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Googol and googolplex

Infinity and strange math · level 2, lesson 4 of 6

A question to start

Here is a number with a funny name: a googol. It is a 1 followed by one hundred zeros.

That is more than the number of grains of sand on every beach on Earth. It is more than the number of drops of water in all the oceans. Scientists think it is even more than the number of atoms (the tiny bits everything is made of) in all the stars and planets we can see, put together.

So is a googol infinite? And if not, can you think of a number that is even bigger? (You already know a trick for that from level 1.)

The idea

Zeros make numbers grow fast

Every time you put a 0 on the end of a whole number, it becomes ten times as big.

  • 1 with one zero is 10 (ten).
  • 1 with two zeros is 100 (a hundred): ten tens.
  • 1 with three zeros is 1000 (a thousand): ten hundreds.

Why does a zero make it ten times bigger? Think of place value. In 30, the 3 is in the tens place: 3 tens. Put a zero on the end and you get 300, where the 3 has moved to the hundreds place: 3 hundreds. Every digit slides one place to the left, and each place is worth ten times the place before. So the whole number is ten times as big.

110100 100010000 each step up:ten times as big

The zeros ladder

Here are some numbers with names that you may have heard.

Name Written out Zeros after the 1
ten 10 1
hundred 100 2
thousand 1,000 3
million 1,000,000 6
billion 1,000,000,000 9
trillion 1,000,000,000,000 12

The commas are there to help your eyes: they split the zeros into groups of three. A million is a thousand thousands, so it has \(3 + 3 = 6\) zeros. A billion is a thousand millions: \(6 + 3 = 9\) zeros.

Meet the googol

A googol has 100 zeros. Here it is, written out: a 1, then ten rows of ten zeros.

1
0000000000
0000000000
0000000000
0000000000
0000000000
0000000000
0000000000
0000000000
0000000000
0000000000

It fits on a page easily. You could write it in a minute or two. But it is unbelievably big. A trillion has only 12 zeros. A googol has 100. Every one of those 88 extra zeros makes the number ten times bigger again.

The story goes that the name was made up about a hundred years ago by a nine-year-old boy. His uncle, an American mathematician called Edward Kasner, asked him for a name for a 1 followed by a hundred zeros, and the boy said "googol". Kasner put the name in a book, and it stuck.

Grown-ups sometimes write a googol in a short way as \(10^{100}\): "ten, multiplied by itself, with a hundred tens". The little 100 counts the zeros. You do not need this short way yet, but now you know what it means if you see it.

Bigger than the universe (in a way)

Scientists have estimated how many atoms there are in the whole part of the universe we can see: all the stars, all the planets, all the gas between them. Their best guess is a number with about 80 zeros after the 1. Nobody has counted, of course; it is a careful estimate.

A googol has 100 zeros: about 20 more zeros than that. Each extra zero is ten times as big. So a googol is far, far bigger than the number of atoms we think there are. If every atom in all the stars you can see were a sticker, you would not have nearly enough stickers to count to a googol.

How long to count to a googol?

The universe is very old: scientists think about 13.8 billion years. Counting one number every second since the universe began, you would have counted to a number with only 18 digits by now. A googol has 101 digits (the 1 and its hundred zeros). So counting to a googol would take far, far longer than the universe has existed.

But a googol is still a number

Here is the important part. A googol is enormous, but it is finite. It is one particular number. It ends. And you already know how to beat it:

A googol and one is bigger than a googol.

A googol is not even close to infinity. Infinity is not "a really big number". It is the fact that the numbers never end, and a googol is just one of them, with lots more numbers after it.

Googolplex

Here is a much, much bigger number. A googolplex is a 1 followed by a googol zeros. Not a hundred zeros: a googol of them.

Can you write a googolplex down? You would need to write a googol zeros. Even if you wrote each zero on a single atom, there are not enough atoms in all the stars and planets we can see: about 80 zeros' worth of atoms, but you need a googol of zeros. Nobody can ever write a googolplex out in full. There is not enough stuff in the universe.

And yet a googolplex is still finite. It is a real number, with a clear rule that says exactly what it is. A googolplex and one is bigger. Infinity is still infinitely far away.

Worked examples

1. How many zeros? A thousand thousands is a million. Count the zeros: a thousand has 3, so a thousand thousands has \(3 + 3 = 6\) zeros: 1,000,000.

2. Ten times a googol. A googol has 100 zeros. Ten times a googol puts one more zero on the end, so it has 101 zeros. It is a 1 followed by 101 zeros.

3. How many digits? A googol is a 1 followed by 100 zeros. How many digits is that? The 1 is one digit and the zeros are 100 more: \(1 + 100 = 101\) digits.

4. Which is bigger? A trillion trillions, or a googol? A trillion has 12 zeros, so a trillion trillions has \(12 + 12 = 24\) zeros. A googol has 100 zeros. More zeros (with a 1 in front) means bigger, so a googol is much bigger.

5. Writing time. You write one zero every second. How long to write a googol? 100 zeros take 100 seconds, which is a minute and 40 seconds. How long for a googolplex? A googol seconds: much longer than the age of the universe.

Try it

  1. How many zeros does a hundred have? A million?
  2. What do you get if you put a zero on the end of 45? How many times bigger is it?
  3. A billion is a thousand millions. How many zeros does it have? Explain using the zeros of a million and a thousand.
  4. What is one more than a googol? Is it bigger than a googol?
  5. A googol times ten times ten: how many zeros?
  6. Kai says: "A googolplex is so big that it must be infinity." Explain why Kai is wrong.
  7. Which number has more zeros: a million millions, or a billion? How many zeros does each have?
Answers
  1. A hundred has 2 zeros. A million has 6 zeros.
  2. 450. It is ten times as big: the 4 moved from the tens place to the hundreds place, and the 5 from the ones place to the tens place.
  3. A million has 6 zeros and a thousand has 3, so a thousand millions has \(6 + 3 = 9\) zeros: 1,000,000,000.
  4. A googol and one is a 1, then 99 zeros, then a 1 at the end. It is bigger than a googol, by one. So a googol is not the biggest number.
  5. Each "times ten" adds one zero: \(100 + 1 + 1 = 102\) zeros.
  6. A googolplex is a particular number with a clear rule (a 1 followed by a googol zeros). It is finite: a googolplex and one is bigger. Infinity is not a number at all; it is the fact that the numbers never end.
  7. A million millions has \(6 + 6 = 12\) zeros (it is a trillion). A billion has 9 zeros. So a million millions has more zeros, and it is the bigger number.

Common mistake

Thinking a googol (or a googolplex) is infinity. They are huge, but each one is a number with a next number after it. Infinity is not at the end of the ladder of big numbers, because the ladder has no end.

Adding instead of putting zeros on. Ten times a googol is a 1 with 101 zeros, not "a googol and ten". Putting a zero on the end multiplies by ten.

Where it leads