Euler's number, e, has few common real life applications. Instead, it appears often in growth problems, such as population models. It also appears in Physics quite often.

Growth of the Bank

As for growth problems, imagine you went to a bank where you have 1 dollar, pound, or whatever type of money you have. The bank offers you 100% interest every year.

This means that next year you'll have 2 dollars. What a generous bank. Instead of 100% every year, let's say they offer you 50% every 6 months. In 6 months, you'll 1.5 dollars, and in another 6 months you'll have

1.5+50%of 1.5=2.25

This is better, actually! Let's take it further.

Now, they give you 25% interest once every 3 months. If you still have 1 dollar in the bank, now you will have

In three months:

1+25%of 1=1+1/4=1.25In another three months: 1.25+25%of 1.25=1+1/4+1/4⋅(1+1/4)=(1+1/4)(1+1/4)=(1+1/4)2Yet again: (1+1/4)2+1/4⋅(1+1/4)2=(1+1/4)(1+1/4)2=(1+1/4)3

If we repeat the process, at the end of the year you will have (1+1/4)4 dollars.

We can see a pattern! If we take a general case, say you get 100/n% interest every 12/n months and you begin with 1 dollar, at the end of the year you will have

(1+1n)n dollars.

So, we saw that it was advantageous to get a smaller interest over shorter intervals of time. Let's confirm this; let f(n) define how much money you get after one year with 100/n% interest over 12/n months:

f(1)=2f(2)=2.25f(3)≈2.37f(4)≈2.44f(5)≈2.49

Yes, it does increase, but it seems to be slowing down, converging to a value even. But what is this value? Well, let's say your bank does the impossible and offers you an interest with n going to infinity basically every nanosecond (in fact, much much faster than that). By the end of the year, you'll have:

limn→∞f(n)=limn→∞(1+1n)n=e

This is one of the definitions of e. But this is not exactly practical, because real life banks don't work this way. However, it does offer us a pretty good image of how e impacts growth.

Population Models

Suppose you have a population with p people and that this population doubles every 30 years. After 180 years, say, the population will double 180/30=6 times.

So the number of people after 180 years, which we will denote as P, is

P=2⋅2⋅2⋅2⋅2⋅2⋅p=26p

Now, we wish to find the instantenous rate of growth of the population. If we find it, it will be helpful to maybe compare it to former rates and form a pretty good impression of what the future holds. This is where e comes in handy.

The population after t years is going to be P=2t/30p.Now, the instanteneous rate of change represents how much the population will have grown in an infinitesimal amount of time.Basically, we ask what will P be after a really, REALLY small period of time, like t=10−100 seconds?

If we denote the infinitesimal interval of time to be dt and the effect it has on P be dP (which is also an infinitesimal unit), instanteneous the rate of change will be

dPdt=p⋅loge230⋅2t/30=p⋅ln230⋅2t/30

In mathematics, we usually just write loge as ln, the natural logarithm. Also, d is not a constant, but rather a symbol which declares that dP and dt are infinitesimals.