**Approximating e using the digits 1–9**

The number e (approximately 2.718281828459045) is one of the most important irrational numbers in mathematics, being the base of the natural logarithm. Remarkably, it is possible to get a stunningly accurate approximation to e using the digits 1 up to 9,

**exactly once each**, together with the standard operations of addition, negation, multiplication and exponentiation. According to the formula in the picture, e is close to

(1 + 9^{–4^{7x6}})^{3^{2^{85}}}.

So why does this work? The key to understanding what is going on is the number N=3^{2^{85}}, an astronomically huge number. The number N appears more subtly in another place in the formula, because 4^{7x6}, or 4^{42}, is equal to 2^{84}. Since 9 is 3^2, it follows that 9^{4^{7x6}} is equal to {3^2}^{2^{84}}, which is equal to 3^{2x(2^{84})}, or 3^{2^{85}}. This means that 9^{–4^{7x6}} is equal to 1/N.

In summary, the formula in the picture is equal to (1+1/N)^N for a really large value of N. The limit of this formula, as N tends to infinity, is well known to be exactly equal to e.

I'd be interested to know where this formula originally came from. I found it on twitter via Colin Beveridge and Chris Smith. Chris Smith says that the approximation is accurate to over 1.8x10^{25} decimal places. I haven't checked this, but it is certainly a

*very*accurate approximation.

Wikipedia has much more about the number e here: http://en.wikipedia.org/wiki/E_(mathematical_constant)

#mathematics

View 42 previous comments

- Astronomically huge is putting it lightly I think.Dec 3, 2014
- +Hugh The finite set of permutations using those digits and the standard operations +,-,*,\ should be exhaustible by a computer program.Feb 17, 2015
- It troubles me that the approximation omits the digit zero. Let's fix it:

e ≈ 0 + (1 + 9^{–4^{7x6}})^{3^{2^{85}}}

There you go. Much better!Sep 29, 2015 - +Richard Green . I like the exact sequence, this one too is interesting. You have very interesting posts and again I am glad to see meet you!39w
- we can see this formula in this page : " Problem of the Month (August 2004) " --> http://www2.stetson.edu/~efriedma/mathmagic/0804.html7w
- Numberphile did a YouTube video on this recently ad well.7w