* If you invest $1 at 100% interest for 1 year, you get $2 at the end
* Compounded 2 times in a year, you get 100/2 = 50% interest every 1/2 year, which amounts to $2.25
* Compounded 4 times in a year, you get 100/4 = 25% interest every 1/4 year, which amounts to $2.44
* Compounded n times in a year, you get 100/n percent interest every 1/n year, which amounts to (1+1/n)^n dollars
* So continuous compound interest is the limit as n approaches infinity, which amounts to $2.71828 at the end of the year
(This is a great problem to give to pre-calc students to see if they can figure out the calculation for themselves.)
A few months later I talked to a super advanced math genius kid who had a signature that said e^(i*pi)+1=0 and I asked him if that was Euler's number. He was a super quiet skittish guy that rarely talked. His eyes lit up and he spent the next 2 hours teaching me about Taylor series and showed me how to prove it.
It remains the most fascinating math equation I have ever seen.
He had a lot of issues and dropped out because he couldn't pass a history class. I found that guy on Facebook 20 years later and thanked him. He didn't remember that but he was so happy that he made such a big impact on me.
e arises when you ask the question: is there a function that is its own derivative? And it turns out the answer is yes. It is this infinite series:
1 + x + x^2/2 + x^3/6 + ... x^n/n! ...
which you can easily verify is its own derivative simply by differentiating it term-by-term. When you evaluate this function at x=1, the result is e. In general, when you evaluate this function at any real value of x, the result is e^x.
But the Euler equation e^iπ = -1 has nothing to do with exponentiating e, it's just a notational convention that is defined to be the series above. When you evaluate that series at x=iπ, the result is -1. In general, the value of the series for any x=iy is cos(y) + i*sin(y).
It's that simple.
The other constants fundamental to science, like the gravitational constant or the speed of light, can only be measured, not discovered from nothing. We aren't even sure how constant they actually are, there might be extremely tiny variations in either time or space that our instruments just can't measure yet. In theory, other universes could exist where these constants are "set" a little bit differently; whether we could live in such universes is another matter entirely.
E, on the other hand, comes from pure mathematics. As long as fractions, addition and exponentiation work the same way in another hypothetical universe, this strange E number is going to have the same strange value.
Neither of those things are sort of empirically measured, and in any formula where they show up, you could theoretically absorb them into other constants -- and in fact the Einstein gravitational constant does exactly that -- it's defined as (8*pi*G)/c^4, absorbing pi into newton's gravitational constant (for historical reasons, when using plank units, they set G and c to 1, so it ends up just being 8pi -- _reduced_ planck units set the whole constant to 1). It's just frequently easier to separate out e and pi for the purposes of actually working out the math.
Relationship to pi; Euler's formula:
ix
e = cos x + isin x
d x x
-- e = e
dxThis is perhaps the most unnatural equation (well identity) in maths. It doesn't fall out anywhere, you would never write it down and solve for e, it's a special case of a more general result you would get first, and it's symbol soup for precisely the reason that the identity itself confers no understanding.
exp/log are natural because you almost can't help but discover them as they appear in so many different seemingly unrelated places.
That's certainly one way, but you can also define exp via its power series (which is easily proven to be convergent everywhere). Then, all the properties of exp, as well as Euler's formula, are actual theorems, not just definitions.
We want a^(i pi) + 1 = 0. Now,
a^(i pi) = e^(ln(a) i pi) = e^(i ln(a) pi) = cos( ln(a) pi) + i sin( ln(a) pi),
so we want cos( ln(a) pi) = -1, sin( ln(a) pi) = 0,
so ln(a) = 1, 3, 5, so a = e, e^3, e^5, ...
Thus indeed e^(i pi) + 1 = 0.
But also ln(a) = -1, -3, -5 work, so for example for
a = 1/e = 0.3678794412... > 0, we have
a^(i pi) + 1 = 0;
and of course for a = e^-99 = 1.0112214926104485... × 10^-43 etc.
(Off-by-one errors, they're not just for programmers!)
1 - x <= e^{-x} so e^x <= 1/(1 - x) for x < 1
(1 + x/n)^n <= e^x <= (1 - x/n)^{-n} for x < 1
Letting n go to infinity gives e^x = \sum_{n=0}^infy x^n/n! using Newton's binomial formula.
The number is what it is cause it's "increment of increment" is the same as it's "increment" when you are using exponentiation.
Why do we measure angles in radians? Because then d/dx (sin x) = 1 at x = 0, and sin x ≈ x for small x.
In my opinion drilling down too much on conventions misses the point of math.
A more practical way to measure angles would be in rotations. 0° = 0, 360° = 1, 90° = 0.25, etc. It would remove a ton of 2π and 4π² factors from a lot of equations in physics.
For example, try to work out the Taylor series for sin(x) using degrees (or rotations). It's awful.
e pops up quite often when taking limits on a surprising number of varied phenomena. It is much more than a mere convention, unless you subscribe to the nihilistic, anti-epistemological notion that all of mathematics is merely convention. It seems to be the center of the conceptual space particularly around questions of relative and absolute scale.
It's true you can use any base for computing things but some are more natural than others in that specific parametrizations have natural interpretations especially when it comes to physics (timescales, information-theoretic optimality, etc.).
90 degrees is one fourth of a circle, it would be so much more intuitive if we'd use "1/4th of something" rather than "1/2 radians" to express this
What is "the point of math?"
So I'm curious to know about this other... non-conventional point.
Something of a trinity
(Wink from Diety)