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What makes e natural? (2004)

komal.hu

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Re: What makes e natural? (2004)

#41

Post by Tainnor »

moefh wrote: post #33
lisper wrote: post #27

Earlier quoted context omitted.

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 hav…

> 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!).

a^b, for positive a and irrational b can also be defined as lim (x -> b, x € Q) a^x - which is possible because Q is dense in R. This is a pretty natural way of extending a function to the reals.

The way we extend exponentiation to complex exponents is IMHO much less straightforward.

Re: What makes e natural? (2004)

#43

Post by jgalt212 »

Tainnor wrote: post #40
lisper wrote: post #27

Earlier quoted context omitted.

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.

Fair enough, but IIRC students are first taught about converging and non converging series in pre calc. So then you get to calc, and work on the e proof, but first you say, "if you paid attention last year, you'll remember this series converges."

Re: What makes e natural? (2004)

#44

Post by Tainnor »

jgalt212 wrote: post #43
Tainnor wrote: post #40

Earlier quoted context omitted.

> 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.

Fair enough, but IIRC students are first taught about converging and non converging series in pre calc. So then you get to calc, and work on the e proof, but first you say, "if you paid attention last year, you'll remember this series converges."

Mathematics education seems to be really different depending on the country (I'm in Europe). I barely saw a proof (except maybe some geometry proofs) in any of my high school maths classes and had to learn what a mathematical proof is in university. But I think this also depends on what focus area / profile you have in high school (mine was languages).

Re: What makes e natural? (2004)

#45

Post by diffeomorphism »

lisper wrote: post #27
mihaic wrote: post #23

Earlier 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…

The exponential function arises from multiple sources:

- the solution of the ode you just stated.

- compound interest.

- The defining property of exponential functions is f(x+y)=f(x)f(y) with some normalization.

- Moving on the unit circle is given by an exponential function because rotation is a group, i.e. a^(i(x+y)). Now choose the basis a such that you move with unit speed.

- ...

The nice thing is that all of these very different motivations lead to the same thing.

The "has nothing to do with exponentiating e" I would strongly disagree with. It has everything to do with exponentiating and is exactly the only way exponentiation can work. So afterwards you can pretend you didn't know that and define exponentiation by using e. Same for matrix exponentials, semigroups etc.

Re: What makes e natural? (2004)

#46

Post by miki123211 »

munchler wrote: post #20

This is a frustrating article because it never explains why e is the natural logarithm base. To me, the easiest way to understand it is via continuous compound interest: * 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, whi…

This is why I find e so fascinating, frustrating and puzzling.

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.

Re: What makes e natural? (2004)

#47

Post by diffeomorphism »

moefh wrote: post #33
lisper wrote: post #27

Earlier quoted context omitted.

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 hav…

> but you can't do that when n is a fraction

Sure you can. You know what 2^n is and you want that 2^(1/3) 2^(1/3) 2^(1/3)= 2^1=1. That uniquely defines the exponential function on the rationals. For the real numbers you need some amount of continuity or measurability, but then it is also uniquely determined.

> 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!).

No, you don't just "say" that. You prove it. Big difference.

Re: What makes e natural? (2004)

#48

Post by miki123211 »

ralferoo wrote: post #34
lisper wrote: post #27

Earlier quoted context omitted.

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 gr…

See the great article on e from Better Explained:

https://betterexplained.com/articles/an-intuitive-guide-to-e...

Re: What makes e natural? (2004)

#49

Post by Tainnor »

Spivak wrote: post #17
cperciva wrote: post #6

e is natural because it's the smallest positive root of the equation e^(i pi) + 1 = 0 .

Nonsense, this equation doesn't even make any sense without a well defined notion of the exponential function, and then a well defined extension of said function into the complex numbers. You will already have e by the time you reach this equation because defining what exp(z) even means requires you already know the properties of e^x over the reals you wish to preserve. And Euler's formula comes from finding such a f…

> And Euler's formula comes from finding such a function and then defining it to be exp(z).

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.

Re: What makes e natural? (2004)

#50

Post by Tainnor »

diffeomorphism wrote: post #47
moefh wrote: post #33

Earlier quoted context omitted.

> 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 hav…

> but you can't do that when n is a fraction Sure you can. You know what 2^n is and you want that 2^(1/3) 2^(1/3) 2^(1/3)= 2^1=1. That uniquely defines the exponential function on the rationals. For the real numbers you need some amount of continuity or measurability, but then it is also uniquely determined. > 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…

> No, you don't just "say" that. You prove it. Big difference.

Whether a^b=exp(b*log(a)) is a definition or a proof really depends on how exactly you define certain terms (e.g. exp). What's certainly a theorem that requires a proof is that the definition of a^b (for irrational b) via limits of rational exponents and the one via exp are equivalent.

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