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# what is e?

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

Aug 5, 2023

#1
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"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)   Aug 5, 2023
#2
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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.
Aug 5, 2023