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Reductio Ad Absurdum

Reductio ad absurdum is the process by which an hypothesis is proved by showing the opposite to be impossible. This method was first devised by the ancient Greeks.

In his book, Mathematical Mysteries, Calvin C. Clawson illustrates how this can be done, "We begin by supposing the opposite of what we want to prove." In this case that the square root of 2 can be "represented by some ratio of whole numbers", that is some whole number that can be expressed as a fraction. He goes on to show that this is impossible. The ratio expressed as either the numerator or the denominator must be an odd number because a fully reduced fraction must have an odd number in it even if the resulting number is even (for example the number 4 can be expressed as 4/1). The negative assumption can then be expressed as: p/q = square root of 2.

Using algebra in a proof we can then square both sides, multiply both sides by q2 and get: p2 = 2q2. Since any number times 2 is an even number this means that p must be even. To continue along the same lines, if p is even, then q must be odd to make our original statement true. But alas, this is not the case, as Clawson's proof shows: We now substitute 2r for p because being even it must be two times some number. This gives us: (2r)2 = 2q2. This can also be expressed as: 4r2 = 2q2. If we divide both sides of the equation by 2 the result is 2r2 = q2. The denominator, then, must be even as well as the numerator.

We have shown that for the square root of two to be "represented by some ratio of whole numbers" the denominator of our ratio must be both even and odd. It is clearly impossible for a number to be both even and odd, so the square root of two must not be "some ratio of whole numbers".

The term reductio ad absurdum actually comes from Medieval Latin and basically means "reduction to absurdity". Even so, its roots are certainly Greek. Aristotle gives the method a thorough workout in his philosophical writings. The method can and has been used for more than mathematics. It has been applied in science, logic, definitions, debate topics, and even in common suppositions as when a conversationalist uses such a phrase as, "If such-and-such is true, then I am a monkey's uncle." However, this last construction is a rather loose use of the "reductio ad absurdum" argument as it does not bother with a proof. -----

Interesting Links:
Thurlow Weed's Mnemonic
Tuchman's Law
Prime Numbers
Aristotle on Rhetoric


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