Proof by Contradiction
The Pythagorean Brotherhood
It is possible that it was a group who called themselves The Pythagoreans, after their founder Pythagoras, who came up with the idea of proving something by contradiction.
The Pythagoreans had a large following and the Pythagorean brotherhood were sworn to secrecy. Its members were dedicated to the pursuit of truth and using mathematical proof.
The truths they discovered then hold true to this day.They succeeded in revealing the objective and timeless nature of Mathematics but because they took and oath of secrecy most of their discoveries died with them.
reductio ad absurdum
The Latin name for “Proof by Contradiction” is reductio ad absurdum and it has proved to be one of the more useful principles of Mathematics to date.
Proof by Contradiction is based on the principle that if you want to prove something is true you first suppose that it is false and then proceed until you find something in your argument to contradict your supposition that it is false.
Irrational
This article is dedicated to one particular problem which dates back to the Pythagoreans and how the problem was solved using Proof by Contradiction.
The Problem:
Is there a fraction which when squared will give an answer of exactly two? Stated mathematically, do the whole numbers x and y exist where y is never zero such that (x/y)2 = 2
QED
We could keep going forever getting and unending sequence of equations always finding a positive number to satisfy each one then this contradicts the notion that any decreasing sequence of positive whole numbers must come to an end.
The fact that we are getting an unending sequence of equations we have proved by contradiction that we cannot find two whole numbers x and y with ration x/y that when squared will give 2 precisely.
Assumptions
There were three assumptions used in this proof because their validity had been proved in earlier.
The assumptions were;
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The square of an odd number is always odd
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If an integer is not odd then it must be even
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Every decreasing sequence of positive integers must end.
Video
Following is a slightly different proof but it still uses the principle of Proof by Contradiction.
Proof by Contradiction in Action
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