### What is the biggest number?

School: Thomas Jefferson Elementary School
Teacher: Mrs. Matyas
Hobbies/Interests: Playing sports

Riley Assistant Professor

Research area: Geometric group theory
Family: 1 spouse and 2 cats
Interests/hobbies: The environment, cards, movies and Do-It-Yourself projects

People have been asking this very question for thousands of years. We can find the answer to your question by playing a game.  For this game, I'll give you a number.  I'll give you as big a number as I can thing of.  Then your job will be to see if you can find a bigger number.  "I'll start with the number 1,000," I say.  You can think of a bigger number though, so you respond, "1,000 is less than 1,001."  So I give you the number 100,000,000.  That's a pretty big number, but you are pretty clever, and you quickly find the number 100,000,001 is bigger. "But what about the number 1,000,000,000,000?"  To which you quickly reply, "1,000,000,000,001 is even bigger!"  "Well," I say, "I appear to be losing this game.  What about a 1 followed by a zillion zeros?"  You respond, "A bigger number is: a 1 followed by a billion zeros, plus 1!"

No matter what number I give you, you can always take the number and add one to it.  If there were a biggest number, you could find a bigger number by just adding 1 to it, but that contradicts the fact that I was supposed to give you the biggest number.  That proves there can be no biggest number!

What we've just figured out with our game is an example of mathematical proof.  Mathematicians have a special name for something that you can prove must be true: it's called a theorem.  We have proven that "there is no biggest number".

Finding anything that can show is absolutely 100% true is very hard. Even with the proof we just found, we had to start with an assumption: that we are able to add 1 to any number.  Basic assumptions like this are called axioms, and every science has them, even math!

While there is no biggest number, there are some very big numbers out there.  The number with a 1 followed by a hundred zeros is called a googol (the search engine Google was named after this number).  A googol is already bigger than the number of atoms in the universe (which is around a 1 followed by 80 zeros).  A bigger number is 1 followed by a googol zeros is called a googolplex.  And a 1 followed by a googoplex zeros is a googolplexian.  Maybe you can think of a name for a 1 followed by a googolplexian zeros?

A googoplexian is unbelievable enormous, and may not be a number that we can use to count anything, but it still exists and it is still not infinite! Infinity comes from the Latin word for "unbounded", and refers to a quantity that never ends.  Infinity itself is a concept, but it is not an actual number, so our theorem is still true: "there is not biggest number," not even infinity.

Last Updated: 9/18/13