How is e considered as 2.17?
The mathematical constant e is the unique real number such that the function ex has the same value as the slope of the tangent line, for all values of x. More generally, the only functions equal to their own derivatives are of the form Cex, where C is a constant. 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 (see representations of e, below).
The number e is one of the most important numbers in mathematics,[3] alongside the additive and multiplicative identities 0 and 1, the constant π, and the imaginary unit i.
The number e is sometimes called Euler's number after the Swiss mathematician Leonhard Euler. (e is not to be confused with γ – the Euler–Mascheroni constant, sometimes called simply Euler's constant.)
Since e is transcendental, and therefore irrational, its value cannot be given exactly as a finite or eventually repeating decimal. The numerical value of e truncated to 20 decimal places is:
2.71828 18284 59045 23536...
by:kennedy perida
source: http://answers.yahoo.com/question/index?qid=20080504015446AABuhLG
The
first time the number e appears in its own right is in
1690. ......the notation e made its first appearance in a letter Eulerwrote to Goldbach in
1731. He made various discoveries regarding ein the following
years, but it was not until 1748 when Euler publishedIntroductio in Analysin
infinitorum that he
gave a full treatment of the ideas surrounding e. He showed that
e = 1 + 1/1! +1/2! +1/3! + ...
and
that e is the limit of (1 + 1/n)n as n tends to infinity. Euler gave
an approximation for e to 18 decimal places,
e = 2.718281828459045235
by
Irrah C. Dimayuga
Complex numbers (numbers involving the
"imaginary" number "i" which is the square root of -1) have
connections to many other parts of mathematics. A particularly striking example
comes from the work of Euler. In 1748 he discovered the amazing identity
This is true
for any real number x.
Such a close connection between trigonometric functions, the mathematical constant "e", and the square root of -1 is already quite startling. Surely, such an identity cannot be a mere accident; rather, we must be catching a glimpse of a rich, complicated, and highly abstract mathematical pattern that for the most part lies hidden from our view.
In fact, Euler's formula has other surprises in store. If you substitute the value
for x in Euler's formula, then, since
cos
= -1 and sin
= 0, you get the identity
Such a close connection between trigonometric functions, the mathematical constant "e", and the square root of -1 is already quite startling. Surely, such an identity cannot be a mere accident; rather, we must be catching a glimpse of a rich, complicated, and highly abstract mathematical pattern that for the most part lies hidden from our view.
In fact, Euler's formula has other surprises in store. If you substitute the value
Rewriting this as
you obtain a simple equation that connects the five most
common constants of mathematics: e,
, i , 0 , and 1.
Not the least surprising aspect of the last equation is that the result of raising an irrational number to a power that is an irrational imaginary number can turn out be a natural number. Indeed, raising an imaginary number to an imaginary power can also give a real-number answer. Setting x =
/ 2 in the equation at the top of the page,
and noting that cos
/ 2 = 0 and sin
/ 2 =1 , you get
Not the least surprising aspect of the last equation is that the result of raising an irrational number to a power that is an irrational imaginary number can turn out be a natural number. Indeed, raising an imaginary number to an imaginary power can also give a real-number answer. Setting x =
and, if you raise both sides of this identity to the power
i, you obtain (since
= -1)
Thus, using a calculator to compute the value of
, you find that
By: irrish c.dimayuga
| What is e? Who first used e? How do you find it? How many
digits does it have?
e is usually defined by the following equation: e = limn->infinity (1 + 1/n)n.Its value is approximately 2.718281828459045... and has been calculated to 869,894,101 decimal places by Sebastian Wedeniwski (you'll find the first 50 digits in a Table of constants with 50 decimal places, from the Numbers, constants and computation site, by Xavier Gourdon and Pascal Sebah). 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. 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). As a result, sometimes e is called the Euler Number, the Eulerian Number, or Napier's Constant (but not Euler's Constant). An effective way to calculate the value of e is not to use the defining equation above, but to use the following infinite sum: e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + ...If you need K decimal places, compute each term to K+3 decimal places and add them up. You 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. Even though there are infinitely many of them, they will not change the decimal places you have already calculated. Now the last decimal place or two of the resulting sum may be off due to truncation or rounding of each term, but the first K places should be correct. That is why the computation uses extra decimal places. As an example, here is the computation of e to 22 decimal places: 1/0! = 1/1 = 1.0000000000000000000000000
1/1! = 1/1 = 1.0000000000000000000000000
1/2! = 1/2 = 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
-----------------------------
2.7182818284590452353602875
Then to 22 decimal places, e = 2.7182818284590452353603, which is
correct. (Actually,it's correct to 25 places, but that was luck!).
There have been recent discoveries of even more efficient ways of computing
e, one of which was used for setting the record mentioned above.
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. It is also true that e is a transcendental number (a fact first proved in 1873 by the French mathematician Charles Hermite), which means that e is not the root of any polynomial with rational number coefficients. These are properties that e shares with pi. The Dr. Math archives contain one proof of The Irrationality of e, and on the Web is another by Kevin Brown. e is also the base of natural logarithms. The natural logarithm function ln(x) is defined that way: ln(x) = limk->0 (xk-1)/k.Another example from calculus is that if Note: The term Euler's Constant is usually reserved for another number also first studied by Euler, 0.5772156649... = by: Izelle Joy D. Clerigo http://mathforum.org/dr.math/faq/faq.e.html |
