As the title says, how come the value e is special? it has functions and rules just dedicated to it, but what sets it apart from another random number?
In: 397
The things where it’s special are all related.
It’s the constant of “self growth”. If you grow your something with a slope that is equal to the current value it will multiply by e within one time unit.
All other cases where e matters can in some way be related to exactly that situation
Look at the function: (1+(1/x))^x and think about what happens as x grows. The whole term will grow exponentially. However, (1/x) will get smaller and smaller until it is essentially 0. These growing and shrinking fight eachother as x grows, making it hard to determine what x will be. It turns out, that function converges to e as x approaches infinity. That is the definition of e, and is what makes it a special number. This is useful because that function is used to describe situations where things grow exponentially. (Lots of things grow exponentially, so it comes up often.)
e has cool math properties that math peeps geek out about. I once had a college TA who mocked the class for getting a problem wrong because we solved it assuming log base 10 when it should have been the natural log, i.e. log base e. Then he said, “remember in math we have e number of fingers.”. I love math, but I rolled my eyes hard.
Imagine you have a snowball rolling down a hill. As it collects snow, it grows.
Let’s say that the amount of snow it collects depends on how much snow it already has – so it’s getting bigger faster and faster! What would a plot of its size look like?
It turns out that you’ll find an ‘e’ in the equation you’ll need to describe this slope – e is 2.7 and change. The equation y=e^(x) creates a curve whose slope (rate of change) is equal to its value at every point.
So lots of things in nature involve this kind of growth – like a colony of bacteria where the rate of growth depends on the number of bacteria in it. It’s very special in calculus, as y=e^(x) is unaffected by derivatives. The questions ‘what’s the value of this function’ and ‘how fast is this function changing (i.e. what is its derivative?)’ have the same answer!
another thing that is very closely related to the number e is the exponential function, usually denotes as e^x or exp(x). this function has very interesting properties for complex numbers (numbers which contain an “imaginary part” in addition to the real part, basically extending the numbers line with a second axis to cover a plane).
This function is at the heart of the famous equation e^i ^pi -1 = 0 and is very prominent through all of engineering and math. so many things where e pops up actually refer to the exponential function an not (directly) to the nunber itself.
It’s a constant related to a quantity that grows at a rate proportional to its own size, like a mythical dragon that eats constantly and that gets bigger when it eats more, and eats more when it gets bigger. In particular the dragon grows M times faster than normal when its size is M, so by the time it’s double in size it’ll eat twice as fast which means it’ll grow twice as fast and be able to eat even faster, and then grow even faster, and on and on in a positive feedback loop.
Let y be the size of the dragon (say in metric tons), and x be how much time has passed (say in years). For the rate of growth we’ll use metric tons per year, and the rate of growth updates continuously (the split moment the dragon gets bigger, it also eats faster). If we chart the size of the mythical dragon as
y = 2^(x)
then it grows at a rate ‘less than’ its own size.
In the case of
y = 3^(x)
it grows at a rate ‘greater than’ its own size.
However, when
y = e^(x)
it grows at a rate exactly ‘equal’ to its own size.
(Equal meaning the number parts are equal, so 4 metric tons and 4 metric tons per year are ‘equal’)
I’ll try to do it without equations.
Imagine you have money in a high risk savings account. Let’s say $1000, which is earning 100% interest per year. If interest is calculated once, at the end of the year, you would have $2000, right?
Suppose I now say the annual rate is still 100% but I’ll calculate the interest and add it on to your original amount every six months. (So 50% return each half year).
This would make you $250 better off! (As you now *also* get the second half of the year’s 50% on the first half of the year’s $500 interest.)
So, the more often we calculate the interest, the better – even though the yearly return is still 100%
Let’s take it further. If the interest calculation was monthly, then the yearly return would be $1613. So 2.61 times your starting amount by me just splitting your 100% return into smaller slices.
So the question is – how far can we take this? What effective return would we get if we recalculated daily? hourly? every minute?
Well it turns out that the answer is ‘e’. If we could recalculate the 100% interest every second or less then we would get end up with $2718.28
I was playing zombicide and decide to make an excel sheet to tell me the probability that I would get a 1 on a roll of dices at least one time.
Then I was rising the number of dices with the number of sides on those dices.
2 2d has 75% of chance
3 3d has 70,3%
And further I go it comes closer to 63.2%
So infinite rolls of dices with infinite sides has 1-1/e chance (63.21205588285%) of landing at least one number 1
Do what you want with this information.
I am not all that familiar with what is going on and would like to know more.
Am i the only one who thinks OP just meant why did mathematicians choose the letter “e” ? Like you always hear e=mc2 . Why not j=mc2?
Or am i just an idiot?
> it has functions and rules just dedicated to it, but what sets it apart from another random number?
It makes a lot more sense if you flip that around.
You have a whole bunch of interesting questions. Because the questions are related to each other, the answers to those problems are themselves (unsurprisingly) related. Because the questions are all “numbers problem” sort of equations, the answers are all number answers. Put the two together, and you’re going to find the same numbers cropping up over and over again, because you can describe the answer to one problem in terms of the answer to another problem: “_foo(x)_ is the same as _√(bar(x^2))_”, that sort of thing.
What you’re left with is a collection of numbers that are notable for one reason or another in specific branches of mathematics. When you’re talking about problems related to circles and angles, the number that keeps cropping up is _π_. In computer science, it’s 2 (and not just because we use binary computers, it’s more fundamental than that). When you’re talking about things that involve exponentials, that number is _e_.
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