Graham's Number

My cousin, when he was a toddler, would often combine the biggest number he would know and sandwich it together thinking it would be an even bigger number. I did this in my elementary years as well.

"...gazillion billion billion million trillion..."

Try to think of a really big number. The biggest number you can think of. Add however many zeroes you want but I bet you it will never be as big as something in mathematics called, "Graham's Number". Graham's number is a super big natural number designed by mathematician Ronald Graham. It is soooo big that if one was to write every single digit, it would not even fit within our universe. The scientific notation does not even come close to capturing the immensity of Graham's number. Instead, we use the "up-arrow" as the new system of notation.

An arrow between two numbers means that the first number is multiplied by itself the second number of times. 

So a 3 ↑ 3 is written as 3 * 3 * 3 = 27. 3 ↑↑ 3 is written as 3 ↑ (3 ↑ 3) so it is 3 multiplied by itself 27 times. 3 ↑↑↑ 3 has a total of 7625597484987 3s. You can see just how massive the number becomes. 

We can then begin to layer this number. 

g1 = 3 ↑↑↑↑ 3

g2 = 3 ↑↑↑↑...↑↑↑↑ 3, where there are g1 arrows.

g3 = 3 ↑↑↑↑↑ ... ↑↑↑↑↑ 3, where there are g2 arrows.

So what is Graham's Number?

Graham's number is g64, where there are g63 arrows! 

In the case of pi, the constant has no end so we often write it as, "3.1415926...". Graham's number is so huge we only know the last digits, "...262464195387".

So the next time you get into an argument with a five-year old, just say Graham's number. Game over.