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Irrational ProofA Rational Number is a number that can written exactly as the ratio of two integers:
The Greeks once thought that every number that could be constructed by the methods of classical geometry would be rational. However, it was with great dismay that the hypotenuse of an easily constructed right triangle with unit sides having a value √2 could not be written as the ratio of two integers. The proof of this is not hard to understand and follow. It is a proof by contradiction. We assume a certain statement is True then show that it leads to a contradiction. Therefore the statement must be False. We start by asserting that √2 is indeed a Rational Number; that is, it can be written as the ratio of two integers that have been reduced to lowest terms:
We assert that this statement is True. By multiplying both sides by the denominator, we arrive at another True statement (assuming the original is True): n·√2 = m Squaring both sides will bring us to another True statement (if the original is True): n2·2 = 2·n2 = m2 Since the left-hand-side of this equation contains a factor of two, the right-hand-side must be even. This means m must be even. Note, an even number squared always produces an even number; whereas, an odd number squared always results in an odd number. Only even numbers when squared are even. Since m is even, we can write it as 2s where s is an integer; that is, every even number can have a 2 factored out evenly. Substituting this for m, we get: 2·n2 = 4·s2 This must be another True statement (as long as the original is True). Now we can divide both sides of this equation by 2 to obtain yet another True equation (assuming the original is True): n2 = 2·s2 Since the right-hand-side contains a factor of 2, it is even; hence n must be an even number (since only even numbers when squared give an even number). Stop! Wait just a minute! Hold the presses! We have established that both m and n are even. Therefore, our original fraction cannot have been reduced to lowest terms. Our original ascertion cannot be True. We have a contradiction! The value √2 cannot be written as the ratio of two integers reduced to lowest terms. Therefore, √2 is not a Rational Number. |
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© 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008 by Lawrence Turner |
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