaring both sides gives a2b2=2. and multiplying by b2 gives a2=2b2.
       

      This means that a2 is an even number: it's a multiple of 2. Now if a2 is an even number, then so is a (you can check for yourself that the square of an odd number is odd). This means that a can be written as 2c for some other whole number c. Therefore, 2b2=a2=(2c)2=4c2. Dividing through by 2 gives b2=2c2. This means that b2 is even, which again means that b is even. But then, both a and b are even, which contradicts the assumption that they contain no common factor: if they are both even, then they have a common factor of 2. This contradiction implies that our original assumption, that 2 can be written as a fraction a/b must be false. Therefore, 2 is irrational.

      This proof is famous example of something called a proof by contradiction.  You can read more about this type of proof in the article Something from nothing?

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      Comments

      Matt Lehman

      10 November 2016

      Permalink

      There's a shorter proof which requires unique factorization of integers, while ignoring the assumption that a and b have no common factors.

      Given a^2 = 2b^2, neither have the same number of 2s as a factor, therefore they can't be equal.

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      Thomas Lang

      2 September 2019

      In reply to Irrationality of sqrt(2) by Matt Lehman

      Permalink

      That's the problem though: the proof through unique factorisation assumes the Fundamental Theorem of Arithmetic, which needs to be built from the ground up first. Either way, the irrationality of root 2 is a great piece of mathematics both for us and the ancient humans.

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      Bill Dixon

      13 September 2020

      In reply to With a caveat by Thomas Lang

      Permalink

      But the Euclid proof also depends on the Unique factorization Theorem, otherwise it would not be possible to decide whether a/b was in its lowest form

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      E. Macías

      22 March 2021

      In reply to Irrationality of root 2 by Bill Dixon

      Permalink

      Not really, you only have to indicate at the beginning of the proof that the numbers a, b they are not both even, otherwise you should simplify by dividing each one by 2

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      Balkrishna Shetty

      7 November 2021

      In reply to mmm by E. Macías

      Permalink

      Even after dividing by 2, both a and b may still be even! You need mathematical Induction to select the least a( or least b) such that a- square divided by b- square is two.

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