The simplest way that I could put it using words is:
The number is what it is cause it's "increment of increment" is the same as it's "increment" when you are using exponentiation.
What makes e natural? (2004)
31–40 of 95 posts
Re: What makes e natural? (2004)
#32Earlier quoted context omitted.
Exactly the same for me and I'm pretty sure this is how Jakob Bernoulli came to define the number as well, trying to see what the upper bound for infinitesimal compounding was.
That may be how it was arrived at historically, but it is not the best way to explain it. 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…
Note that C + Cx + ... would have also worked. So why the 1? Just simplicity?
Re: What makes e natural? (2004)
#33Earlier quoted context omitted.
Exactly the same for me and I'm pretty sure this is how Jakob Bernoulli came to define the number as well, trying to see what the upper bound for infinitesimal compounding was.
That may be how it was arrived at historically, but it is not the best way to explain it. 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…
> 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.
Can't the same thing be said about using fractions on the exponent? Exponentiation is actually just repeated multiplication (a^n=a*a*...*a, repeated n times), but you can't do that when n is a fraction or irrational anymore than you can do it when it's imaginary.
We have to define what it means for an exponent to be non-integer: for fractions we might define a^(b/c) as the root of the equation x^c=a^b, and to allow irrationals I think you need some real analysis (it's been a while, but I think the usual way is to first define exp and log, and then say that a^b=exp(b*log(a)), which is kind of cheating because we have to define exp first!).
There's a very intuitive way to "see" that e^ix=cos(x)+i*sin(x): all you have to do is to treat complex numbers like you would any other number, and "believe" the derivative rule for complex numbers (so (e^(ix))'=ie^(ix)). Then you can just graph f(x)=e^(ix) for real x by starting at x=0 (when clearly f(x)=1) and from there take small steps in the x axis and use the derivative to find the value of the next step with the usual formula f(x+dx)=f(x)+f'(x)*dx.
Doing that you realize the image of e^(ix) just traces a circle in the complex plane because every small step in the x direction makes e^(ix0) walk a small step perpendicular to the line going from 0 to e^(ix0), simply because multiplying by i means rotating 90 degrees.
Re: What makes e natural? (2004)
#34Earlier quoted context omitted.
Exactly the same for me and I'm pretty sure this is how Jakob Bernoulli came to define the number as well, trying to see what the upper bound for infinitesimal compounding was.
That may be how it was arrived at historically, but it is not the best way to explain it. 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…
It's a pretty good way of explaining it.
I've seen (1+1/n)^n before, but never seen an explanation of why I might ever want to use something of that form. I've used the e^ix notation extensively, but again I've never really cared, because to me it was just a compact representation of sin and cos together. Likewise, all the proofs in that article are still a bit "so at this point on this carefully chosen graph, the gradient is e" and I think "Who cares? You carefully chose the graph to prove a point, I'll never see it in the real world."
And then the example with compounding interest - immediately I can see the application. It's definitely a good way of explaining it, although maybe it'd be even more grounded if it had n=12 and n=365 as examples. When you notice that the actual values seem to be converging, then you can try plugging in ever bigger and bigger numbers. This way you can discover the value of e for yourself and that process of discovery leads to a better understanding than rote learning of an abstract thing you haven't mentally visualised yet. All the other explanations are useful later, and they allow you to see it in different situations, but having at least one "this is why it's tangibly useful" hook at the start is definitely a massive help in understanding something.
Re: What makes e natural? (2004)
#35e is the unique real number satisfying 1 + x 1 - x (1 + x/n)^n Letting n go to infinity gives e^x = \sum_{n=0}^infy x^n/n! using Newton's binomial formula.
Re: What makes e natural? (2004)
#36Earlier quoted context omitted.
Exactly the same for me and I'm pretty sure this is how Jakob Bernoulli came to define the number as well, trying to see what the upper bound for infinitesimal compounding was.
That may be how it was arrived at historically, but it is not the best way to explain it. 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…
> it's just a notational convention
I'd argue it's a bit more, it's a natural way to extend domain of exponentiation, as we do before this with naturals -> integers -> reals.
Re: What makes e natural? (2004)
#37Things like Relationship to pi; Euler's formula: ix e = cos x + isin x derivative is itself: d x x -- e = e dx
Sorry, you cannot use "imaginary" numbers to define something as "natural"!
I guess imagination doesn't come naturally to you, then?
Ah, that explains why you didn't see this obvious one coming.
Re: What makes e natural? (2004)
#38e is the unique real number satisfying 1 + x 1 - x (1 + x/n)^n Letting n go to infinity gives e^x = \sum_{n=0}^infy x^n/n! using Newton's binomial formula.
1+x is the tangent line at x=0. Since the graph of an exponential lies above its tangent line, 1+x <= e^x is another way of saying that the derivative of e^x at x=0 is 1, ie. e^x is its own derivative.
Re: What makes e natural? (2004)
#39Earlier quoted context omitted.
For trigonometry/calculus/physics radians are by far the most practical because they are dimensionless, so no constants appear when differentiating or integrating. (By the way, these constants will involve factors of pi anyway, it's inherent.) For example, try to work out the Taylor series for sin(x) using degrees (or rotations). It's awful.
I don't see how Taylor series specifically would be affected. Differentiation of sin x and cos x is the same independent of the unit of x , and nothing else is used in the series. Fourier transform would have 4π² instead of 2π under the exponent, no big deal. The Euler's formula gets a factor of 2π under the exponent though. Given its wide application, it adds plenty of noise, of course.
If you define variants sint(x) = sin(2πx) and cost(x) = cos(2πx) that takes x in units of turns instead of radians, then d/dx sint(x) = 2π cost(x) etc.
I agree with you that this is completely fine though. I also find it more natural to think of “how many percent of a turn” an angle is than how many “degrees” or “radians” something is, since we use base-10 everywhere else. My workaround is to mostly write everything in terms of sin(2πτ), cos(2πτ), and exp(2πiτ) when I can, where τ measures turns.
Re: What makes e natural? (2004)
#40Earlier quoted context omitted.
Exactly the same for me and I'm pretty sure this is how Jakob Bernoulli came to define the number as well, trying to see what the upper bound for infinitesimal compounding was.
That may be how it was arrived at historically, but it is not the best way to explain it. 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…
> which you can easily verify is its own derivative simply by differentiating it term-by-term.
Yes, but it's not a priori clear that the series actually even converges for all x, and the fact that power series are differentiable term-by-term within their radius of convergence also requires proof.