What is e (2.718…) and why does it literally appear everywhere?
In: 756
You know how pi is the circle number? e is the growth number.
Pi is the “constant of circles.” If you’re working with circles, pi isn’t far away. It appears everywhere that circles can be found. Every ellipse has a constant that will appear whenever you use that particular ellipse. For a circle (an ellipse whose foci are in the same spot) we call that constant pi, because it’s useful.
e is the “constant of self growth.” If any system’s rate of change depends on its current size, then e isn’t far away. It appears everywhere “natural growth” occurs: accruing interest, population growth, probability distributions, etc
It’s just one of the mysteries of the universe why the value is what it is, but the concept behind the value is much easier to explain.
And for any pendants reading, yes it’s true that if e were 3 or 1 then it wouldn’t give the same behavior, but that’s tautological. What I’m saying is *why* that particular value gives that particular behavior is unknown to us, just like why pi is 3.14… and not 2.5.
E: phrasing
other people gave an interesting definition and it’s used. Another interesting fact not mentionned :e=1 + 1/1 + 1/1*2 + 1/1*2*3 + 1/1*2*3*4 + … + 1/n!
well, strictly speaking, e is the number value where the slope of the graph of its exponential function (e^x) is the value of that function at any point on the curve. The slope at any point on the e^x curve will be the value of e^x. The slope and its value are the same everywhere. This is very useful and very convenient for a lot of purposes.
So, it is a special exponential function. Most functions are not like this (the slope of the function does not change at the same rate as the function itself), but math allows any exponential function to be expressed in terms of this special e^x function and some conversion factors, so any phenomenon that has exponential behavior will be able to be expressed by the special function. We like that special function because it is way easier to use when using higher-level mathematics to figure things out. This is why it appears everywhere, because exponential and logarithmic relationships exist everywhere, and the natural exponent and natural log behave mathematically in a simple-to-use way.
Do not need it, but it must exist (there must be some number that satisfies the requirement that the slope of the exponential function is the function itself), and once found, math became a lot easier in many ways, so it appears a lot in math. The value of e itself does not really much matter, the important thing is that there must be such a value. Once we know what that value is, lots of hard calculations become simple. Just plug it in.
I’ve never seen e (2.718) anywhere before, can someone here possibly explain what the question refers to? and eli5 an answer? all these other answers are a bit too complicated.
Here is another definition. Let’s assume there is a saving account that gives you 100% interest rate a year. You put £100 and take £200 at the end of the year.
* Now assume, they give you 50% interest rate for 6 months. You first put all your money and make £150 in the 6th month and put it all again to the bank. At the end of the year, you make £225.
* What would happen if the interest rate was 25% for 3 months, every quarter? You would make, first £125, then £156.25, £195.3125 and £244.1406 at the end of each quarter.
* If you make the same calculation for every 1 month with an interest rate of (100/12)%, you would make £261.3035 at the end of the year.
* If you assume a daily interest rate of 100/365%, you would make £271.45674… at the end of the year.
If you continue dividing a whole year into the smallest time interval you can take, you will converge to this number e = 2.718… This is where the number e comes from.
E is a transcendental number (so its 2.7182818…) it goes on and on and never repeats. Now if you think of exponents and logarithms (10^x, 11^x, log_12(x) log_13(x), etc), you quickly see you could just pick any number, and they all work (remember there are formulas to change from one base/log to another). So how do you pick which number to use? Well different fields do it in different ways (computer scientists like to use 2 because of binary, sometime scientists use 10 cause like decibels are in base 10 for example).
Now the reason why math people use e boils down to the fact that it has some really nice calculus properties.
* The main property is that (d/dx) e^x = e^x or more generally (d/dx) e^f(x) = f'(x) e^f(x). So this let’s you start solving problems like y”- y’ + 5y=0 called ordinary differential equations.
* another really nice properties happens to be that e^i*t = cos(t) + i sin(t), which you might notice that the real and imaginary part of that track the x,y coordinates of the unit circle being traced out from t = 0.. 2*pi. So now we have a cool way to relate circles and angles to exponents. This is called De Moivre’s formula and this formula is how you’d figure out those tables of trig identities you saw in geometry class
So I’ve always been confused by that, but this got me to dive in and I think I have it figured out.
First, e is very similar to pi. It’s simply a mathematical constant that just exists. There’s no real satisfying rhetoric that makes it makes sense, it only really makes sense with mathematical proofs/equations
Second, it’s important to know what a derivative is. It’s basically the slope of a curve, but at a specific point. Think the original line is speed, and the derivative line is acceleration.
So go to Desmos.com/calculator
1. Copy and paste these two equations in two separate boxes (looks crazy, but just paste it in the calculator and it should look clean). These are 2^x and the derivative of 2^x.
fleft(xright)=2^{x}
frac{d}{dx}left(2^{x}right)
Notice that the line for the derivative (the one that has the d/dx) is below the original function.
2. Same thing but with 5^x
fleft(xright)=5^{x}
frac{d}{dx}left(5^{x}right)
Now you’ll notice the derivative line is above the original function.
3. Now with e
fleft(xright)=e^{x}
frac{d}{dx}left(e^{x}right)
These two lines are on top of each other.
Hope that helps make more sense of it. From one layman to another
lots of great answers. But in short – it’s the only number where the derivative of exp(x) equals exp(x). Or more visually, draw a line y = exp(x)
– The value at x is, by definition, exp(x)
– The slope at x will also be exp(x)
– The area under the curve from -infinity to x will be, you guessed it, exp(x)
There is only 1 number that does this. It’s E.
Let’s start with £1.
100% interest once means I end up with £2.
50% interest twice means I end up with £2.25.
10% interest 10 times gets you closer to e.
Share that 100% interest more times and you will end up with e.
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