Question

What is the significance of "e" ?

Answer 1

Euler's number is an irrational number like PI. The number goes like this e =2.71828............and there is no repeat pattern.

You can get the information on the web, but these are some of my thoughts......
1) "e" is the base of Natural Logarithm (ln). Given that x=ln (y), then y = e^x

2) "e" is also used in complex numbers, e^(ix) = cos x + i sin x.

3) "e" has a special place in Calculus, d/dx (e^x) = e^x, d/dx (ln x) = 1/x

4) "e" is also used in the definitions of hyperbolic functions, sinh x , cosh x and tanh x

Answer 2

copied ^^

Answer 3

LOOOOOOOL

Answer 4

i just answered it twice, so thought to copy and paste instead of writing it much! :P
the definition of the natural logarithm is the integral of 1/x. this just so happens to satisfy a property which we would like th natural logarithm to have: ln(x*y) = ln(x) + ln(y). now if y = lnx, then x = e^y, where e is some number satisfying ln(e) = 1 (in other words, the base of the natural logarithm). this is quite an important and useful tool to have lying around - using logarithms we can split products of functions in two. this is especially useful when it comes to integrating in certain situations (especially differential equations) and it is even nicer that the function has a relatively simple derivative: 1/x.

log base 10 of x for example, has a derivative of about 1/(2.30258509*x) which is MUCH uglier to deal with. it has the property that it can slit products of functions in two, but it gives us a nasty number to deal with.

another incredibly useful identity is that the taylor series (an infinite sum of polynomials) of e^x happens to look a heck of alot like the sum of the taylor series' of cosx and sinx. so much so in fact that it turns out that e^(ix) = cosx + i*sinx (i is the square root of negative one) for all values of x. note that this means that e^(pi * i) = cos(pi) + i * sin(pi) = -1 + i * 0 = -1 or:

e^(pi * i) = -1

this very simple identity demonstrates a direct connection between some of the most useful numbers in mathematics. notice that if you take the natural log of both sides you get:

pi * i = ln(-1) which now gives us a definition for logarithms of negative numbers!

many many things can be represented in terms of periodic functions (sin's and cos's) or inverse relationships: y = 1/x.

for example, the motion of a spring can be represented by a periodic function. the rate of growth of a population looks like: change in population = k * population where k is a constant or

P' = k * P

then dP/P = k* dt and integrating both sides,

ln P = kt raising e to both sides:

P = e^(kt)

we could use any other number we wanted for this purpose, but it would make everything messy.

for example,

2^(pi * i) = -0,570233249 + 0,821482831 i which is quite useless

Answer 5

no1 can write that much in a minute.

Answer 6

i think @miteshchvm is using copy+paste command

Answer 7

yaaaaaaaa......

Answer 8

Do you mean is it significant outside of mathematics? Appears in nature, etc?