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Big but not infinite

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

A question to start

Go to a beach and pick up a handful of sand. Look closely: it is made of tiny grains, each one smaller than a crumb. Now look along the beach. Sand, sand, sand, as far as you can see.

People sometimes say: "There are infinitely many grains of sand on the beach." Or: "There are infinitely many stars in the sky."

Are they right? Is a beach full of sand infinite? Or is it just very, very big? What is the difference?

The idea

Two kinds of "lots"

In this level you met things that never end: the counting numbers, a cookie halved forever in your imagination, a loop story. Those are infinite.

But some things are huge without being infinite. The word for "not infinite" is finite (say "FY-nite"). Finite means: it stops somewhere. If you had enough time and never made a mistake, you could count it all and get to the last one.

  • A bag of 20 marbles: finite. You can count them: 1, 2, 3, ..., 20. Then you are done.
  • All the people in your town: finite. It is a lot of counting, but it stops.
  • The counting numbers: infinite. You never get to the last one.

The test: could you finish counting?

Here is a test to tell "big" from "infinite". Ask:

If I counted one by one, very fast, and never got tired, would I ever finish?

If the answer is yes, the thing is finite, however big it is. If there is always one more, forever, it is infinite.

Let us try the test on the sand. A beach is a real thing. It has a start and an end. Each grain of sand takes up a little bit of space, and the beach only has so much space. So there is only so much sand. If you counted grain after grain, very fast, for a very long time, you would reach the last grain. The number of grains is enormous, but it is finite.

Counting big piles in groups

Counting one by one is slow. Here is how people count big piles: they make groups of ten.

onesa bundle of 1010 bundles = 100
  • 10 ones make a ten.
  • 10 tens make a hundred.
  • 10 hundreds make a thousand.
  • A thousand thousands make a million.

Each step is ten times bigger, or a thousand times bigger. With groups, people can write down huge numbers without counting every single thing. But notice: every one of these numbers stops. A million is a million, not one more.

How long is a million?

How long would it take to count to a million, saying one number every second, without stopping to eat or sleep?

A day has 86,400 seconds in it (24 hours of 60 minutes of 60 seconds). A million seconds is about 11 and a half days. So counting to a million, day and night, takes almost two weeks. (Really it takes much longer, because big numbers like "seven hundred and forty-two thousand, three hundred and sixteen" take more than a second to say!)

Counting to a million is hard, but it can be done. A million is finite.

And a whole lifetime? If you counted one number every second for 100 years, without ever stopping, you would only reach about 3 billion. Bigger numbers than that, like a trillion, you could never count to in your life. But they are still finite: they still stop. A trillion and one is bigger than a trillion.

The stars you can see

On a dark, clear night, far from city lights, the sky looks full of stars. It looks like there are too many to count.

But people have counted them. The stars bright enough to see with just your eyes, across the whole sky, number only a few thousand. From one place, on one night, you can see only some of those, because half the sky is below the ground and the stars near the ground are hard to see. A few thousand is a lot, but you could count them in about an hour, if they stood still and you did not lose your place.

With a telescope, you see many more stars. Scientists think our galaxy, the Milky Way, has more than a hundred billion stars. That is a number too big to count in a lifetime. But it is still finite.

An old sand puzzle

About 2,200 years ago, a Greek thinker called Archimedes heard people say that the grains of sand were too many to have a number. He disagreed. He wrote a short book showing that even if you filled the whole universe, as people then imagined it, with sand, the number of grains would be less than a number he could name: a 1 followed by 63 zeros.

His big point was this: "too many to count" is not the same as "infinite". Big things still have a number.

Sand on Mars

Mars is covered in dust and sand. In some places the wind piles the sand into huge dunes, bigger than houses, stretching for kilometres. The rovers driving on Mars roll over sand and send back pictures of single grains.

How many grains of sand are on Mars? Nobody knows the number, and it is an enormous number. But Mars is a ball of a certain size, so it holds only so much sand. The number is finite. If you could count very fast for a very, very long time, you would get to the end.

Worked examples

1. Finite or infinite? Decide for each one.

  • The children in your class: finite. You can count them.
  • The counting numbers: infinite. There is always one more.
  • The leaves on a big tree: finite. There are lots, but the tree is a real thing of a certain size.
  • The numbers you get by halving a pretend cookie forever: infinite. There is always one more halving.

2. Counting by tens. A jar holds beans. You put them into piles of 10. You make 7 piles, and 3 beans are left. How many beans? Count the tens: 10, 20, 30, 40, 50, 60, 70. Then 3 more: 73 beans. Grouping makes big counting easy, and it shows the number stops.

3. How long? You count one number every second. How long does it take to count to 60? 60 seconds, which is one minute. To 3600? 3600 seconds, which is one hour. Counting gets slow fast.

4. Is "a lot" infinite? Zara has a box of 1000 beads. She says "I have infinity beads!" She doesn't. 1000 is a lot, but it is finite: if she counted her beads, she would stop at 1000. And if she got one more bead, she would have 1001.

Try it

  1. Is the number of children in your school finite or infinite?
  2. Is the number of counting numbers finite or infinite?
  3. Ten bundles of ten sticks make how many sticks?
  4. How many tens are in a hundred? How many hundreds are in a thousand?
  5. You count one number every second. How many seconds to count to 120? How many minutes is that?
  6. Noah says: "There are so many stars, there must be infinitely many I can see." What would you tell Noah?
  7. Which is bigger: the number of grains of sand on a beach, or the counting numbers? Explain.
Answers
  1. Finite. A school has a certain number of children, and you can count them all.
  2. Infinite. There is always one more counting number, so the counting never stops.
  3. \(10\) tens make 100 sticks.
  4. There are 10 tens in a hundred, and 10 hundreds in a thousand.
  5. 120 seconds, which is 2 minutes (each minute is 60 seconds, and \(60 + 60 = 120\)).
  6. The stars you can see with your eyes number only a few thousand. That is a lot, but it is finite. People have counted them.
  7. The counting numbers. The sand on a beach is a huge but finite number: the counting stops at the last grain. The counting numbers never stop, so there are more of them than any finite number of things. In fact, whatever the number of sand grains is, that number itself is one of the counting numbers, and there are more counting numbers after it.

Common mistake

Saying "infinite" when you mean "very many". People often say "infinite" for things that are just hard to count: sand, stars, leaves. In math, infinite means never ends. Sand, stars and leaves all end, even if we do not know the exact number.

Thinking a number you cannot count to in your life is infinite. A trillion is too big to count to in your life, but it is still a number. One more than a trillion is bigger.

Where it leads