Graham’s Number Is Too Big to Explain How Big It Is

It needs its own special notation

You’ve probably heard of a googol before. It’s a pretty big number:

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 can be described as 1 followed by one hundred 0s. So, it has 101 digits. Sometimes for really big numbers, we can talk about how many digits the number of digits has. In this case it would be three.

Now Graham’s number, the one this article is all about, is so big that I can’t even tell you how many digits the number of digits has—or even how many digits the number of digits of the number of digits has. That is still too big a number for me to write out. Think about that for a second.

You might be thinking that we’re getting pretty close to infinity at this point and wondering why we don’t just call it infinity and get the article over with. But Graham’s number is not actually anywhere near close! Infinity is infinitely big afterall.

SEE ALSO: How Do We Know That There Is More Than One Type of Infinity?

For numbers as big as this, Mathematician Donald Knuth came up with a new type of notation that you have probably never heard of. It is called “up arrow” notation.

If you think about multiplication, it is really iterated addition: 4*3=4+4+4.

Exponentiation is really iterated multiplication: 4³=4*4*4. You may have seen this written as 4^3 before. The ^ is shorthand for a single arrow.

In up arrow notation, a double arrow (↑↑) denotes iterated exponentiation (titration): 4↑↑3=4^(4^4).

It continues one step further with a triple arrow that denotes iterated titration: 4↑↑↑3=4↑↑(4↑↑(4)).

As you can imagine, there are some cases where even the triple arrow isn’t enough to describe an incredibly huge number. Here, ↑ⁿ can be used where n indicates the number of arrows.

SEE ALSO: Math Trick: Find Any Square Root in 3 Seconds Flat

Now, to construct Graham’s number:

Start by thinking about g₁=3↑↑↑3.

g₂=3↑^g13. This means that there are g1 arrows between the two 3s.

g₃=3↑^g23. I’m going to assume that by this point you can see where this is going. There are now g₂ arrows between the 3s.

To get Graham’s number, we need to continue this process until we get to g₆₄. That’s just insanity! Nobody knows what the first digit of Graham’s number is, but the last digit is 7, in case it ever comes up in dinner conversation.

Why would anyone need a number like this you ask? Mathematician Ronald Graham came up with it when talking to another mathematician named Martin Gardner. Graham was working on a combinatorics question and found a proof involving an extremely large number. However, he found that Graham’s number was much easier to explain, and so he used it instead when talking to Gardner. According to John Baez in a Google+ post, "He said he made up Graham's number when talking to Martin Gardner! Why? Because it was simpler to explain than his actual upper bound — and bigger, so it's still an upper bound!"

Next time you feel overwhelmed by life and think that you have a gazillion things to do, be happy that it’s not a Graham’s number of things!

Watch the Numberphile video below to hear more about up arrow notation and the combinatorics problem that Gardner and Graham were solving: