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What makes e natural? (2004)

komal.hu

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Re: What makes e natural? (2004)

#11

Post by nine_k »

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.

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.

Re: What makes e natural? (2004)

#12

Post by akira2501 »

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?"

Re: What makes e natural? (2004)

#13

Post by __MatrixMan__ »

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?"

I thought it was about finding conventions that are useful, surprising, or pleasing in some way. Or, given a set of conventions, finding new ways that they are useful, surprising, or pleasing.

So I'm curious to know about this other... non-conventional point.

Re: What makes e natural? (2004)

#16

Post by JadeNB »

nine_k wrote: post #9
cperciva wrote: post #8

Earlier quoted context omitted.

No, we measure angles in radians so that e^(ix) = cos x + i sin x.

But this does not depend in the unit.

> But this does not depend in the unit.

It does!

e^z, defined as the series \sum_{n = 0}^\infty z^n/n!, can only be a function of a dimensionless number z.

sin(z) and cos(z), defined as power series, technically also work this way. And that's OK, because angles are dimensionless: a radian is just C/(2πr), where C is the circumference of a circle of radius r. But it is sometimes convenient to pick your favorite number of radians, like π/180 of them, and call that a degree, and then to say that sin(x degrees) is the same as sin(xπ/180 radians).

With this convention, where the left-hand side of e^(ix) = sin(x) + icos(x) is a function of a dimensionless variable, and the right-hand side can be viewed as a function of a dimensioned argument only in the sense written above, it really is the case that the equation written is true, but the equation e^(ix) = sin(x degrees) + icos(x degrees) is false.

(On the other hand, you could make the case that e^(ix) is really a function of an angle, where its value is the complex number that lies on the unit circle at that angle. Then you do recover a "dimensioned" version of e^(ix) = sin(x) + i*cos(x) that's valid even if you measure angles in degrees.)

Re: What makes e natural? (2004)

#17

Post by Spivak »

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 function and then defining it to be exp(z). Multiplication in the complex plane by a unit vector is a rotation. Exponentials "repeated multiplication" by such a vector is spinning. And it turns out spinning at a constant rate satisfies the properties of the exponential function so it makes sense to say that's what exp(ix) means.

This is perhaps the most unnatural equation (well identity) in maths. It doesn't fall out anywhere, you would never write it down and solve for e, it's a special case of a more general result you would get first, and it's symbol soup for precisely the reason that the identity itself confers no understanding.

exp/log are natural because you almost can't help but discover them as they appear in so many different seemingly unrelated places.

Re: What makes e natural? (2004)

#18

Post by kazinator »

nine_k wrote: post #9
cperciva wrote: post #8

Earlier quoted context omitted.

No, we measure angles in radians so that e^(ix) = cos x + i sin x.

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 the unit circle as the measure of the angle.

Re: What makes e natural? (2004)

#19

Post by mb7733 »

nine_k wrote: post #11
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.

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.

Re: What makes e natural? (2004)

#20

Post by munchler »

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, which amounts to $2.44

* Compounded n times in a year, you get 100/n percent interest every 1/n year, which amounts to (1+1/n)^n dollars

* So continuous compound interest is the limit as n approaches infinity, which amounts to $2.71828 at the end of the year

(This is a great problem to give to pre-calc students to see if they can figure out the calculation for themselves.)

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