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What makes e natural? (2004)

komal.hu

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Re: What makes e natural? (2004)

#51

Post by School-Cotton »

nine_k wrote: post #21
mb7733 wrote: post #19

Earlier 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.

> Differentiation of sin x and cos x is the same independent of the unit of x,

That’s not true. If the unit is degrees, d/dx sin(x) = pi/180 * cos(x).

Re: What makes e natural? (2004)

#52

Post by moefh »

Tainnor wrote: post #41
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…

> 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…

Ah, a vaguely wondered if it had something involving Cauchy sequences, I think that was that limit!

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

I think it depends on how much you're used to dealing with complex numbers. In college, I was always taught to prove that Euler formula by replacing ix for x in that series, and then noting that the alternating signs and presence/absence of i in the terms allowed you to separate it into two series for cosine and sine. That always felt awkward, like there was no way anyone could just come up with that naturally.

Many years later I found that construction with graphing e^(ix) by taking small steps using f(x+dx)=f(x)+f'(x)*dx, and everything clicked: how exponentials work in the real axis is pretty different from the imaginary axis, but both are completely intuitive and unavoidable once you understand that.

Even later I "discovered" the connection with group theory[1]; that one still blows my mind.

[1] This is a really nice explanation: https://www.youtube.com/watch?v=mvmuCPvRoWQ

Re: What makes e natural? (2004)

#53

Post by saintradon »

miki123211 wrote: post #46
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…

That's the same with pi if you think about it, and in my eyes is what makes it a transcendental number, so to speak. The numbers themselves could be different if you used a different base instead of base 10, but they still just represent a particularly useful value for us.

Re: What makes e natural? (2004)

#54

Post by [deleted] »

miki123211 wrote: post #46
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…

[deleted]

Re: What makes e natural? (2004)

#56

Post by 9question1 »

"A base of 2 is useful because there are several small positive integers whose base-two logarithms are also integers." What? No! Base 2 is natural in exactly the same way that base e is natural, except for discrete domains instead of continuous domains. There is a unique family of functions for which the rate of change of the function is equal to the current value of the function everywhere. On discrete domains it's some scaled translation of 2^x, and on continuous domains it's some scaled translation of e^x. "Some scaled translation" here is accounting for the fact that the function is only uniquely exactly 2^x or e^x if we also add the constraint that f(0)=1.

Re: What makes e natural? (2004)

#57

Post by pif »

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.

Re: What makes e natural? (2004)

#58

Post by jerf »

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…

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 course, that doesn't show how it will show up in all sorts of other places; "the one and only function that is its own derivative" strikes me as more likely to be something we encounter everywhere.

Re: What makes e natural? (2004)

#59

Post by empath75 »

miki123211 wrote: post #46
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…

e isn't really a fundamental physical constant and doesn't show up very much in famous scientific formulas. It's a number that has very convenient properties for performing calculus operations and shows up in that context. pi isn't really an important physical constant, either, but it shows up a lot because of two things --- it's useful for calculations involving cycles, and because things being spherical or circular is a useful simplifying assumption (and also because things particularly in astronomy end up being spherical or circular (or nearly) quite often in reality).

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.

Re: What makes e natural? (2004)

#60

Post by jerf »

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…

"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."

That depends on your point of view. You can also view exponentiation as "really" being e's infinite series, and it so happens that that matches what you get in the case of repeated multiplication. The advantage of that is now you can start exponentiating a lot more than just numbers. Here's 3blue1brown on raising e to the power of a matrix: https://www.youtube.com/watch?v=O85OWBJ2ayo

In general there is fruit in viewing infinite series as the fundamental building block of a lot of math and non-infinite series as the special case. I won't claim which is "real" or "correct", though, just point out that there is value in viewing "repeated multiplication" as the special case rather than the "real" thing. Of course you can always view exponentiation as the generalization too.

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