Numbers are used all around the world everywhere you look. From the amount of money in your bank account to the number of words in this newspaper to the distance between here and the sun, you can put a number on just about anything. But how high can you really go? Is there a limit?
Before asking such questions we must start with something simple. You, reading this column, most likely have 10 fingers. Not very much, but better than nothing. Add that to the number of toes you have, that’s 20. Twice as much! But wait … that’s counting each as just an additional one. We could keep going, but there is something we aren’t taking advantage of. Instead of adding once for each count up, we could repeatedly add the value to itself for each count up. That’s multiplication. 20 times 10 is 200, much bigger and faster. But this is still too slow.
We have to abuse the system that we use to make and write numbers to make bigger and bigger values. This is the study of googology, otherwise known as the study of big numbers. When you write a number, like 257 for example, each digit starting from the right side is actually a multiple of 10. In this pattern, seven is seven ones, five is five tens, two is two hundreds, and so on. We can use this to our advantage. Instead of each count up being just plus one, each count up could make our number 10 times bigger. We can write this as 10^n, where n is the number of times multiplication is repeated, and the 10 is the number we are multiplying by itself.
This is part of scientific notation, a way to show bigger numbers without writing all the digits out. As a result, the number we have after plugging in, say, 20, is a one followed by 20 zeros, or 10^20. Look at that! We already have a number 10 times more than the amount of grains of sand on earth! When n becomes 100, we get the number googol, or 10^100, which is what gave this field of math its name, originally being coined by a nine-year-old boy in the 1940s.
We could go on, happily adding zeros and making our numbers bigger and bigger. But looking back on this system, what we are doing is exponentiating, which is repeated multiplication, which is repeated addition. So what if we kept going? Repeating each step instead would be even faster.
I would like to introduce you to our best tool for the job: arrow notation. It works like this: 3^3 is 27, which is repeated multiplication. 3^^3 however, is 7,625,597,484,987, which is repeated exponentiation. This process is called making a power tower, as 3^^3 can be written as 3^3^3 or 3^27 We can continue this. 3^^^3 can be written as… Well, we can’t write that one, as it contains 7,625,597,484,987 ‘3^’s. 3^^^^3, as you might imagine, is way bigger. But now, we will make the biggest number yet: Graham’s number. 3^^^^3 becomes step g1. We take that number, and make g1 the number of arrows between the two threes. That number becomes g2, and so on, until we reach g64. That is one of the biggest numbers ever.