Euler's Number

It's a number similar to pi. The value is approximately 2.718.

Origin - Continuous Compounding of interest

First reference of Euler's number

It was first referenced to show in a specific example mentioned below. Irrespective of how frequently the interest is paid out, it reaches a ceiling that's equal to the Euler's number.

1. $1 is kept in a bank account which gives 100% interest per year.
2. If this interest is paid only once a year, then the value of the money becomes $2 at the end of 1 year.
3. Similarly, the interest can be paid out at shorter intervals, example 50% every 6 months. In this case, the 50% of the first 6 months is also compounded into the interest paid at the end of next 6 months.
4. If we keep getting the interest paid out infinite frequency, the compounding will lead to a ceiling value of around 2.718 at the year.

continuous vs discreet

In case of discreet change, the bank interest is paid out at regular intervals. Whereas in case of continuous change, the same interest is paid out continuously at every millisecond .

Relation to differential calculus

Even though the value 2.718 was established in the continuous compounding context, the other benefits were proven only with calculus.

It was figured out that, $f(x)=e^x$ is the only possible exponential function that has a special property where the derivative is same as the function output.

$\frac{dy}{dx} = y \quad \Rightarrow \quad y = e^x$

what does it mean?

For input x and output y, $f(x)=e^x$ is the only function when plotted as a graph, results in a derivative value at any point in the graph that's equal to the output value at that point.