What is e (2.718…) and why does it literally appear everywhere?
In: 756
Not quite 5 but here’s my explanation for my precal kiddos. So, the basis is an exponential equation. So a linear equation has the same thing added each time right, so like each next x value adds 5 to the y value. Well, an exponential equation multiplies it by the same thing every time. So we take our x value and multiply it by the same value for each new y value.
Ok so, lets use the exponential equation with 2 in it. So, y=2^x well this equation only “compounds yearly” for a problem. As in, its only being multiplied once a year. So what happens if we compound it more often than a year? Well there’s an equation for that but theres not really a need to get into that unless you’d like me to in another comment. So back to compounding, what if we compound it every month, well it will have more money at the end of that year, how about every week? More money at the end of the year, every day? More money. Eventually though, you can only get so much more by compounding more and more often. This number that they came to was 2.718… the number e. So for continuously compounding we use e.
Hope this helped!
You probably know that π is found everywhere in math because of circles. *e* is similar to that but has to do with growth and speed of growth.
Suppose you are driving a car. Now suppose we graph the total distance it travels with relation to total time traveled, that is, we have a graph in which the total distance traveled is on the vertical (y) axis and the time its been traveling is the horizontal (x) axis.
Now, you may know that we can write an equation, in terms of time, to express the distance traveled with respect to time. Lets call this function f(t).
Now, this car you are driving travels at either a constant speed, or is accelerating/daeccelerating. Now, if we were to graph the speed of the car, a flat line would indicate a constant speed, a line with a positive slope would indicate an accelerating car, and a line with negative slope would indicate the car is slowing down. Lets call the function for the speed of the car f'(t)
Now, what if I were to tell you that the speed of the car is equivalent to its distance traveled? That is, when it travels x miles, it is also traveling at x speed?
This is called the exponential function. Its defined to be the function for which its speed of growth is equivalent to itself, namely f'(t) = f(t).
Now, in physics, and the rest of math, all things are measured as they change. The rate of change and speed of things and their relation to each other (in this case speed and displacement) are the fundamental backbone of physics, because we measure how natural phenomena change with regards to each other. Due to this, the exponential function is the absolute *backbone* of physics and other areas of math and science.
Now, how does e tie into this? Well, if exp(x) is the exponential function, exp(1) is e.
[removed]
There’s a lot of stuff in this thread but want to add a way of explaining what e^x is in calculus. In maths there is a concept called an identity element for different kind of functions.
For addition and subtraction, the identity element is 0. If I add or subtract 0 I get the same as I started. 5 + 0 = 5
For multiplication and division, the identity element is 1. 5 / 1 = 5
For a similar function, x^y, 1 is again an identity element for x and for y. 5^0.1 gets smaller, 5^2 gets bigger, but 5^1 stays the same. 1^y = 1 no matter what y is.
e^x is an identity element for differentiation and integration. Differentiation is a way of telling how fast something is changing. For example, If you have an equation for the speed of the car, differentiation gives us the acceleration. Integration goes the other way: if you had an equation for the rate of acceleration you could calculate the speed of the car. (Or to be precise, how many units it has increased by).
If your car is travelling at e^x m/s and x is measured in seconds, then it is also accelerating at e^x. The rate of change of acceleration is also e^x, and the rate of change of that rate of change is also e^x. You get the same as you started. The same is true of integration.
(fun bonus maths fact: logarithms were known and used for hundreds of years before e was discovered. It was at least 80 years after e was discovered that it is some kind of opposite for logs.)
(Even weirder fun bonus fact: Euler’s Identity gives us relations for e that appear to have nothing to do with growth. Why?)
(Yes, I forgot c. Just give me half marks.)
It’s less useful to think of the number e than the function f(x)=e^x. This is the unique function whose slope is always equal to its value. The constant e is just this function evaluated at x=1.
This function is important because a lot of things grow at a rate proportional to their current value, because each thing is producing more of the thing (think population of some animal).
e is the number that solves the property that e^x is the derivative of e^x . As suggested in another comment, the derivative of 2^x is less than 2^x while the derivative of 3^x is more than 3^x.
It’s the Fibonacci sequence
There, I just saved you 3 boobless hours from trying to read all these comments
Latest Answers