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An amazing approximation to e
Euler’s number, e, is an important number in mathematics. It is an irrational number, so it cannot be expressed exactly as a fraction or decimal number. But there are many ways to calculate its approximate value, as accurately as you like.
In this article, we will look at a clever (and amazingly accurate) approximation that emerged in response to a simple mathematical challenge.
My book Calculus is available on Amazon.
The challenge
The challenge appeared in 2004 on Eric S. Friedman’s personal website, Math Magic. The site is no longer active, but an archive can be found on github.
The problem was stated as:
This month’s problem is to approximate famous mathematical constants using only the first k digits (each used exactly once) and the mathematical symbols + — × / ( ) and ^. For each k, what is the best approximation you can get to the following constants?
The problem listed several mathematical constants. In a little more detail:
- You must use the first k digits (1 to k) exactly once each.
- They can be used as single-digit numbers (eg 1).
- Or they can also be used to make multi-digit numbers (eg 2 and 3 can be used to make twenty-three).
