ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

Proof by Contradiction

Updated on February 9, 2018

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 square 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


    0 of 8192 characters used
    Post Comment
    • kathryn1000 profile image


      6 years ago from London

      I used to teach that.. good on you.

    • Spirit Whisperer profile imageAUTHOR

      Xavier Nathan 

      7 years ago from Isle of Man

      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.

    • Glenn Stok profile image

      Glenn Stok 

      7 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.

    • topquark profile image


      7 years ago from UK

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

    • Spirit Whisperer profile imageAUTHOR

      Xavier Nathan 

      7 years ago from Isle of Man

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

    • phdast7 profile image

      Theresa Ast 

      7 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. :)

    • Mr. Happy profile image

      Mr. Happy 

      7 years ago from Toronto, Canada

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

    • Spirit Whisperer profile imageAUTHOR

      Xavier Nathan 

      7 years ago from Isle of Man

      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 

      7 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 imageAUTHOR

      Xavier Nathan 

      7 years ago from Isle of Man

      You are very welcome James. Thank you.

    • James A Watkins profile image

      James A Watkins 

      7 years ago from Chicago

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

    • Spirit Whisperer profile imageAUTHOR

      Xavier Nathan 

      7 years ago from Isle of Man

      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.

    • Genna East profile image

      Genna East 

      7 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 imageAUTHOR

      Xavier Nathan 

      7 years ago from Isle of Man

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

    • Spirit Whisperer profile imageAUTHOR

      Xavier Nathan 

      7 years ago from Isle of Man

      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!

    • Alastar Packer profile image

      Alastar Packer 

      7 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.

    • aravindb1982 profile image

      Aravind Balasubramanya 

      7 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

    • Spirit Whisperer profile imageAUTHOR

      Xavier Nathan 

      7 years ago from Isle of Man

      Nice one Bob!

    • diogenes profile image


      7 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


    This website uses cookies

    As a user in the EEA, your approval is needed on a few things. To provide a better website experience, uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

    For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at:

    Show Details
    HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
    LoginThis is necessary to sign in to the HubPages Service.
    Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
    AkismetThis is used to detect comment spam. (Privacy Policy)
    HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
    HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
    Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
    CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
    Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the or domains, for performance and efficiency reasons. (Privacy Policy)
    Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
    Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
    Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
    Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
    Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
    VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
    PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
    Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
    MavenThis supports the Maven widget and search functionality. (Privacy Policy)
    Google AdSenseThis is an ad network. (Privacy Policy)
    Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
    Index ExchangeThis is an ad network. (Privacy Policy)
    SovrnThis is an ad network. (Privacy Policy)
    Facebook AdsThis is an ad network. (Privacy Policy)
    Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
    AppNexusThis is an ad network. (Privacy Policy)
    OpenxThis is an ad network. (Privacy Policy)
    Rubicon ProjectThis is an ad network. (Privacy Policy)
    TripleLiftThis is an ad network. (Privacy Policy)
    Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
    Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
    Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
    Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
    ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
    Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)
    ClickscoThis is a data management platform studying reader behavior (Privacy Policy)