What is the significance of the number e?

Answer 1

e is not the same as from e = mc^2 (e there stands for energy and in fact should be a capital E).

e comes from mathematicians looking at "growth curves". These are functions on the (x,y) plane of the form y = ab^x (a and b are constants, only x is a variable).

This is a curve, nearly flat and close to the x-axis when x is very negative, then it grows, at x=0 y=a, and the slope at any point is proportional to the value of y. It is found in things like interest-rate curves for savings, or population growth in humans. The change in the function is proportional to where y already is. Proportionally the increase is a constant, but in absolute numbers the growth can become monstrous (since the growth can grow itself in turn).

To figure out the slope at any point, you need to use calculus.

The slope of the function y = ab^x is dy/dx (the ratio of an infinitesimal change in y for every infinitesimal change in x)

dy/dx = ab^x * ln b

ln b is a function called "the natural logarithm of b", used as a kind of correction factor. There is a certain value of b where the slope will be EXACTLY the same as the y-value. So if y is 5 the slope is 5, etc.

It turns out that the value where this happens is a number called e, which is about 2.718 281 828 459 045... It's a transcendental number, it can never be expressed as a simple fraction and the digits go on forever.

Knowing the value where the rate of change, or slope, equals y, you can define every other growth curve in terms of how far above or below e that the factor b is. If b equals e^2, then the slope will always be twice y.

Answer 2

It's the base of the natural logarithm. It crops up in many nifty ways.

{For all x|x is a real number} d (e^x)/dx = e^x
so at every point, the slope of a tangent to the curve for this function is equal to the function itself. Which means that this function increases in value "exponentially." Note that the inverse of the logarithmic curve is the exponential curve.

Another nifty property of e (I used to have a T-shirt with this formula written on it):

e^(i*pi) + 1 = 0

a sngle equation that relates the five fundamental constants of maths,

God must be a mathematician,

Answer 3

It shows up as the base for "natural" logs as the result of the solution of a basic integral. It is rampant in physics and shows up in solutions too differential equations. Semiconductor current flow uses this number directly and when used in a complex domain it can represent sine waves. It is to physics as pi is to geometry.

Answer 4

The function f(x) = e^x has the property such that, at x = 0, the slope of the tangent of the exponential is 1.

Answer 5

One of the great numbers in mathematics, also called Euler's number, has applications in probability, compound interest, calculus. Irrational, transcendental.

See good article in link below.

Answer 6

e is the base of the 'natural' logarithm system. It also has a number of interesting analytical properties as well. You'll get to them in due time ☺

Doug

Answer 7

lol ln e = 1
never mind that just means e^1 = e, but any way
Derivatives and integrals don't change it
Also there is the best formula of all
e^(i pi) +1 =0
where i is imaginary and pi is the pi we all know and love

Answer 8

ln e = 1

Answer 9

In e^x, the y value equals the gradient at that point, woohoo!

so when you are differentiating or integrating e^x, you get e^x

Answer 10

its like how pi relates to circles e relates in the same since to logarithms