One 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.
What makes e natural? (2004)
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Re: What makes e natural? (2004)
#62Earlier 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…
Have you happened to have read Visual Complex Analysis?
If not I think you might like it.
And a tangent from your username: I quite like how complex numbers and functions like CIS made it into Common Lisp.
Re: What makes e natural? (2004)
#63Earlier 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…
> But the Euler equation e^iπ = -1 has nothing to do with exponentiating e
It has, that's the beauty of it. You can define as usual the function x -> e^x on the real line. Now, complex analysis tells us that if this function can be extended to a holomorphic function on the whole complex plane, then the extension is unique. And, in fact, this function does admit such an extension, so you can compute e^z for any complex number z, and in this way one gets e^iπ = -1.
Re: What makes e natural? (2004)
#64This 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…
We did that in 11th grade pre-calculus class. We got the number 2.71828 and then our teacher said "Now go to that button on your graphing calculator that says e^x and type e^1" We did and our minds were blown. Why was this strange number that we "discovered" in our calculator already?
A few months later I talked to a super advanced math genius kid who had a signature that said e^(i*pi)+1=0 and I asked him if that was Euler's number. He was a super quiet skittish guy that rarely talked. His eyes lit up and he spent the next 2 hours teaching me about Taylor series and showed me how to prove it.
It remains the most fascinating math equation I have ever seen.
He had a lot of issues and dropped out because he couldn't pass a history class. I found that guy on Facebook 20 years later and thanked him. He didn't remember that but he was so happy that he made such a big impact on me.
Re: What makes e natural? (2004)
#65Earlier quoted context omitted.
Sorry, you cannot use "imaginary" numbers to define something as "natural"!
TIL a two dimensional plane isn't natural.
C 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.
Re: What makes e natural? (2004)
#66Earlier quoted context omitted.
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.
There's a good number of constants, and some of them are quite fun: https://en.wikipedia.org/wiki/List_of_mathematical_constants
One of my favorites is https://en.wikipedia.org/wiki/Feigenbaum_constants . Pops up in many situations, and as I understand it, it is not even well-understood why it pops up in as many places as it does. Intuitively one would expect that many of the places it pops up ought to have their own local constant of some sort, but instead this one keeps popping up. 7-year-old Numberphile video on it: https://www.youtube.com/watch?v=ETrYE4MdoLQ
Re: What makes e natural? (2004)
#67Earlier 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…
b^x = (e^ln(b))^x = e^( ln(b) * x ),
and its derivative then is (by the chain rule)
ln(b) exp( ln(b) * x ) = ln(b) * b^x.
Re: What makes e natural? (2004)
#68e is the unique real number satisfying 1 + x 1 - x (1 + x/n)^n Letting n go to infinity gives e^x = \sum_{n=0}^infy x^n/n! using Newton's binomial formula.
> e is the unique real number satisfying 1 + x I think if you replace e by 2e, this still holds. Therefore your definition of e is not unique.
Re: What makes e natural? (2004)
#69Earlier 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, and nothing else is used in the series.
Implicit in that statement is the use of the series definition of sine, which is a "meaningful" or "natural" definition insofar as it represents some function we care about. I'll address at the end what happens if we assume that definition regardless of its independent plausibility, but first consider:
The semantic meaning of sine, at least from its historical roots and how you might independently uncover it from earlier fields like geometry instead of later fields like differential equations, is that given an angle (in some units, we'll touch on that in a moment) we'd like to know the ratio two sides of a particular triangle associated with that angle inscribed in a circle. Given a choice of units for the angle, the triangle is fixed, and so the result (that ratio of side lengths) is also fixed.
Suppose you want to know how that ratio varies with respect to the angle. You can imagine a change of coordinates `y = cx` and consider the derivative of `sin(x)` vs `sin(y/c)`. The latter will have a numeric value `1/c` times less than the former. E.g., imagine a whole circle represented `1` angle instead of `2pi`. Then converting from our normal radians baseline to that new unit you have `y = (1/2pi)x`, and the derivative of the semantic ratio we're considering with respect to the new measure of angle is multiplicatively `2pi` greater than the original.
Going back to your series definition, suppose we pick that series as the definition of sine, independent of units. The problem that arises is that particular uses of sine do have units, and converting from the problem you care about to your particular from-on-high chosen definition of sine will run into the exact sort of problem our `y = cx` paragraph above touched on. The derivative with respect to the quantity of interest still has an extra `1/c` factor, and the fact that our God-blessed choice of sine is independent of units didn't actually solve anything in the composite problem.
Re: What makes e natural? (2004)
#70As I remember it e^x is the only function which is its own derivative, which I guess makes e “natural” in some sense.