e

There is a number in maths that can be agreed upon by everyone to be real...

e, or 2.7182818284...

It's irrational and transcendental, so is thus in its own unique category, and is also a very interesting number.

One definition of e is this:

The summation of inverse factorials converges to e. 

e also equals this:

The limit as n approaches infinity of (1+1/n)^n. This relates to the idea of compound interest - indeed, you will notice that the limit itself resembles the compound interest formula of pq^r (where p is the starting value, q is the percentage change, and r is the time elapsed).

 

That's all nice and good. But what is truly so interesting about e?

Well, for starters, if you differentiate e^x, you get e^x as the output - the only function that this will work with. For the same reason, integrate e^x and you get e^x (+c) back. In other words, the gradient at any point along the graph e^x will be equal to e^x. This is as d/dx a^x = a^x ln(a), and (as explained below), d/dx e^x = e^x ln(e), or e^x (since ln(e) by definition is equal to 1).

For this reason, the function: loge(x) is given its own separate notation - ln(x); if you differentiate ln(x), you get 1/x, which is another interesting differential.

 

e is also an interesting number when looking at complex numbers. Euler's formula tells us that e^ix = cos(x) + i sin(x), which is a result of the Taylor series expansion of e^x (which is similar to the inverse factorial definition at the beginning). If we let π equal x in this formula, we get that e^iπ = cos(π) + isin(π), or e^iπ = -1, so e^iπ + 1 = 0, which is known as Euler's identity.  

For the same reason, e^i2π -1=0, as you're simply altering the value of x.

Indeed, e is quite an interesting number...

Irrelevant footnote:

I tried to include images of the formulas equal to e, but apparently Blogger doesn't accept large base64 images, which I find annoying, as they were snips from formulae from Math Input Panel. It's not really annoying, I suppose, but I'd prefer to post the formulae here rather than snipping them from Wikipedia, for example. Something similar seemed to affect my i post with Schrodinger's formula, though that was directly copying the formula from Wikipedia. Apologies if you have been inconvenienced, even if very slightly, by that.