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What makes e natural? (2004)
- smitty1e 2y agoe, 3, and pi be Something of a trinity (Wink from Diety)
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- RicoElectrico 2y agoMuch 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.
- uoaei 2y agoWithout delving too far into the philosophy of math as concerns existence vs convention... e pops up quite often when taking limits on a surprising number of varied phenomena. It is much more than a mere convention, unless you subscribe to the nihilistic, anti-epistemological notion that all of mathematics is merely convention. It seems to be the center of the conceptual space particularly around questions of relative and absolute scale. It's true you can use any base for computing things but some are more natural than others in that specific parametrizations have natural interpretations especially when it comes to physics (timescales, information-theoretic optimality, etc.).
- programjames 2y agoNo, we measure angles in radians so that d^4/dx^4 (sin x) = sin x.
- cperciva 2y agoNo, we measure angles in radians so that e^(ix) = cos x + i sin x.
- nine_k 2y agoBut this does not depend in the unit.
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- JadeNB 2y ago> 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.)
- kazinator 2y agoIf 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.
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- nine_k 2y agoWe 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.
- mb7733 2y agoFor 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.
- nine_k 2y agoI 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.
- setopt 2y agoIf you define variants sint(x) = sin(2πx) and cost(x) = cos(2πx) that takes x in units of turns instead of radians, then d/dx sint(x) = 2π cost(x) etc. I agree with you that this is completely fine though. I also find it more natural to think of “how many percent of a turn” an angle is than how many “degrees” or “radians” something is, since we use base-10 everywhere else. My workaround is to mostly write everything in terms of sin(2πτ), cos(2πτ), and exp(2πiτ) when I can, where τ measures turns.
- School-Cotton 2y ago> 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).
- akira2501 2y ago> drilling down too much on conventions misses the point of math. What is "the point of math?"
- __MatrixMan__ 2y agoI 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.
- ithkuil 2y agoWell, the point, in math, is a zero dimensional object.
- Aardwolf 2y agoOther 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
- pif 2y agoNo! We measure angles in radians because it's the simplest way to link the length of an arc to the radius of the circle.
- aidenn0 2y agoBut 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.
- cperciva 2y agoe is natural because it's the smallest positive root of the equation e^(i pi) + 1 = 0.
- mr_toad 2y agoIn high school we were taught that it is because The slope of the function e^x is equal to e^x. Growth being proportional to magnitude is natural.
- Spivak 2y agoNonsense, 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.
- Tainnor 2y ago> 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.
- SassyBird 2y ago
- kazinator 2y agoThings like Relationship to pi; Euler's formula: ix e = cos x + isin x derivative is itself: d x x -- e = e dx
- setopt 2y agoAnd e^x is the inverse of \int dx/x, which pops up a lot.
- pif 2y agoSorry, you cannot use "imaginary" numbers to define something as "natural"!
- kazinator 2y agoI guess imagination doesn't come naturally to you, then? Ah, that explains why you didn't see this obvious one coming.
- raincole 2y agoTIL a two dimensional plane isn't natural.
- Tainnor 2y agoC is a plane with multiplication. I would say, the fact that there is even a way of making this work is surprising (it doesn't work for any R^n, n > 2, at least not if you want a field). The particular way how we multiply complex numbers is also IMHO not self-evident (neither in the algebraic, nor in the geometric interpretation). It just turns out that doing it this way gives us really nice properties.
- kazinator 2y ago> how we multiply complex numbers is also IMHO not self-evident Multiplication of complex number in the x + iy form treats them exactly like any other binomial factors, using the "FOIL" rule: F O I L (a + bi)(c + di) = ac + adi + bci + bdi^2 = ac + (ad + bc)i - bd = ac - bd + (ad + bc)i I don't know about self-evident; you pretty much have to do it that way. If you hand (a + bi)(c + di) to someone who knows basic algebra, but has no idea what i is, they will come up with the first FOIL expression above; after that, we have to know that i^2 is -1. > nor in the geometric interpretation. That isn't self-evident, but when you gemoetrically work out what the multiplication is doing, it's just "add the arguments (angles), multiply the moduli (distances from origin)". Very simple!
- munchler 2y agoThis 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.)
- mihaic 2y agoExactly 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.
- lisper 2y agoThat 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.
- psini 2y agoSorry 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
- reliablereason 2y agoThe simplest way that I could put it using words is: The number is what it is cause it's "increment of increment" is the same as it's "increment" when you are using exponentiation.
- keithalewis 2y agoe is the unique real number satisfying 1 + x <= e^x for all x. 1 - x <= e^{-x} so e^x <= 1/(1 - x) for x < 1 (1 + x/n)^n <= e^x <= (1 - x/n)^{-n} for x < 1 Letting n go to infinity gives e^x = \sum_{n=0}^infy x^n/n! using Newton's binomial formula.
- edflsafoiewq 2y ago1+x is the tangent line at x=0. Since the graph of an exponential lies above its tangent line, 1+x <= e^x is another way of saying that the derivative of e^x at x=0 is 1, ie. e^x is its own derivative.
- queuebert 2y ago> e is the unique real number satisfying 1 + x <= e^x for all x. I think if you replace e by 2e, this still holds. Therefore your definition of e is not unique.
- FabHK 2y ago1 + (-0.1) = 0.9 (2e)^(-0.1) = 0.844243... The former is not <= the latter.
- 9question1 2y ago"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.
- User23 2y agoOne of my favorite parts if the excellent book Visual Complex Analysis is where he explains that e is defined in terms of the derivative of the exponential function. E is e because that’s what it has to be for the exponential function to be its own derivative.
- bjornsing 2y agoAs I remember it e^x is the only function which is its own derivative, which I guess makes e “natural” in some sense.
- rjbwork 2y agoTrivial, but y=0 is also its own derivative.