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Play: Does it ever end?

Infinity and strange math · step 1 of 5

A question to start

What is the biggest number you know? A million? A billion? A googol, which is a 1 followed by a hundred zeros?

Whatever number you say, a friend can say "that number, plus one", and win. And then you can add one to theirs. The game never ends. There is no biggest number. The numbers go on and on, forever. Mathematicians have a word for "going on forever": infinite. And the idea of something endless is called infinity.

The idea

Infinity is not a number at the end

It is tempting to think of infinity as a giant number sitting at the very end of the number line. But there is no end. Infinity is more like a direction, or a promise: you can always keep going. When we say "there are infinitely many numbers", we mean you could count for your whole life, and the lives of everyone after you, and never run out.

Lots of things are infinite in this way.

  • The counting numbers 1, 2, 3, 4, ... never stop.
  • The even numbers 2, 4, 6, 8, ... never stop either. Neither do the odd numbers.
  • Stand between two mirrors facing each other. You see a copy of yourself, and a copy of the copy, and a copy of that, getting smaller and smaller. In a perfect mirror the copies would never end.

Getting closer and closer, forever

Here is a different kind of endless. Stand 2 metres from a wall. Walk half the way, so you are 1 metre away. Walk half of what is left: now half a metre. Then half again, and again.

halfquarter8th

Each step takes you closer, and each step is smaller than the one before. On paper, you never quite reach the wall: there is always a little bit left, half of which you walk next. And yet all those steps together fit inside 2 metres. Infinitely many pieces, but not an infinite distance. Hold on to this thought; it comes back in the Build step, where it becomes one of the most useful ideas in mathematics.

A strip of paper with only one side

A sheet of paper has two sides, a front and a back. So does a ring made by taping the ends of a strip together. But a small twist makes something strange.

AB BA give one end a half twist, then tape A to A and B to B

This shape is a Möbius strip (say "MER-bee-us"), named after the German mathematician August Möbius, who studied it in the 1800s. It looks like an ordinary twisted loop, but it has only one side and only one edge. You can check this yourself with paper, tape and a crayon.

Worked example

Make a Möbius strip and test it.

  1. Cut a strip of paper about 3 centimetres wide and as long as your paper allows.
  2. Bring the two ends together as if to make a ring, but before you tape them, turn one end over (half a turn). Tape the ends.
  3. Put a crayon in the middle of the strip and draw a line along it, without lifting the crayon. Keep going.
  4. Your line comes back to where it started, but only after going round twice as far as you might expect, and the line is now on "both sides" of the paper. There was only ever one side. An ant could walk over every part of the strip without crossing an edge.
  5. Now run your finger along the edge. It also comes back to its start only after going round twice. There is one edge, not two.

Now cut it. Cut along the line you drew, all the way round. With an ordinary ring you would get two thin rings. With the Möbius strip you get one long loop, twice as long. (This new loop has two sides; the cutting changed it.) Try it and see. Then try cutting a new Möbius strip one third of the way in from the edge instead of down the middle, and be ready for another surprise: you get two loops, linked together like a chain.

Strange shapes like this belong to a part of mathematics called topology, which studies what stays the same when you bend and stretch things without tearing them.

Try it

  1. Is there a biggest even number? Explain how you know.
  2. Mia has a chocolate bar. On day 1 she eats half of it. Each day after, she eats half of what is left. Will she ever finish it, if the chocolate can be cut as small as she likes?
  3. Make an ordinary ring with no twist, draw a line down the middle and cut along it. What do you get? How is it different from cutting the Möbius strip?
  4. How many sides and how many edges does a Möbius strip have? How many does an ordinary ring have?
  5. Which do you think there are more of: counting numbers (1, 2, 3, ...) or even numbers (2, 4, 6, ...)? Try pairing them up: 1 with 2, 2 with 4, 3 with 6, ... Does anyone get left without a partner?
Answers
  1. No. Whatever even number you pick, adding 2 gives a bigger even number. So no even number can be the biggest.
  2. No, she never finishes. Each day there is still some left, and she only eats half of it. But the bit left gets tiny very fast: after 10 days, less than one thousandth of the bar is left.
  3. You get two separate thin rings. The Möbius strip gives one long loop. The half twist makes all the difference.
  4. A Möbius strip has 1 side and 1 edge. An ordinary ring has 2 sides (inside and outside) and 2 edges (top and bottom).
  5. Every counting number gets an even partner (its double), and every even number gets a counting-number partner (its half). Nobody is left out. So, strange as it sounds, there are just as many even numbers as counting numbers. This is the puzzle of the next step.

Common mistake

Treating infinity like an ordinary number. "Infinity plus one is bigger than infinity" sounds right, but there is no last number to add one to. Infinity is not a place you can reach on the number line, so it does not follow the usual rules of adding and taking away. The next steps show what rules it does follow.

Where it leads