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Going round and round

Infinity and strange math · level 1, lesson 3 of 6

A question to start

A little ant is walking along the rim of a round plate. It walks and walks. It never turns back. It never stops.

Where does the ant reach the end of the rim?

Now think about a stick. An ant walking along a stick soon gets to the end, and has to stop or turn round. What is different about the plate? Think about it, then read on.

The idea

Lines with ends

Take a stick, a pencil or a piece of string pulled straight. You can point to its ends: one here, one there. A straight piece of line like this has two ends.

2 ends no ends

A circle has no end

Now bend the string round and tie its two ends together, so it makes a circle (a perfectly round loop). Where are the ends now? They are gone! The knot is just one more spot on the loop. The ant can walk past it without even noticing.

A circle has no end. It also has no start. You can start walking anywhere on a circle, and every spot is just like every other spot. If you walk and walk in the same direction, you come back to where you started. Then you can keep going, round and round, forever.

That is why the ant on the plate never reaches an end. There is no end to reach.

No end, but not big

Here is a puzzle that tricks lots of people. The ant can walk round the plate forever. Does that make the plate infinitely big?

No! The plate is small. It fits on a table. The rim of the plate is short: you could measure it with a tape measure. The ant walks forever only because it goes round and round the same path, again and again.

So there are two different ideas here, and it is important to keep them apart:

  • No end: you can keep going without ever reaching an edge. A circle has no end.
  • Infinitely long: the path itself never stops, and you never come back to where you were. The number line is like this.

A circle has no end, but it is not infinitely long. The number line from the first lesson is infinitely long. These are two kinds of "never ending". Mathematicians are very careful about which one they mean.

The race track

A race track is a loop. Runners start at the start line, run all the way round, and come back to the same line. Going round once is called one lap.

start and finish line

A race has to say how many laps, like "4 laps". Why? Because the track itself never tells you to stop! There is no end to run into. If nobody said "4 laps", you could run lap after lap after lap. People have to choose when to stop.

And here is something lovely: the number of laps can keep going up: 1 lap, 2 laps, 3 laps, ... That is the counting numbers again, and they never end. So a short loop and the endless counting numbers work together: a small path, gone round forever, gives a count that never stops.

Things that come back round

Lots of things in life go round in a loop and come back to the start.

  • The clock. The hour hand goes 1, 2, 3, ... up to 12, and then starts again at 1. The clock never runs out of hours.
  • The days of the week. Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday, and then Monday again.
  • The seasons. Spring, summer, autumn, winter, and then spring again.
  • Planets. Earth goes round the Sun once a year. Mars goes round the Sun too, on a bigger loop, and takes almost two Earth years to go round once. The planets have been going round and round for a very, very long time.

Songs that go round

Some songs come back to their start, so you can sing them round and round. Row, row, row your boat is a song like this. It is often sung as a round: one group starts, then a second group starts the same song a little later, then a third. When each group gets to the end, it can start again at the beginning. The singing could go on and on, until someone decides to stop.

A song like this is a loop, just like a circle. The end joins the beginning, so there is no place where the song has to finish.

Worked examples

1. The ant's lap. An ant's loop takes 10 steps to go round once. The ant walks 25 steps. How many laps is that, and where does it end up?

10 steps is one lap. 20 steps is two laps. That leaves \(25 - 20 = 5\) more steps. So the ant goes round 2 whole laps, then 5 steps more. That is halfway round the third lap, because 5 is half of 10.

2. The clock. It is 9 o'clock. What time will it be in 5 hours?

Count on: 10, 11, 12, then the clock starts again: 1, 2. That is 5 hours. It will be 2 o'clock. The clock went past 12 and came back round.

3. The days. Today is Friday. What day is it in 3 days?

Count on: Saturday, Sunday, Monday. It will be Monday. The week went round to the start.

4. Seven days later. Today is Tuesday. What day is it in 7 days?

Count on 7 days: Wednesday, Thursday, Friday, Saturday, Sunday, Monday, Tuesday. After 7 days you are back where you started, because the week has 7 days. One full trip round the loop brings you home.

5. Endless laps. Ria runs round the track 3 times. Raj runs round 4 times. Who ran further? Raj, because each lap is the same length and he ran one more. The track did not get any longer, but the count of laps did.

Try it

  1. Does a circle have a start? Does it have an end?
  2. A pencil has how many ends? A rubber band has how many ends?
  3. It is 11 o'clock. What time will it be in 2 hours?
  4. Today is Saturday. What day will it be in 2 days?
  5. A track takes 4 minutes to run round once. How many laps can you run in 12 minutes?
  6. An ant's loop takes 6 steps. The ant walks 13 steps. How many whole laps does it walk, and how many steps more?
  7. Sara says: "A circle goes on forever, so it must be infinitely big." What would you tell Sara?
  8. Can you think of something else that goes round in a loop and comes back to its start?
Answers
  1. No start and no end. Every spot on a circle is just like every other spot. You can start anywhere.
  2. A pencil has 2 ends. A rubber band has no ends: it is a loop.
  3. 12, then 1. It will be 1 o'clock.
  4. Sunday, then Monday. It will be Monday.
  5. 4 minutes is one lap, 8 minutes is two laps, 12 minutes is 3 laps.
  6. 6 steps is one lap, 12 steps is two laps. \(13 - 12 = 1\). So 2 whole laps and 1 step more.
  7. A circle has no end, but it is not infinitely big. You go on forever only by going round the same short path again and again. A small plate has a circle round its rim, and the plate is small.
  8. Many answers are right: a merry-go-round, a wheel turning, the months of the year, a toy train on a round track, a hamster wheel, day and night.

Common mistake

Mixing up "no end" and "infinitely long". A circle has no end, but you can measure all the way round it. A number line is infinitely long: it never comes back. Before you say "it goes on forever", ask: does it come back round, or does it go on and on without returning?

Forgetting that the clock and the week start again. After 12 o'clock comes 1 o'clock, not 13 o'clock. After Sunday comes Monday.

Where it leads