Skip to main content
Home
plus.maths.org

Secondary menu

  • My list
  • About Plus
  • Sponsors
  • Subscribe
  • Contact Us
  • Log in
  • Main navigation

  • Home
  • Articles
  • Collections
  • Podcasts
  • Maths in a minute
  • Puzzles
  • Videos
  • Topics and tags
  • For

    • cat icon
      Curiosity
    • newspaper icon
      Media
    • graduation icon
      Education
    • briefcase icon
      Policy

      Popular topics and tags

      Shapes

      • Geometry
      • Vectors and matrices
      • Topology
      • Networks and graph theory
      • Fractals

      Numbers

      • Number theory
      • Arithmetic
      • Prime numbers
      • Fermat's last theorem
      • Cryptography

      Computing and information

      • Quantum computing
      • Complexity
      • Information theory
      • Artificial intelligence and machine learning
      • Algorithm

      Data and probability

      • Statistics
      • Probability and uncertainty
      • Randomness

      Abstract structures

      • Symmetry
      • Algebra and group theory
      • Vectors and matrices

      Physics

      • Fluid dynamics
      • Quantum physics
      • General relativity, gravity and black holes
      • Entropy and thermodynamics
      • String theory and quantum gravity

      Arts, humanities and sport

      • History and philosophy of mathematics
      • Art and Music
      • Language
      • Sport

      Logic, proof and strategy

      • Logic
      • Proof
      • Game theory

      Calculus and analysis

      • Differential equations
      • Calculus

      Towards applications

      • Mathematical modelling
      • Dynamical systems and Chaos

      Applications

      • Medicine and health
      • Epidemiology
      • Biology
      • Economics and finance
      • Engineering and architecture
      • Weather forecasting
      • Climate change

      Understanding of mathematics

      • Public understanding of mathematics
      • Education

      Get your maths quickly

      • Maths in a minute

      Main menu

    • Home
    • Articles
    • Collections
    • Podcasts
    • Maths in a minute
    • Puzzles
    • Videos
    • Topics and tags
    • Audiences

      • cat icon
        Curiosity
      • newspaper icon
        Media
      • graduation icon
        Education
      • briefcase icon
        Policy

      Secondary menu

    • My list
    • About Plus
    • Sponsors
    • Subscribe
    • Contact Us
    • Log in
    • Maths in a minute: Counting numbers

      25 February, 2014
      Cantor

      Georg Cantor

      Are there more irrational numbers than rational numbers, or more rational numbers than irrational numbers? Well, there are infinitely many of both, so the question doesn't make sense. It turns out, however, that the set of rational numbers is infinite in a very different way from the set of irrational numbers.

      As we saw here, the rational numbers (those that can be written as fractions) can be lined up one by one and labelled 1, 2, 3, 4, etc. They form what mathematicians call a countable infinity. The same isn't true of the irrational numbers (those that cannot be written as fractions): they form an uncountably infinite set. In 1873 the mathematician Georg Cantor came up with a beautiful and elegant proof of this fact. First notice that when we put the rational numbers and the irrational numbers together we get all the real numbers: each number on the line is either rational or irrational. If the irrational numbers were countable, just as the rationals are, then the real numbers would be countable too — it's not too hard to convince yourself of that.

      So let's suppose the real numbers are countable, so that we can make a list of them, for example 1. 0.1234567… 2. 1.4367892… 3. 2.3987851… 4. 3.7891234… 5. 4.1415695… and so on, with every real number occurring somewhere in the infinite list. Now take the first digit after the decimal point of the first number, the second digit after the decimal point of the second number, the third digit after the decimal point of the third number, and so on, to get a new number 0.13816…. Now change each digit of this new number, for example by adding 1. This gives the new number 0.24927…. This new number is not the same as the first number on the list, because their first decimal digits are different. Neither is it the same as the second number on the list, because their second decimal digits are different. Carrying on like this shows that the new number is different from every single number on the list, and so it cannot appear anywhere in the list.

      But we started with the assumption that every real number was on the list! The only way to avoid this contradiction is to admit that the assumption that the real numbers are countable is false. And this then also implies that the irrational numbers are uncountable.

      It's easy to see that an uncountable infinity is "bigger" than a countable one. An uncountable infinity can form a continuum, such as the number line, in a way that a countable infinity can't. Cantor went on to define all sorts of other infinities too, one bigger than the other, with the countable infinity at the bottom of the hierarchy. When he first published these ideas, Cantor faced strong opposition from some of his colleagues. One of them, Henri Poincaré, described Cantor's ideas as a "grave disease" and another, Leopold Kronecker, went so far as to denounce Cantor as a "scientific charlatan" and "corrupter of youth". Cantor suffered severe mental health problems which may have resulted in part from the rejection his work had met with. But we now know that his work had simply come too soon: 150 years on, Cantor's ideas form a central pillar of mathematics and many of his results can be found in standard textbooks.

      See our infinity page to find out more about this and other things to do with infinity.

      Read more about...
      infinity
      what is infinity
      Maths in a minute
      • Log in or register to post comments

      Anonymous

      2 April 2015

      Permalink

      In our base-ten system, there is a range of square numbers, 1,4,9,16,25 etc. Of these very many consist of a mixture of odd and even digits. Is this subset countable? There is a much smaller subset, 4, 64, 400, 484, etc, which consists of only even digits. Is this countable? And there is a very special subset which is definitely countable, consisting of the squares with only odd digits - 1 and 9. As far as I can see, there is no relationship between these subsets, except that they add together to the "master"set.

      • Log in or register to post comments

      Leslie.Green

      10 September 2018

      In reply to Countable infinities by Anonymous

      Permalink

      According to mathematicians who follow Cantor's idiocy, the set of all square numbers is the same size as the set of counting numbers. In fact they go even further and declare that the set of rational numbers is the same size too. They have a fundamental problem with their definition of the infinity symbol. By declaring that twice infinity is the SAME infinity, they can't actually compare the sizes of different infinite values. Consider some value N which we will soon increase towards infinity. N² is much bigger than N, which is bigger than sqrt(N), which is bigger than log10(N). Put N = 10^10. N² has 19 digits, N has 10 digits, log10(N)=10. These values do no get closer as we approach infinity, they diverge more. It is useful to compare different infinites and they have set themselves up to fail.

      It's time we ditched this Cantor-induced insanity: http://lesliegreen.byethost3.com/articles/new_maths.pdf

      • Log in or register to post comments

      max

      3 October 2022

      In reply to sizes of infinities by Leslie.Green

      Permalink

      I looked at your enumeration of the continuum and it appears that you have enumerated only the terminating decimals (equivalently decimals that end in infinite trailing zeros or 9's). You have not enumerated all of the real numbers. If we focus on the enumeration of [0, 1) , how does your list produce the real number 1/3 = 0.333...? This real number is not a terminating decimal (nor does it end in infinite trailing zeros or nines). It will never be in the list - even if you are given an infinite amount of time to list numbers. Said another way, the real number 1/3 is not in the 'eventually-we-will-get-there' list, since your enumeration or listing method has no way of producing 0.333..., it is not in the list at all. A similar argument can be made for √2, but I will use 1/3 as a simpler counterexample.

      • Log in or register to post comments

      max

      3 October 2022

      In reply to sizes of infinities by Leslie.Green

      Permalink

      I looked at your enumeration of the continuum and it appears that you have enumerated only the terminating decimals (equivalently decimals that end in infinite trailing zeros or 9's). You have not enumerated all of the real numbers. If we focus on the enumeration of [0, 1) , how does your list produce the real number 1/3 = 0.333...? This real number is not a terminating decimal (nor does it end in infinite trailing zeros or nines). It will never be in the list - even if you are given an infinite amount of time to list numbers. Said another way, the real number 1/3 is not in the 'eventually-we-will-get-there' list, since your enumeration or listing method has no way of producing 0.333..., it is not in the list at all. A similar argument can be made for √2, but I will use 1/3 as a simpler counterexample.

      • Log in or register to post comments

      math.nights

      11 March 2016

      Permalink

      To Arabic: https://goo.gl/LTmQUn

      • Log in or register to post comments

      Read more about...

      infinity
      what is infinity
      Maths in a minute
      University of Cambridge logo

      Plus Magazine is part of the family of activities in the Millennium Mathematics Project.
      Copyright © 1997 - 2025. University of Cambridge. All rights reserved.

      Terms