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.

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



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

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Comments 19 comments

diogenes profile image

diogenes 5 years ago from UK and Mexico

Hi You lost me with the math; I just can't concentrate these days. But the method of proving things by contradiction makes sense. Probably why no one could ever prove that a god exists...Bob

Spirit Whisperer profile image

Spirit Whisperer 5 years ago from Isle of Man Author

Nice one Bob!

aravindb1982 profile image

aravindb1982 5 years ago from Puttaparthi, India

I was not able to understand why both the numbers must be positive. Squaring a negative number also yields a positive integer right?

Is it maybe because, for the next step, you will not be get the root for a negative fraction?

If that gets clarified, it will be wonderful......Very interesting thought.....

Thanks for the hub

Alastar Packer profile image

Alastar Packer 5 years ago from North Carolina

Hi Spirit. If the teachers had made Algebra this fun & interesting might have done better in it. Thanks for the bit of history on the Pythagorean Brotherhood too.

Spirit Whisperer profile image

Spirit Whisperer 5 years ago from Isle of Man Author

Hi Aravind, the problem arose when the Pythagoreans went to measure the diagonal of a 1X1 square and found they could not find a rational number for this measurement. The squares they used did not have negative lengths LOL!

Spirit Whisperer profile image

Spirit Whisperer 5 years ago from Isle of Man Author

Thank you Alastar for the visit and your compliment is very much appreciated.

Genna East profile image

Genna East 5 years ago from Massachusetts, USA

Math is not my forte, but the premise, here, is fascinating. (I keep kicking myself for not wanting to understand algebra better in my younger years. I did not realize until later that it taught us different ways of thinking and understanding the relationships between variables. ) Thank you for this very interesting read.

Spirit Whisperer profile image

Spirit Whisperer 5 years ago from Isle of Man Author

Thank you Genna. Algebra is beautiful. It summarises all the research done by mathematicians looking for patterns and connections between numbers. It is never too late to learn and perhaps you migh take a look at another hub I wrote called Marcy Mathworks

This is a fabulous resource and learning tool.

James A Watkins profile image

James A Watkins 5 years ago from Chicago

I enjoyed your Hub on "reductio ad absurdum." It is well done and thought-provoking. Thank you!

Spirit Whisperer profile image

Spirit Whisperer 5 years ago from Isle of Man Author

You are very welcome James. Thank you.

Mr. Happy profile image

Mr. Happy 5 years ago from Toronto, Canada

This is a little off topic but do you think the study of math stuns creativity? (I think it somewhat does ...)

Sorry I cannot comment on the problem/equation: for me, numbers no longer have much of the meaning that mathematics imposes on them.


Spirit Whisperer profile image

Spirit Whisperer 5 years ago from Isle of Man Author

Maths for me is just another way of expressing thought. So when you ask if it stunts creativity I would say that unlike art and music it does not evoke emotion but nurtures logic. Thank you for commenting.

Mr. Happy profile image

Mr. Happy 4 years ago from Toronto, Canada

Great observation Mr. Spirit Whisperer (about the emotion part). Thank you!

phdast7 profile image

phdast7 4 years ago from Atlanta, Georgia

Excellent Hub. I am a historian, not a mathematician, and Algebra was never my forte, although Geometry was, but I teach a History of Science course and we briefly discuss the secretive Pythagoreans. What great intellectual fun and right before Christmas! Thank you. :)

Spirit Whisperer profile image

Spirit Whisperer 4 years ago from Isle of Man Author

Thank you phdast7. I appreciate you taking the time to read my hub and I am glad you enjoyed it.

topquark profile image

topquark 4 years ago from UK

Yours is much more thorough than mine! I should have checked first. Nice job :)

Glenn Stok profile image

Glenn Stok 4 years ago from Long Island, NY

I enjoyed your Hub. It brought back memories of of my college days. I remember finding it fun trying to solve algorithms. I once had a final exam where the proof had no final outcome. I think it was proof by contradiction. Your Hub made me remember that and understand it better than I was taught back then.

A number of times in my life since college I would use algebra to figure something out. It's a useful tool for a different way of thinking. You might almost think of algebra as a language. Every language offers a unique way of thinking or carrying on a thought process. That's something I've recently been studying.

Your Hub was very enlightening. Voted up.

Spirit Whisperer profile image

Spirit Whisperer 4 years ago from Isle of Man Author

Thank you Glenn Stok for your thoughtful comment and I am always happy when what I write is of some use to those who read it. I admire your desire to keep learning and your openness to new entertain ideas outside your comfort zone.

kathryn1000 profile image

kathryn1000 4 years ago from London

I used to teach that.. good on you.

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