How people invented irrational numbers?

e first came about during John Napier's studies of logarithms, although the first calculations of it appear to have been in the work of Bernoulli:

lim (as n -> inf) of (1 + 1/n)^n

In fact, outside the sciences, the place where e most often occurs naturally is in the financial world.

When interest on an investment is compounded, we talk about the "compounding period": interest can be compounded annually, monthly, daily.  The formulas for the change in value of an investment is:

A = P·(1 + r/n)^(nt)

where A = the final amount, P = the principal, r = the annual interest rate, n = the number of compounding periods per year, and t = the number of years.

When money is compounded annually, n = 1; when it's compounded monthly, n = 12; and when it's compounded daily, n = 365 (though sometimes banks use 360, to simplify calculations).

See how n is going up?  So what happens if we compound every hour?  every minute? every second?  In fact, if we "compound continuously," then we're letting n approach infinity, and the function actually becomes

A = P·e^(rt)

There are a couple articles linked below that give more info.  Hope that answers your question!

It seems that Napier sort of made it up, perhaps working on a limit for the formula S = P (1 + r / n)^nt  [compound interest]

The irrational numbers are a necessity to explain certain phenomena of mathematics. For example, it is very simple to demonstrate that the square root of two cannot be a rational number (a number that one can write as the quotient of two integers).
The proof goes like this:
Assume that there is such a number, i.e. that (p/q)^2 = 2. We assume here also that p and q share no factor (if they do, we merely remove it to obtain two numbers that share no factors).
Then by simple rules of high school algebra we can rearrange a little to get p^2 = 2*(q^2)
Thus p^2 is even.
This implies that p is even (since the square of an odd number must be odd).
Then we can write: p=2*r where r is simply half of p and a whole number.
Thus we can write: (2*r)^2 = 2 * (q^2), which reduces to
4*(r^2)=2*(q^2)
and further to 2 * (r^2) = q^2.
But by the same reasoning, this implies that q is even. So we have both p and q being even, which implies that they share a factor and can be reduced. This contradicts our starting hypotheses, so we can conclude that there does not exist a rational number whose square is two. Thus there must be at least one irrational number. Ultimately is is relatively simple to show that there are, in a mathematical sense, "more" irrational numbers than rational ones.

As for e, e can be derived in many ways. It can be written as the sum of one over n factorial as n goes from zero to infinity. It can be written as the limit that people have already described. Most significantly however, the existence of "e" can be derived as the solution to the problem of finding a function whose derivative is itself. Most students usually encounter this process in their Real Analysis courses in college. The actual process is somewhat laborious and tedious, but suffice to say that the existence of "e" is strongly supported by centuries of mathematical analysis.

As others have already explained, "e" has been shown to have a wide range of applications, even outside of mathematics, in fields such as economics, engineering, and other science-related disciplines.

The way I remember it, you start out trying to solve the  f'(x) = 1/x.  If you start looking at the properties of f(x) you find that it is a logarithmic function. The next question is "What is the base?" solving that leads to e.

The concept of irrational numbers goes back to the Greeks who proved that the solution to the equation x^2 - 2 = 0 couldn't be represented as a fraction.

Calculate (1 + 1/n)^n.
The bigger you choose n the more the answer goes to Euler's number e = 2.71828... .

Calculate 1/0! + 1/1! + 1/2! + 1/3! + ... .
0! =1! =1. -----  5! = 1x2x3x4x5
The further you go, the close you come to e.

The slope of the graph of the function f(x) = e^x = exp(x) is in each point equal to the value of that function. In short: for any x is f(x) = f'(x).