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What makes e natural? (2004)

komal.hu

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Re: What makes e natural? (2004)

#81

Post by lisper »

jerf wrote: post #58
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…

Another way to explain it, which I think is a "best" explanation in its own right, is that if you start differentiating exponential functions you can't help but discover e . The general differentiation of b^x, for b a constant and x the variable, is ln(x)*b^x. And of course to understand ln you need e . Or, to put it another way, you will blunder into this somewhere around month two of calculus 1, unavoidably. Of cou…

> The general differentiation of b^x, for b a constant and x the variable, is ln(x)*b^x.

That's true, but it's begging the question because you can't define ln without already knowing about e.

You have to go back to first principles:

d(b^x) := lim(∂->0):(b^(x+∂) - b^x)/∂ = ((b^x)(b^∂) - b^x)/∂ = (b^x)(b^∂-1)/∂

So d(b^x) is itself multiplied by lim(∂->0):(b^∂-1)/∂. But now what? How do you evaluate that limit? How do you show that e is the magic value of b that makes that limit turn out to be 1? And in particular, how do you show that to someone whose only background knowledge is how to differentiate polynomials?

IMHO it's a lot easier to see that e is the value to which the polynomial series that is its own derivative converges at x=1.

Re: What makes e natural? (2004)

#82

Post by lisper »

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…

> Can't the same thing be said about using fractions on the exponent?

Yes, it can. In fact, it is really useful to think of the "usual" definition of exponentiation as repeated multiplication as nothing more than a special case of a much more general concept, which is evaluating a function whose defining property is that it is its own derivative.

Re: What makes e natural? (2004)

#83

Post by lisper »

miki123211 wrote: post #48
ralferoo wrote: post #34

Earlier quoted context omitted.

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...

> e is the base rate of growth shared by all continually growing processes.

I don't think that's a very good explanation because all the heavy lifting is being done by the phrases "base rate" and "continually growing", nether of which are well defined. "Continually growing" could reasonably be interpreted to mean "monotonically increasing" in which case f(x)=x qualifies and the whole explanation falls apart.

I think the idea of finding a function that is its own derivative, discovering that there is a polynomial series that meets that requirement, and evaluating that series at x=1 is a much more natural explanation.

Re: What makes e natural? (2004)

#85

Post by cperciva »

FabHK wrote: post #73
cperciva wrote: post #6

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

Not really. Your equation has no unknowns, but if you consider the "e" in your equation as the unknown, and solve it (here I use "a" instead as the unknown, and use "e" in its usual meaning): 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)…

Oops, I meant the smallest root greater than 1, not the smallest root greater than 0.

(Off-by-one errors, they're not just for programmers!)

Re: What makes e natural? (2004)

#86

Post by Tainnor »

lisper wrote: post #79
mihaic wrote: post #29

Earlier quoted context omitted.

For me these concept seem intuitively linked, simply because derivation measures the rate of change, and that compounding definition aims to find the compounding function with constant growth. The factorial definition only seems to make for an easier proof of the derivation of e^x.

That's kind of the point. The only background knowledge you need to understand my definition is how to differentiate a polynomial. Anyone capable of understanding what it even means to find a function that is its own derivative is almost certainly going to know that.

a power series is not a polynomial though.

Re: What makes e natural? (2004)

#87

Post by earthboundkid »

lisper wrote: post #83
miki123211 wrote: post #48

Earlier quoted context omitted.

See the great article on e from Better Explained: https://betterexplained.com/articles/an-intuitive-guide-to-e...

> e is the base rate of growth shared by all continually growing processes. I don't think that's a very good explanation because all the heavy lifting is being done by the phrases "base rate" and "continually growing", nether of which are well defined. "Continually growing" could reasonably be interpreted to mean "monotonically increasing" in which case f(x)=x qualifies and the whole explanation falls apart. I think…

You don't need or even want well defined terms when you're trying to explain why a mathematical formula is useful or naturally applicable to real world situations.

Re: What makes e natural? (2004)

#88

Post by aidenn0 »

pif wrote: post #57
RicoElectrico wrote: post #4

Much in math is a matter of convention. But, to paraphrase - all conventions are possible, but some are useful. 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.

No! We measure angles in radians because it's the simplest way to link the length of an arc to the radius of the circle.

But we have an inconsistency because pi relates the diameter to the circumference, not the radius to the circumference, which leaves an annoying 2 in the 2pi radians in a circle.

Re: What makes e natural? (2004)

#89

Post by lisper »

earthboundkid wrote: post #87
lisper wrote: post #83

Earlier quoted context omitted.

> e is the base rate of growth shared by all continually growing processes. I don't think that's a very good explanation because all the heavy lifting is being done by the phrases "base rate" and "continually growing", nether of which are well defined. "Continually growing" could reasonably be interpreted to mean "monotonically increasing" in which case f(x)=x qualifies and the whole explanation falls apart. I think…

You don't need or even want well defined terms when you're trying to explain why a mathematical formula is useful or naturally applicable to real world situations.

Maybe you don't want well-defined terms in your pedagogy, but I do.

Re: What makes e natural? (2004)

#90

Post by lisper »

Tainnor wrote: post #86
lisper wrote: post #79

Earlier quoted context omitted.

That's kind of the point. The only background knowledge you need to understand my definition is how to differentiate a polynomial. Anyone capable of understanding what it even means to find a function that is its own derivative is almost certainly going to know that.

a power series is not a polynomial though.

That's splitting a pretty fine technical hair for pedagogy at this level.
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