e

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e

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What is e, and who cares? Lots of people care, and you are about to?

Here is the definition (the upper equation on the left). The lower equation is equivalent, and may be easier to deal with on occasion. These equations use Limits. Don't be fooled by the notation; it's fairly easy. Choose a big number for n, in the first equation, like 1000. Then, using a calculator that can do powers, (1+1/n)ⁿ is (1+1/1000)¹⁰⁰⁰ or 2.71692... For n=1,000,000, we get 2.718280469... The actual value is 2.71828 18284 59045 23536 02874 71352 66249 77572 47093 69995 ... e is also equal to this famous series:

e=1+1/1!+1/2!+1/3!+1/4!+...

4! is 4x3x2x1. The series converges rapidly to e. That series comes from this series:

e^x=1+x/1!+x²/2!+x³/3!+x⁴/4!+...

That apparently comes from this interesting equation:

e is the base of the natural Logarithms. If x=e^y, then y=ln(x), where ln means natural logarithm.

All of that may not seem very useful. We can calculate e very rapidly using the series. But base e logarithms are difficult to work with. It turns out that e and its logarithms simplify many equations in calculus.

And in algebra, e helps us answer the question, "What is something to a complex power?" See Imaginary Numbers. It turns out that e^{i pi}=-1. With this information, we can simplify some of the most horrendous equations.

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