what is the purpose of the value e?


what is the purpose of the value e?

In: Mathematics

It’s the value of C where the gradient of the curve C^x is equal to C^x .

In other words at e^x the rate of growth/decay is equal to the size of the population/sample.

For other values of C^x there are modifiers you have to do to calculate the gradient.

In practical terms this makes e inherent to any sort of equations that relate to growth/decay where the growth/decay is continuous (eg radioactive decay/bacterial growth) rather than calculated incrementally (eg compound interest calculated monthly)

It also crops up in more complex and less intuitive applications, like the solutions to differential equations.

e is a rate of growth (or decay) shared by all continually growing (or decaying) functions.

So say we start with a value of x, and x is increasing at a rate of x • 2 over a period of time. You might think that over that period of time x would double to 2, but that’s not the case, because as x grows larger it will start increasing faster, so the actual value you end up with ends up slightly larger than 2.

Through a lot of math, it can be found that the true value of that idea is approximately 2.718, which is e.

The mathematical constant e is referred to as representing natural growth. Unlike other popular constants like pi, e does not have a handy visual to explain it. Instead it can be thought of using banks and interest:
(For true ELI5, interest means “the bank will add ‘this much’ to your money at the specified time)

If we have a bank that generously offers 100% interest every year, and you give them $1, then after a year you will have $2.

But wait! A rival bank wants to offer 50% interest TWICE a year if you bank with them. At first, you may think that 50% twice = 100% once, but that isn’t the case. After 6 months with this bank, your total + interest will be $1.50. Then another 6 months later it would be $2.25, getting $0.75 from 50% of the $1.50.

So it would seem that even if the % interest adds to the same 100%, getting it more often leads to giving you more money! So what about a bank that offers 25% interest every 3 months? Or 8.3% every month? How about an inconceivablely small percentage interest at every fraction of a millisecond throughout the year?

You’ll find that the final amount tends toward the value of e if you were able to gain interest at a “natural” rate of growth.

100% once a year: $2

50% twice a year: $2.25

25% four times a year: $2.4414

8.3% twelve times a year: $2.613035

Every possible moment in a year: $2.71828 (approaching the value of e)

e is also important in Calculus and the natural log, but those are waaaay beyond a 5 year old.

e (Euler’s Number) is also extremely important in complex number theory and representing quantities as a complex vector. A complex vector can be used to represent a real thing with real properties such as an electromagnetic wave. As such, the math around e and complex numbers is used constantly in the study of electromagnetism and the signal processing associated for communication systems.

In algebra, you learn about functions and you’re mainly concerned with what the output is (y) for any given input (x).

In calculus, you are now concerned with the rate of change of the output (dy/dx) for any given input (x). In calculus, dy/dx is called “the derivative of y with respect to x”. Mathematically speaking, it is the slope of the function’s tangent line for any given input (x).

So what if we could make a function f(x) whose rate of change was the same as its output for all inputs? Well experimentally we can find that this is true of some specific number raised to the x power. That number is e (2.71…..).

For example: consider again the function y=e^x. let’s plug in 2 for our input x. This gives y = 7.39. What makes our function unique is that our output (y) is the same as our rate of change (dy/dx). So y= dy/dx = 7.39 This pattern will continue for all possible inputs.

So to recap, what we have created is a function whose rate of change increases exactly as its output does. In math terms, e^x is a function that is its own derivative. As the other commenters have pointed out, that has extremely important applications in compounding interest and estimating all sorts of cool stuff.

Calculus sounds much more intimidating than it is, and if you’d like a very very well explained and animated version of what calculus actually accomplishes, check out [this video ](https://youtu.be/WUvTyaaNkzM). Good question OP.

I think some of these responses are missing the point. I would say almost no one cares about the number e, but we end up writing it down anyway so some math students start thinking they should care about its value.

Real ELI5 here: adding numbers is easy, multiplying them is hard. There is a function, called the natural logarithm, which turns multiplying into adding–the hard thing into the easy thing. This is great! But it makes numbers look different, so we want to be able to “undo” this. There is another function, called exp(x) which undoes the natural logarithm. So if you have some complicated mathematical expression to deal with, frequently a good strategy is

A) take the natural log

B) deal with the (now easier to deal with–all multiplication became addition, after all) new expression

C) take the exp() to make it look like the kind of number we started with.

The number e just happens to be what exp(1) is, but there isn’t any reason to expect exp(1) to be a very interesting or important number.

The ‘value’ e itself not nearly as interesting as the exponential function associated with that number, which happens to be the ‘most natural’ exponential form. This exponential function has some incredible properties which, along with some other fundamental assumptions, essentially lay the entire foundation for a significant chunk of modern mathematics. At face value, this function connects addition with multiplication, allowing efficient interchanging of the two operations which itself has tons of obvious applications. In fact, it goes deeper than that because the exponential function and its basic properties can be used to derive all of the fundamental relations involving, i.e. to define, the functions sine, cosine, etc… and essentially everything involving complex numbers has intimately to do with the connection between the exponential and trigonometric functions. I’m not a mathematical expert, but I know enough to recognize that the function is of tremendous importance, probably to a degree we don’t even fully comprehend yet.