HISTORY OF NUMBER “e”
I. INTRODUCTION
In mathematics, we know many symbols or notations that are used to solved many problems, such as phi, sigma, epsilon, delta, e, etc. In this paper, we have chosen one of them that is number of e. We think that the number e was very important in the past time to historical development mathematics until now. “e” is important simply because it has all those nice properties we have been studying. Whenever we take the derivative of e^x (that's e to the x), we get e^x back again. It's the only function on Earth.
II. A. What is “e”…?
e is a real number constant that appears in some kinds of mathematics problems. The mathematical constant e is the unique real number such that the value of the derivative (slope of the tangent line) of the function f(x) = ex at the point x = 0 is equal to 1. The function ex so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base e. The number e is also commonly defined as the base of the natural logarithm (using an integral to define the latter), as the limit of a certain sequence, or as the sum of a certain series
B. Who First Used e?
The number e first comes into mathematics in a very minor way. The number e was first studied by the Swiss mathematician Leonhard Euler in the 1720s, although its existence was more or less implied in the work of John Napier, the inventor of logarithms, in 1614. Napier's definition did not use bases or algebraic equations. Algebra was not advanced enough in Napier's time to allow for our modern definition. Logarithmic tables were constructed, even tables very close to natural logarithmic tables, but the base, e, did not make a direct appearance till about a hundred years later.Euler was also the first to use the letter e for it in 1727 (the fact that it is the first letter of his surname is coincidental). Leonhard Euler was a Swiss mathematician who made enormous contributions to a wide range of mathematics and physics including analytic geometry, trigonometry, geometry, calculus, and number theory. Leonhard was sent to school in Basel and during this time he lived with his grandmother on his mother's side. This school was a rather poor one, by all accounts, and Euler learnt no mathematics at all from the school. He entered the University in 1720, at the age of 14, first to obtain a general education before going on to more advanced studies. As a result, sometimes e is called the Euler Number, the Eulerian Number, or Napier's Constant (but not Euler's Constant).
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 started to use the letter e for the constant in 1727 or 1728, and the first use of e in a publication was Euler's Mechanica (1736). Some assume e was meant to stand for "exponential"; others have pointed out that Euler could have been working his way through the alphabet, and the letters a, b, c, and d already had common mathematical uses. What seems highly unlikely is that Euler was thinking of his own name, even though e is sometimes called Euler's number. While in the subsequent years some researchers used the letter c, e was more common and eventually became the standard.
C. How do you find it?
An effective way to calculate the value of e is to use definition of e for infinite sum, that is :
e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + ...
If we want K decimal places, calculate each term to K+3 decimal places and add up them. We can stop adding after the term 1/n! where n! > 10K+3, because to K+3 decimal places, the rest of the terms are all zero. Actually, there are infinitely many of them, they will not change the decimal places we have already calculated. That is why the calculation uses extra decimal places. As an example, here is the calculation of e to 22 decimal places:
1/0! = 1/1 = 1.0000000000000000000000000
1/1! = 1/1 = 1.0000000000000000000000000
1/2! = ½ = 0.5000000000000000000000000
1/3! = 1/6 = 0.1666666666666666666666667
1/4! = 1/24 = 0.0416666666666666666666667
1/5! = 1/120 = 0.0083333333333333333333333
1/6! = 1/720 = 0.0013888888888888888888889
1/7! = 1/5040 = 0.0001984126984126984126984
1/8! = 1/40320 = 0.0000248015873015873015873
1/9! = 1/362880 = 0.0000027557319223985890653
1/10! = 1/3628800 = 0.0000002755731922398589065
1/11! = 0.0000000250521083854417188
1/12! = 0.0000000020876756987868099
1/13! = 0.0000000001605904383682161
1/14! = 0.0000000000114707455977297
1/15! = 0.0000000000007647163731820
1/16! = 0.0000000000000477947733239
1/17! = 0.0000000000000028114572543
1/18! = 0.0000000000000001561920697
1/19! = 0.0000000000000000082206352
1/20! = 0.0000000000000000004110318
1/21! = 0.0000000000000000000195729
1/22! = 0.0000000000000000000008897
1/23! = 0.0000000000000000000000387
1/24! = 0.0000000000000000000000016
1/25! = 0.0000000000000000000000001
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2.7182818284590452353602875
Then to 22 decimal places, e = 2.7182818284590452353603, which is correct. It is a fact that e is an irrational number, so its decimal expansion never terminates, nor is it eventually periodic. Thus, no matter how many digits in the expansion of e we know, the only way to predict the next one is to calculate e using the definition of e above making more accuracy.
D. Application of number e
Examples of such problems are those involving growth or decay (including compound interest), the statistical "bell curve," the shape of a hanging cable (or the Gateway Arch in St. Louis), some problems of probability, some counting problems, and even the study of the distribution of prime numbers. It appears in Stirling's Formula for approximating factorials. It also shows up in calculus quite often, wherever you are dealing with either logarithmic or exponential functions. There is also a connection between e and complex numbers, via Euler's Equation. e is usually defined by the following equation:
e = limn->infinity (1 + 1/n)n.
Euler's interest in e stemmed from the attempt to calculate the amount that would result from continually compounded interest on a sum of money. The limit for compounding interest is, in fact, expressed by the constant e. If you invest $1 at a rate of interest of 100% a year and in interest is compounded continually, you will have $2.71828 . . . at the end of the year.
Euler devised the following formula to calculate e:
e= 1+ 1/2 + 1/(2 x 3) + 1/(2 x 3 x 4) + 1/(2 x 3 x 4 x 5) + . .
III. CONCLUSION
The symbol e for the base of natural logarithms (2.71828 . . . ) was first used by the Swiss mathematician Leonhard Euler (1707-83) in a 1727 or 1728 manuscript called Meditatio in Experimenta explosione tormentorum nuper instituta (Meditation on experiments made recently on the firing of cannon). He was all of 21 at the time. Euler also used the symbol in a letter written in 1731, and e made it into print in 1736, in Euler's Mechanica. Euler was not the inventor of the number e, even though he gave mathematicians the symbol e. The existence of e is implicit in John Napier's 1614 work on logarithms, and natural logarithms are sometimes inexactly dubbed Napierian logarithms. John Napier, the inventor of logarithms, is credited with discovering this constant. Leonhard Euler is credited with popularizing the use of the letter e for this number.
The constant 2.71828 . . . was referred to in Edward Wright's English translation of Napier's work in 1618. It is a fact (proved by Euler) that e is an irrational number, so its decimal expansion never terminates, nor is it eventually periodic. Thus no matter how many digits in the expansion of e you know, the only way to predict the next one is to compute e using the method above using more accuracy.
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