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.
This article is dedicated to one particular problem which dates back to the Pythagoreans and how the problem was solved using Proof by Contradiction.
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
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.
There were three assumptions used in this proof because their validity had been proved in earlier.
The assumptions were;
The aquare of an odd number is always odd
If an integer is not odd then it must be even
Every decreasing sequence of positive integers must end.
Following is a slightly different proof but it still uses the principle of Proof by Contradiction.
Proof by Contradiction in Action
- Life and Death Mathematics
As a boy I saw numbers as soldiers. There were two distinct armies and every number belonged to one army or the other. The armies were the plus army and the minus army. I could distinguish between the soldiers by the uniform and emblem they wore on..
- Solving Algebraic Equations Part 2
In Solving Algebraic Equations Part 1 I show how to solve equations with two terms using an escalator. In this the second part of my mini series I will show you how to solve equations with three terms. To do this I employ a “wall”. When I was...
- Solving Algebraic Equations Part 1
In order to simplify what I was being taught in Algebra my mind came up with cartoon like descriptions for the equations I was expected to solve. I found that there were only two kinds of algebraic equations. In my mind I saw every equation I was...
More by this Author
The therapeutic ritual developed by Phyllis Krystal to enable people to sever their negative attachments and dependence on parents and things.
When I was first introduced to algebra as a boy I was very confused. The teacher had gone from talking about numbers to now doing mathematics with the letters of the alphabet. At the time the leap was too great and I...
The Photoelectric Effect was once referred to as the Hertz effect after Heinrich Hertz who first observed the phenomenon in 1887. Hertz observed that a metal plate illuminated with ultra violet light spontaneously...