Dave Hulbert's Today I Learned (TIL)

The Unimaginable Sizes of Graham's Number and Other Large Numbers

I was talking to my kids about large numbers like a googol (10^100) and a googolplex. I knew bits and pieces about bigger number but not enough to confidently explain it until now.

Addition, multiplication, iteration and tetration

First, let's see how operations are just repeats of the previous operation.

Writing a Googolplex

A googolplex is an enormous number defined as 10^10^100. That's a 1 followed by a googol (10^100) zeros! It's impossible to write down in normal notation due to its sheer size.

Beyond Exponential Notation

There are numbers so big that they can't be expressed using exponential notation. These require other methods, such as Knuth's up-arrow notation, which allows us to describe numbers that are exponentially bigger than a googolplex.

You can see how ↑ notation can quickly lead to large numbers, even with just 4 of them.

Graham's Number

Then there's Graham's number, perhaps the most famous of very very large numbers. Here's how it builds:

We start off with 3↑↑↑↑3. Remember that this number takes 30,000 years to describe the the height of the power tower: the power tower isn't just billions or googols tall, it's more than the number of atoms in the universe tall.

We call this g₁.

The next level is g₂. This is 3, with g₁ up arrows then a other 3. You saw how quickly the numbers grew when we went from 2, to 3, to 4 up arrows. Now we're at an impossibility large number of up arrows.

This process continues, where we take a number that we can't even describe with up arrows, then use that many up arrows between two 3's.

We keep doing this 64 times to get Graham's number.