# Which is larger 3^3^3^3 or 100 billion factorial?

Yesterday we ended up encountering some large numbers when asked some questions about patterns on an Othello board:

Looking at patterns on an Othello board

Today I thought it would be fun to revisit large numbers a little bit, so we looked at a few really large numbers on Wolfram Alpha and talked informally about logarithms. The spirit here is hopefully along the lines of Jordan Ellenberg’s description of logarithms in How not to be Wrong – as I was thinking about what to talk about today, his discussion of the “flog-arithem” inspired me to try this project.

First up, exploring logarithms and factorials and seeing what patterns we could find. A few Fibonacci numbers showed up in the beginning, but that pattern didn’t continue – wouldn’t it be cool if the number of digits in n! was related to the nth Fibonacci number!?! We did see the connection between the number of digits and the base 10 logarithm, though.

Next up we started looking at some large factorials and then moved on to other large numbers. We also ended up stumbling on some interesting properties of the logarithm function sort of by accident. At the end we looked at a pretty neat problem: which was larger 3^3^3^3 or 100 billion factorial?

As I was writing this up, both Dan Anderson and Burnheart123 on twitter realized there was an easy way to estimate the number of digits of 100 billion factorial – wish I would have realized their point when I was talking with the boys:

In the last video, the kids asked me about logs with bases other than 10. That led to a fun discussion about logs with a few other bases and we eventually arrived at base e. One fun surprise in this discussion is that 100 billion factorial has roughly the same number of digits in binary as 3^3^3^3 does in base 10.

The last bit of our talk was about the relationship between logs and prime numbers. This is the part specifically inspired by Jordan Ellenberg’s discussion in How not to be Wrong. Even if we can’t go into any details that he does his book, it is neat to show the kids this surprising connection.

Also, sorry here – the camera seems to have cut off in the middle of the discussion. In the part that got cut off, we checked the formula for approximating the number of twin primes.

So, a fun little discussion today piggy backing off of yesterday’s discussion about patterns on an Othello board. I’m also really happy that I can share some of Ellenberg’s discussion / ideas about logs and primes with the boys (even if that sharing is very informal). Also happy to have stumbled on the fun question about 3^3^3^3 and 100 billion factorial.

## 2 thoughts on “Which is larger 3^3^3^3 or 100 billion factorial?”

1. Somewhat inspired by this, my middlest was teasing our oldest by asking how large 3 triple arrow 3 is. In the middle of the taunting, he suddenly looked worried, came to me, and whispered, its about 7.6 trillion, right? We talked for a minute, realized the mistake, then went right back to “you don’t even know 3 double arrow 3 (repeated 3^3 times)”

1. You might enjoy some of our prior projects on Graham’s number. Challenging to talk through with kids, but also pretty fun.