
So now that we now that the slope of two to the power of X at X equals zero is slightly less than one,

and the slope of three to the power of X at X equals zero is slightly larger than one,

so it's not hard to imagine that somewhere in between two and three, there is a number called E

Euler's numberso that E to the power of X has a slope of exactly one at X equals zero.

This function is called The Exponential Function. Whereas two to the power of X and three to the power of X are just exponential functions.

As this exponential function, The Exponential Function is so important, it gets a special name;

It's not just E to the power of X. It's called exp of X. This is the natural exponential function.

And we already know that its basis, E, is a number between two and three.

Now we can start from this idea and compute Euler's number.

E is nothing but E to the first power. Great. Let me write this one as 10,000 divided by 10,000.

Up to now, this looks completely crazy. Where am I going? I'm going to apply the laws for powers.

This is E to the one over 10,000 to the power of 10,000. Remember that ruleif you take a power of a power,

you multiply the exponents. This is what you get. But E to the power of one over 10,000,

that's something pretty close to X equals zero on this blue curve.

So rather than looking up the value on that blue curve, I'm using the red line, the tangent line, to compute that value,

to approximate that value, that is. The slope is one, so we can just put X in here, plus the Y intercept is one as well.

And now I'm using this equation for the tangent line to approximate this value.

The value of E to the X at X equals one over 10,000. So this is approximately equal to one over 10,000 plus one.

We simply looked up that value using the tangent line. Let me plug that in. Then we have E is approximately equal to

I changed the order here to make it look nicerone plus one over 10,000, to the power of 10,000.

Guess what? We have an equation, well, an approximate equation, for Euler's number.

The official definition for Euler's number is the following: E equals the limit of N goes to infinity of one plus one over N

to the power of N. I hope that you can now easily guess where this comes from.

The larger this number gets, the better these approximations get. And the result is 2.7 something.

Footnote: This equation is pretty elegant when it comes to deriving E, but when it comes to computing the actual value of E,

this equation is pretty inefficient.