E
lastmod 2018-08-22created 2015-09-0118 edits
The constant e is a specific real number derived from the slope of the tangent curve; it is irrational and transcendental. It is also called Euler's number after the Swiss mathematician Leonhard Euler, and Napier's constant in honor of the Scottish mathematician John Napier, who introduced logarithmic calculation. Also, because e is the base of the natural logarithm, it is called the natural constant. e is treated as one of the important constants of mathematics, along with π, 1, 0, i, and others.
It is a number created to express growth (the continuous growth of nature - growth -).
Definition
\[ \displaystyle e={ \lim_{n\to\infty}}\left(1+{1 \over n} \right)^{n}={ \lim_{x\to0}}\left(1+x\right)^{1 \over x} \]
\[ \displaystyle e={\sum_{n=0}^{\infty}\frac{1}{n!}=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\cdots} \]
When you first learn \(e\) in high school, you may wonder why a number that is not natural at all (isn't it a number defined in an awfully unnatural way!) is called the natural constant, but that is not so. Only the definition that humans use is unnatural; in practice, using \(e\) in calculus and throughout mathematics makes notation remarkably simple.
\[ \displaystyle \frac{d}{dt}e^{t}=e^{t} \]
\[ \displaystyle \frac{d}{dt}\log_{e}t=t^{-1} \]
\[ \displaystyle \int_{1}^{e}t^{-1}dt=1 \]
Derivative
Differentiating \(e^{x}\) gives \(e^{x}\). (It is the same.) To prove it, use the derivative of the logarithmic function to prove the derivative of the exponential function.
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What is e
The Natural exponential function
- [Recommended] Youtube: Why we need the natural constant e
According to it, e is defined as follows.
- The maximum amount of growth attainable with a growth rate of 100% and one continuous growth.
Referring to the natural exponential function, if you know two of the growth rate, the number of growths, and the amount of growth, you can find the third.
\[ e^{성장률 * 성장횟수} \]
Example
Assume the following scenario.
In this case, the following amounts and growth rates apply.
| Number of growths | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
| Once | Amount | 1 | - | 1 + 1 | ||||||||
| Amount of growth | 1 + 0 | - | 1 + 1 | |||||||||
If the growth above is split into two growths, it grows as follows.
| Number of growths | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
| Twice | Amount | 1 | - | (1) + (0.5) | - | (1 + 0.5) + (0.5 + 0.25) | ||||||
| Amount of growth | 1 + 0 | - | (1 + 1/2) | - | (1 + 1/2) * (1 + 1/2) | |||||||
(The last amount of growth for two growths is (1 + 1/2) * (1 + 1/2) because the grown result is 1.5x * 1.5x.)
In the same way:
- For 3 growths:
(1 + 1/3) * (1 + 1/3) * (1 + 1/3) - For 4 growths:
(1 + 1/4) * (1 + 1/4) * (1 + 1/4) * (1 + 1/4) -
...
So the following formula holds.
\[ \displaystyle e={ \lim_{n\to\infty}}\left(1+{1 \over n} \right)^{n}={ \lim_{x\to0}}\left(1+x\right)^{1 \over x} \]