where

*e*=2.718281828... (Euler's number)

Mainly a note to self, just so I don't have to remember it again...

Prompted by a couple of posts by Adrian McMenamin on his blog

Not entirely sure this is entirely robust or complete – but I think it’s basically the way I learned it and have remembered and rehearsed it in the past. Any comments/corrections from those more in the know welcome.

Define the function

*f*(

*x*) as follows:

Then

*f*(

*x*) is uniquely defined (for all real non-zero positive values of

*x*) and

and

Also, for a variable

*y*(independent of

*x*but on the same domain)

and

where

*C*is some constant value.

To find the value of

*C*let

*x*= 1. Then, since

*f*(1) = 0

Now define the function

*g*(

*x*) to be the inverse of the function

*f*(

*x*). i.e.

and

So

And from the previous result that

we get

i.e. the function

*g*satisfies the basic exponential identity

So,

*g*(

*x*) can be expressed as some number (go on, let's call it

*e*) raised to the power of

*x*.

Since

*f*(

*x*) is the inverse of

*g*(

*x*),

*f*(

*x*) can be expressed as the logarithm to the base

*e*of

*x*. i.e.

So

That's the first bit done (I think)...

**...but what's the value of**

*e*?First, look at the properties of the derivative of the function*

i.e.

Remember (from above)

So

Remember too that

[This result allows for the derivation of both the sum of a series and limit definitions of Euler's number. What follows here is the derivation of the sum of a series definition - the limit definition is derived below]

It follows directly from the result

that all higher derivatives of

can be expressed as follows:

for any positive integer

*n*.

So

is infinitely differentiable and

where

represents the value of the nth derivative of

with respect to

*x*at

*x*= 0.

So

can be expressed as a Maclaurin series:

So set

*x*= 1 to calculate

*e*to arbitrary precision using

__LIMIT DEFINITION OF EULER'S NUMBER__

Consider the limit

Now

and

and

for any

*y*and

is defined, therefore L'Hopital's rule applies. i.e.

Substitute

Then

and as

Therefore

Now

therefore (since the exponential function is continuous*)

Or alternatively if

then

__*POSTSCRIPT__

I've used

*g*(

*x*) to denote the exponential function here to try and avoid preconceptions but it's obviously more commonly denoted by

I think I'm right in saying that,

*without*a definition of the exponential function:

and exponential identities such as

can only be defined where

*x*(and

*y*) is rational.

However, because

is (by definition) continuous and differentiable (with non-zero derivative) at all points on its domain of all non-zero positive numbers (range is all real numbers), its inverse

is also continuous and differentiable at all points on its domain of all real numbers (its range is all non-zero positive numbers). Therefore

*e*can be raised to the power of

*any*real number, not just rationals. This leads on to the definition of values for non-zero positive real numbers raised to the power of

*any*real number - and eventually can be completely extended to include negative and complex numbers. A few steps beyond that and ultimately comes the famous and beautiful (and immensely useful) Euler function and identity:

I think... I might post a proof of that too at some point. Weather, time and memory permitting.