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Fifth grader Mario asks, "Is there a limit to the biggest or smallest Number?"


Good question, Mario! Sorry for the long delay in answering; we've been swamped with mail lately! The answer is: no, there's no limit. Numbers keep going on forever in all directions. You might not know the names of the numbers, but that doesn't mean they don't exist!

Big Numbers

What's the biggest number you can name? A million? A trillion? A quadrillion? A googol? (Not a Google, that's is a search engine!) A googol looks like this:

10 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000

It's a pretty big number! (And it's also called ten duotrigintillion, but I don't even know how to pronounce that, so I'm going to stick to saying "googol.") A googol is a one followed by one hundred zeroes!

But is that the largest number? It isn't! After all, I could add one to it, couldn't I? Then I would have:

10 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001

Better than that, I could multiply it by ten:

100 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000

Or by one hundred:

1 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000

Wait a minute! I could even multiply it by itself!

100 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000

I have no idea what to call that number (except I could call it "a googol times a googol"). But even though I don't have a fancy name for it, that doesn't mean it's not a real number, right? In fact, could I keep multiplying it by itself? Sure! I could even multiply it by itself a googol times, which would have so many zeroes that I couldn't possibly display it on this page.

No matter how big a number is, I can make it even bigger by tacking more zeroes on the end of it! So even though I don't know the names of all those numbers, there is no limit to how big a number can be!

Negative Numbers

When you said "how small" you might have meant a couple different things - you might have meant negative numbers, or you might have meant numbers close to zero. I'll address the question of how close you can get to zero in a second.

But first, in case you were talking about negative numbers, here's something you should know: For every positive number, there is a corresponding negative number. Take any positive number and stick a negative sign in front of it, and you've got the corresponding negative number. 7 and -7. 100 and -100. 1 googol and -1 googol. 

You get the idea, right? If there is no limit to how big the positive numbers can be, that means there's no limit to how big the negative numbers can be, either!

Close to Zero

But maybe, when you said "smallest number" you were talking about how close to zero you can get. And the answer is still "No, there is no limit."

For this, let's grab a calculator (if you have one handy) and check out a couple things. Calculate the following for me: 

  • 1/2 = ?
  • 1/3 = ?
  • 1/4 = ?
  • 1/5 = ?

These are called reciprocals. When you divide one by a number, you get that number's reciprocal. Many calculators have a reciprocal button; on my calculator it looks like this:

x-1

You should get something that looks like this:

  • 1/2 = 0.5
  • 1/3 = 0.33333333
  • 1/4 = 0.25
  • 1/5 = 0.2

Notice that the bigger the number, the closer its reciprocal is to zero. Let's try a few more numbers to make sure. This time we'll pick numbers that are powers of ten (a one, followed by some zeroes).

  • 1/10 = 0.1
  • 1/100 = 0.01
  • 1/1000 = 0.001
  • 1/10000 = 0.0001

Oh, I'm seeing a definite pattern here! The number of zeroes after the decimal is one less than the number of zeroes in the number we took the reciprocal of. What happens if we divide 1 by a googol?  Well, you don't want to try to enter a googol in your calculator, but we can use the pattern to figure out the answer. (That just shows that you're actually smarter than a calculator!)

1/googol = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 1

Wow! That number is tiny! But is it the tiniest? No! Because there are numbers even bigger than a googol that we could take the reciprocal of. And the bigger the number, the closer its reciprocal is to zero! And since there is no limit to how big numbers can be, there's no limit to how close to zero we can get, either.

Another way of looking at it: No matter how close to zero a number is, we can always make it smaller by sticking another zero after the decimal point.

Now, Mario, when you get into high school or college, you'll probably take a calculus class, and you'll start learning more about these kinds of ideas. You'll start hearing teachers talk about "limits" and you'll think, "But wait! Long ago, Professor Puzzler told me there were no limits!" Don't worry. Keep listening. You'll realize that the way your teacher is using the term "limit" is different from the way you used it back when you were in fifth grade. And you'll also realize that we just scratched the surface of these ideas. When it comes to understanding math...

...the sky is the limit!

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