many people tell me about imaginary numbers, such as i and e, but I do not know what e means. Sombody pls answer.

jonathanldong Aug 5, 2023

#1**+1 **

"e" is NOT an "imaginary" number.... is the base for the natural logarithm (just as 10 is the base for the "normal" log)

It equals ≈ 2.71828..... (it is irrational so it has a non-repeating, non-terminating decimal part )

This number comes in very handy for many scientific and financial applications (such as growth/decay functions)

CPhill Aug 5, 2023

#2**+3 **

The concept of \(e\) comes from compound interest. Suppose a very generous bank offers you a \(100\%\) interest rate, which would give you twice as much as what you put in every year. If you compound the interest quarterly, you would recieve \(25\%\) of your \(\bf{current}\) amount every three months, eventually gaining around \((1.25)^4\approx 2.44\) times of what you invested.

So suppose that you compounded your money even further, recieving interest every day, hour or even second. Would you achieve infinte wealth? Sadly, no. \(e\) is defined as the limit of this growth, which is

$$\lim_{n\to \infty}\left(1+\frac1n\right)^n = 2.71828182846\dots$$

Despite its apparent disconnection from other areas of mathematics, \(e\) turns up quite a lot in calculus and complex numbers. For example, you may have heard of the famous equation \(e^{i\pi}=-1\).

Some other facts about \(e\):

- It is both irrational and transcendental, that is, it cannot be expressed as \(\frac mn\) where \(m\) and \(n\) are relatively prime integers. Nor is it the root of any polynomial with rational coefficients.
- The growth of the function \(e^x\) is exactly itself, that is, the slope of the tangent line at any point \(\left(x,e^x\right)\) is exactly equal to \(e^x\), or in other words \(\frac d{dx}\left(e^x\right)=e^x\). You will learn more about it when you get to calculus.

plaintainmountain Aug 5, 2023