Inverse Functions, Iterated Logs, and Persistence
A bell curve meme, enormous numbers, and casting out nines
The inverse of an approximation is an approximation of the inverse. With some caveats. For example, the inverse of a great approximation might be a mediocre approximation if the problem is sensitive (“poorly conditioned”).
This post on inverses and approximations leads into why it’s true that
dx/dy = 1/(dy/dx)
even though the expressions above are not fractions.
One use of logarithms is to bring huge numbers down to an understandable scale. Sometimes taking the log once isn’t enough, and one way to get a grasp on enormous numbers is to count how many times you have to take the log before you get a number below 1.
This post applies this idea to some big numbers, such as a googolplex and the largest known prime.
One way to test whether a number is divisible by 9 is to add all its digits together. If the digit sum is a big number, you can take the digit sum again. Eventually you’ll get a single number. The number of times you have to repeat the process in order to get a single digit is called the “additive persistence” of a number.
You can also repeatedly take the product of digits until you get a single digit number, and the number of steps is called the “multiplicative persistence” of that number. Multiplicative persistence turns out to be more interesting that additive persistence. More on that here.
Enjoy!