Hello everyone, you will get some interesting results that might be not known to many of you. You will get that results in the end of this page. Many people are wondered about e but only few of them are literally know about e and it's origin. In this post I will show you What is e, It's origin, Proof of e.
Origin of e (Information taken from Wikipedia)
The number e, also known as Euler's number, is a mathematical constant approximately equal to 2.71828, and can be characterized in many ways. It is the base of the natural logarithm. It is the limit of (1 + 1/n)^n as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series.
The first known use of the constant, represented by the letter b, was in correspondence from Gottfried Leibniz to Christiaan Huygens in 1690 and 1691. Leonhard Euler introduced the letter e as the base for natural logarithms, writing in a letter to Christian Goldbach on 25 November 1731. Euler started to use the letter e for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons, while the first appearance of e in a publication was in Euler's Mechanica (1736). Although some researchers used the letter c in the subsequent years, the letter e was more common and eventually became standard.
The first known use of the constant, represented by the letter b, was in correspondence from Gottfried Leibniz to Christiaan Huygens in 1690 and 1691. Leonhard Euler introduced the letter e as the base for natural logarithms, writing in a letter to Christian Goldbach on 25 November 1731. Euler started to use the letter e for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons, while the first appearance of e in a publication was in Euler's Mechanica (1736). Although some researchers used the letter c in the subsequent years, the letter e was more common and eventually became standard.
How the Value of e come from?
Here I will shoe you Euler Number Derivation (Value of e)
There is a interesting thing behind the value of e but only few people are know about that. Here I will explain about it's value.
Let us consider initial value of 1 and it will double in every fixed time period. Let that time period is 1 year. We can say that it shows 100% growth in 1 year. So that after 1 year we get 2 instead of 1.1 ⟹ 2 (After 1 Year)
Now we consider time period is 6 month in this time period growth will be 50%. By this condition after 1 year you will get 2.25 instead of 1
1 ⟹ 1.5 (6 Month) ⟹ 2.25 (1 Year)
Now we consider time period is 3 month in this time period growth will be 25%. By this condition after 1 year you will get 2.4414 instead of 1
1 ⟹ 1.25 (3 Month) ⟹ 1.5625 (6 Month) ⟹ 1.9531 (9 Month) ⟹ 2.4414 (1 Year)
By this one can thin think that what happen if there are infinite intervals. Will the value of e will goes to infinity or there is a limit for max value of this. Actually there is a limit for this and that limit is known as e (Euler's number). Let us know how we can find that value.
For general case we can create a formula
\[\left[1+\frac{1}{n}\right]^{n}\]
where n is number of partitions created of time period.
For example,
If n=1, we get 2
If n=2, we get 2.25
If n=3, we get 2.37
If n=4, we get 2.4414
.
If n=10, we get 2.5937
.
If n=100, we get 2.704813
.
If n=365, we get 2.714567
.
If n=10000, we get 2.7181459
As you can see this value coming close to actual value of e.
If n tends to infinite then that final value will be known as e.
We can write
\[e = \lim_{n \rightarrow \infty}\left[1+\frac{1}{n}\right]^{n}\]
\[e = \lim_{n \rightarrow \infty}\left[1+\frac{n}{1!} \times \frac{1}{n}+\frac{n(n-1)}{2!} \times \left(\frac{1}{n}\right)^{2}+......upto (\infty)\right]\]
\[e = \lim_{n \rightarrow \infty}\left[\frac{1}{0!}+\frac{1}{1!} +\frac{(1-\frac{1}{n})}{2!}+\frac{(1-\frac{1}{n})(1-\frac{2}{n})}{3!}+......upto (\infty)\right]\]
\[e = \left[\frac{1}{0!}+\frac{1}{1!} +\frac{(1-0)}{2!}+\frac{(1-0)(1-0)}{3!}+......upto (\infty)\right]\]
\[e = \left[\frac{1}{0!}+\frac{1}{1!} +\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+\frac{1}{6!}+......upto (\infty)\right]\]
\[e = \sum_0^\infty\frac{1}{n!}\]
\[e = 2.7182818284590......\]
Here is value of e up-to 100 decimals
2.
71828 18284 59045 23536 02874 71352 66249 77572 47093 69995 95749 66967 62772 40766 30353 54759 45713 82178 52516 64274
Click Here to get value of e up-to 10000 digits.
Some fun facts about e
1. (π^4 + π^5)^1/6 = 2.718281809
and e = 2.718281828
∴ (π^4 + π^5)^1/6 ≈ e
2. Here is a easy to learn sequence which will give approximate value of e
\[e = 2+\frac{1}{1+\frac{2}{2+\frac{3}{3+\frac{4}{4+\frac{5}{5+\frac{6}{6+\frac{7}{7+\frac{8}{8+............}}}}}}}}\]
3. e can also be written as
\[e = \lim_{n \rightarrow \infty}\left[1+\frac{1}{n}\right]^{n}=\lim_{n \rightarrow 0}\left[1+{n}\right]^\frac{1}{n}\]
4. It's actually Euler's Identity
\[e^{i\pi}+1=0\]
It's all about value of e (Euler's number). If you know any other interesting fact about e then you can comment it down. I will update that information in this page. Comment your views about this article. If you want any article on a specific topic then you can contact me on Gmail which is provided in the contact info. If you have any Maths puzzle then you can send me. I will solve it I upload it with including name of sender of that puzzle. I will glad to solve that puzzles.
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