Why are we using "Euler's Number" constantly?

Question

I am a student at technical university.We are currently studying Calculus and I am really curious about why we are using Euler's Number.

Answer 1

Below is one of my favourite examples of compound interest, we will illustrate that the maximum amount of interest we can receive over interest has an upper bound in a way. Note that this answer is just a peak at the wonderful world of maths and not meant to scare away any readers, it is by no means rigorous, it is supposed to be a light read.

Suppose we have a euro and over some amount of time we receive 100% interest over this euro:

$$ 1+\frac{1}{1}=2$$ Nice, can we get more money if we ask the bank folks to just give us 50% interest, but over half the time (interest over interest?) $$ (1+\frac{1}{2})^2=1.5^2=2.25$$ Nice, more money! Does this go on forever $\dots$ ? $$ (1+\frac{1}{100})^{100}=1.5^2=2.7048\dots$$ Hmm, it seems like this number has a ceiling, in fact: $$ (1+\frac{1}{n})^{n} \rightarrow e$$ If you would like to know why, read up on some real analysis, it's a truly marvellous subject.

Another great illustration is the polynomial that is its own derivative, let's define the polynomial by a funny name: $$ \exp(x)= 1+ x+ \frac{1}{2}x^2 + \frac{1}{6}x^3 + \frac{1}{24}x^4\dots $$ Notice that if we derive term by term we get: $$ \frac{d}{dx} \exp(x)= 1+ x+ \frac{1}{2}x^2 + \frac{1}{6}x^3 \dots $$ How peculiar, what would $\exp(1)$ be? $$\exp(1) \approx 1+ 1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24}=2.708 \dots$$ Truly, this number has such an interesting property in nature, if it would be the base of an exponent it would correspond to one of the most natural things to define, maybe we could even define some sort of logarithm for it, wouldn't that be the natural thing to do?

Also see: http://math.wikia.com/wiki/Euler%27s_number

https://www.quora.com/What-are-the-most-fascinating-facts-about-Eulers-number-e

Actually Jacob Bernoulli was possibly the first to discover $e$, not Euler, who was a student of Jacob Bernoulli: https://en.wikipedia.org/wiki/Jacob_Bernoulli

Answer 2

Because it is a special number, indeed it is the only base for the exponential function $f(x)=a^x$ such that the derivative is equal to the function itself, that is

$$f'(x)=f(x)$$

Refer also to

• Applications - Calculus (Wiki)