About natural number e
The first known use constant E, Leibniz to Huygens communication, expressed by b in 1690 and 1691. 1727 began with Euler constant e to indicate this; while e first used in the publication is 1736 Euler's "mechanics" (Mechanica). Although later researchers also represented by the letter c, e, but more commonly used, and finally become the standard.
Indeed the reason indicated by e is unknown, but probably because e is the "index" (exponential) the first letter of the word. Another view called a, b, c, and d often have other uses, and e is the first free letter. However, the reason for the letter Euler election is unlikely because this is the first letter of his name Euler, because he is a very humble person, always properly affirmed the work of others.
From $ e ^ {i \ pi} + 1 = 0 $ seen Euler's formula, the letter e is defined as a natural number, and there is a direct relationship between Euler. I pour not believe Wikipedia says the reason for the letter Euler election is unlikely because this is the first letter of his name Euler, because he is a very humble person, always properly affirmed the work of others.
\[e=\lim\limits_{x \to \infty }(1+\frac{1}{x})^x\]
In fact, this formula is still too early to tell e = 2.71828, who initially think this formula would limit the number of e it is natural, what is the importance of the natural number e.
But we can from the logarithm and exponential to contact the importance of e.
\ [f (x) ^ { g (x)} = e ^ {g (x) lnf (x)} \ quad into A \ cdot B-type form of the function is multiplied by \]
using squared logarithms of properties a characteristic multiplied, we know that in the natural number of operands is the most common base number.
For operands of: \ (log_ab = N \)
for the index derivation \ ((a ^ x) '
= a ^ x \ cdot lna \) then if \ (lna = 1 \) like, equal to the number A, will make the \ (lna = 1 \) it?
Happens to a natural number e time, lne = 1.
Thus, a = e may be brought into index derivation formula:
\ [(e ^ X) '= e ^ X \]
itself to the function derivative remains that, this is a good property.
е involved mainly in local growth, such as economic growth, population growth, radioactive decay, etc., it can be said to represent the beauty of nature е rate of.
For example, a city population of 1.2 million, the annual population growth rate of 20%:
one year after the Population: 1,000,000 +100 Wan Wan x20% = 100 (1 + 20%) = 1.2 million
after two years of Population: 1.2 million +120 Wan Wan x20% = 120 (+ 20%) = 1000000 (1 + 20%) (+ 20%) =
\ [= 1000000 (1 + 20 \%) ^ 2; 2 index into a corresponding through 2 years of population growth \]
population after three years: = \ [= 1 million (1 + 20 \%) ^ 3 \]
after four years population: = \ [= 1 million (1 + 20 \%) ^ 4 \ ]
the X-years population: = \ [= 1 million (1 + 20 \%) ^ X \]
When the population growth rate of 20% has remained impossible, because of the limited living space, the growth rate should be reduced over time. And X is inversely proportional to the time rate of growth is assumed, i.e. an increase of \ (\ frac {1} { X} \)
Then the above population growth mathematical model can be abstracted as:
\ [f (the X-) = (1+ \ FRAC {1} {X}) ^ X \]
when we want to know the number of people many years later, that time X tends to infinity \ (\ infty \) of the population limit the time you can get:
\[ \lim\limits_{x \to \infty}(1+\frac{1}{x})^x=e \]