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What makes e natural? (2004)

komal.hu

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Re: What makes e natural? (2004)

#21

Post by nine_k »

mb7733 wrote: post #19
nine_k wrote: post #11

Earlier quoted context omitted.

We measure angles in radians because it's an easy way to measure the length of the circle sector in the units of the circle's radius. It feels neat in many cases. 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 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.

Re: What makes e natural? (2004)

#22

Post by kazinator »

kazinator wrote: post #18
nine_k wrote: post #9

Earlier quoted context omitted.

But this does not depend in the unit.

If we use a degrees version of sin and cos (call them sind and cosd), then we cannot have e on the left side without a conversion factor. (iπx/180) e = cosd x + i sind x ix π/180 -> e = cosd x + i sind x π/180 -> let f = e ix f = cosd x + i sind x Probem is, f doesn't have nice properties like: d x x - f /= f dx There is something uniquely special about the unit circle, and about using the unscaled distance around th…

Radians have the property that if we step x by some tiny amount δ, then the cos/sin coordinates will move by that same distance around the unit circle:

   |[cos(x+δ) + i sin(x+δ)] - [cos(x) + i sin(x)]| = δ
This is also related to how we can estimate sin(x) = x for small values next to zero, if using radians.

In radians, the derivative sin'(x) is cos(x), and cos'(x) is -sin(x). Derviation just shifts the waveform left by ninety degrees. In units other than radians, we get wacky constant terms that change at each step.

That's related to how e^x is its own derivative.

Re: What makes e natural? (2004)

#23

Post by mihaic »

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…

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.

Re: What makes e natural? (2004)

#26

Post by ithkuil »

akira2501 wrote: post #12
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.

> drilling down too much on conventions misses the point of math. What is "the point of math?"

Well, the point, in math, is a zero dimensional object.

Re: What makes e natural? (2004)

#27

Post by lisper »

mihaic wrote: post #23
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…

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 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.

Re: What makes e natural? (2004)

#28

Post by psini »

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…

Sorry but as a layman high-school educated guy, GP's explanation was a lot easier to follow with it having a "natural" real-life example

Re: What makes e natural? (2004)

#29

Post by mihaic »

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…

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.

Re: What makes e natural? (2004)

#30

Post by Aardwolf »

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.

Other chosen conventions are less great, like not having the circle constant be 6.283185... so that we don't need to deal with factors or divisors of 2 all the time when using radians or almost anything else involving pi

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

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