simply-disconnected

[simply-disconnected]

in the physiced lab strait up "jorkin it." and by "it," haha, well. let's justr say. the third derivative of position.

[simply-disconnected]

mathematics is a pure and beautiful branch of study, like a unicorn. Calculus is the act of forcing mathematics to do useful work for us, like putting chains and a yoke upon a unicorn and making it plow a muddy field

[simply-disconnected]

Potentially a hot take but the whole point of mathematics, especially pure mathematics is to be pedantic. We want to be sure what we're doing makes logical sense.

Sure you have experiments to back up your flimsy mathematical arguements but we care about details because that's what maths is.

[simply-disconnected]

I feel like I'm just being an idiot and I'm missing the obvious, but why are complex numbers (and related concepts) formulated using 1 as a basis with the whole real and imaginary parts thing? Would it make any difference if both parts of the basis were given names, like {u,v} or something instead of {1,i}, with relevant definitions for relating the parts akin to i² = -1.

[simply-disconnected]

A LaTeX mod to draw coffee cup rings on your technical papers

LaTeX is the venerable, gold-standard layout package favored for scholarly papers, especially technical papers; back in 2009, Hanno Rein released LaTeX Coffee Stains, an extension to draw a variety of coffee-cup rings on your paper; the code has been improved by community contributions over the years and is very robust and full-featured! (via Evil Mad Scientist Labs)

https://boingboing.net/2018/04/30/hanno-rein.html

[simply-disconnected]

Speak for yourself, many of my friends are perfectly spherical

[simply-disconnected]

after spending a long time trying to do the math myself (as you do when you have an interesting math question) I finally looked up what the cardinality of the set of surreal numbers is (like, "has it been proven that there are more surreal numbers than real numbers?") and uh. holy hell

[simply-disconnected]

I had a math question come to me in a dream, which had never happened to me before. The question was: how do you characterize a number with exactly 3 even factors? So I immediately woke up and started thinking, and it turns out to be quite simple, n has exactly 3 even factors if and only if it has the form 2*p^2 for some prime p. One direction is obvious: if it has the given form then its even factors are 2, 2*p and 2*p^2, which is 3. The opposite direction is still easy: since n has even factors then 2 must be one of them, so it has to be of the form 2*m. If m has two different prime factors p and q, then 2, 2*p and 2*q and 2*p*q are all different even factors, so m must be a prime p raised to some power b. If b is 3 or greater, then 2, 2*p, 2*p^2 and 2*p^3 are different even factors, so b must be 2 or smaller. if b is 0, then we get 2 which only has 1 even factor. If b is 1, then only 2 and 2*q are even factors.

[simply-disconnected]

It's a funny and unfortunate coincidence that in S2E12 of Star Trek: TNG (aired 1989) they talk about Fermat's Last Theorem as this untractable problem that even hundreds of years into the future people are still trying to crack. Fermat wrote down the conjecture around 1637, so when the episode aired it had been an open problem for 352 years. Little did they know that Wiles would finish the proof in 1994!

[simply-disconnected]

To expand on this, the idea is that most fractals by definition have a non-integer dimension. You might have been told that fractals are self-similar, but that's not actually enough (or even necessary) - For instance a square is self similar, as it is composed of 4 smaller versions of itself that are a quarter of the size, but you can see why that's not considered a fractal. What is actually meant by non-integer dimension can be boiled down to not considering dimension in the sense of size of the basis of the vector space, but rather by the log of the scaling factor. Examples: -If I scale a square up by a factor of 2, then it has 4 times the area, so its dimension is log2(4)=2 -The sierpinski triangle is made of 3 versions of it that are half the size, so If I scale it up by a factor of 2, it has 3 times as much "stuff", so its dimension is log2(3). Now, that means its dimension is strictly less than 2, which means we can't meaningfully talk about its area, in much the same way as we can't talk about the area of a line segment (if anything, it's zero). But conversely, its dimension is strictly greater than 1, so we can't meaningfully talk about its length, in much the same way we can't talk about the length of a square (if anything, it's infinite).

Fractals don't just have finite area.

They have zero area.

[simply-disconnected]

what's the 3-dimensional number thing?

Well I'm glad you asked! For those confused, this is referring to my claim that "my favorite multiplication equation is 3 × 5 = 15 because it's the reason you can't make a three-dimensional number system" from back in this post. Now, this is gonna be a bit of a journey, so buckle up.

Part One: Numbers in Space

First of all, what do I mean by a three-dimensional number system? We say that the complex numbers are two-dimensional, and that the quaternions are four-dimensional, but what do we mean by these things? There's a few potential answers to this question, but for our purposes we'll take the following narrative:

Complex numbers can be written in the form (a+bi), where a and b are real numbers. For the variable-averse, this just means we have things like (3+6i) and (5-2i) and (-8+3i). Some amount of "units" (that is, ones), and some amount of i's.

Most people are happy to stop here and say "well, there's two numbers that you're using, so that's two dimensions, ho hum". I think that's underselling it, though, since there's something nontrivial and super cool happening here. See, each complex number has an "absolute value", which is its distance from zero. If you imagine "3+6i" to mean "three meters East and six meters North", then the distance to that point will be 6.708 meters. We say the absolute value of (3+6i), which is written like |3+6i|, is equal to 6.708. Similarly, interpreting "5-2i" to mean "five meters East and two meters South" we get that |5-2i| = 5.385.

The neat thing about this is that absolute values multiply really nicely. For example, the two numbers above multiply to give (3+6i) × (5-2i) = (27+24i) which has a length of 36.124. What's impressive is that this length is the product of our original lengths: 36.124 = 6.708 × 5.385. (Okay technically this is not true due to rounding but for the full values it is true.)

This is what we're going to say is necessary to for a number system to accurately represent a space. You need the numbers to have lengths corresponding to actual lengths in space, and you need those lengths to be "multiplicative", which just means it does the thing we just saw. (That is, when you multiply two numbers, their lengths are multiplied as well.)

There's still of course the question of what "actual lengths in space" means, but we can just use the usual Euclidean method of measurement. So, |3+6i| = √(3²+6²) and |5-2i| = √(5²+2²). This extends directly to the quaternions, which are written as (a+bi+cj+dk) for real numbers a, b, c, d. (Don't worry about what j and k mean if you don't know; it turns out not to really matter here.) The length of the quaternion 4+3i-7j+4k can be calculated like |4+3i-7j+4k| = √(4²+3²+7²+4²) = 9.486 and similarly for other points in "four-dimensional space". These are the kinds of number systems we're looking for.

[To be explicit, for those who know the words: What we are looking for is a vector algebra over the real numbers with a prescribed basis under which the Euclidean norm is multiplicative and the integer lattice forms a subring.]

Part Two: Sums of Squares

Now for something completely different. Have you ever thought about which numbers are the sum of two perfect squares? Thirteen works, for example, since 13 = 3² + 2². So does thirty-two, since 32 = 4² + 4². The squares themselves also work, since zero exists: 49 = 7² + 0². But there are some numbers, like three and six, which can't be written as a sum of two squares no matter how hard you try. (It's pretty easy to check this yourself; there aren't too many possibilities.)

Are there any patterns to which numbers are a sum of two squares and which are not? Yeah, loads. We're going to look at a particularly interesting one: Let's say a number is "S2" if it's a sum of two squares. (This thing where you just kinda invent new terminology for your situation is common in math. "S2" should be thought of as an adjective, like "orange" or "alphabetical".) Then here's the neat thing: If two numbers are S2 then their product is S2 as well.

Let's see a few small examples. We have 2 = 1² + 1², so we say that 2 is S2. Similarly 4 = 2² + 0² is S2. Then 2 × 4, that is to say, 8, should be S2 as well. Indeed, 8 = 2² + 2².

Another, slightly less trivial example. We've seen that 13 and 32 are both S2. Then their product, 416, should also be S2. Lo and behold, 416 = 20² + 4², so indeed it is S2.

How do we know this will always work? The simplest way, as long as you've already internalized the bit from Part 1 about absolute values, is to think about the norms of complex numbers. A norm is, quite simply, the square of the corresponding distance. (Okay yes it can also mean different things in other contexts, but for our purposes that's what a norm is.) The norm is written with double bars, so ‖3+6i‖ = 45 and ‖5-2i‖ = 29 and ‖4+3i-7j+4k‖ = 90.

One thing to notice is that if your starting numbers are whole numbers then the norm will also be a whole number. In fact, because of how we've defined lengths, the norm is just the sum of the squares of the real-number bits. So, any S2 number can be turned into a norm of a complex number: 13 can be written as ‖3+2i‖, 32 can be written as ‖4+4i‖, and 49 can be written as ‖7+0i‖.

The other thing to notice is that, since the absolute value is multiplicative, the norm is also multiplicative. That is to say, for example, ‖(3+6i) × (5-2i)‖ = ‖3+6i‖ × ‖5-2i‖. It's pretty simple to prove that this will work with any numbers you choose.

But lo, gaze upon what happens when we combine these two facts together! Consider the two S2 values 13 and 32 from before. Because of the first fact, we can write the product 13 × 32 in terms of norms: 13 × 32 = ‖3+2i‖ × ‖4+4i‖. So far so good. Then, using the second fact, we can pull the product into the norms: ‖3+2i‖ × ‖4+4i‖ = ‖(3+2i) × (4+4i)‖. Huzzah! Now, if we write out the multiplication as (3+2i) × (4+4i) = (4+20i), we can get a more natural looking norm equation: ‖3+2i‖ × ‖4+4i‖ = ‖4+20i‖ and finally, all we need to do is evaluate the norms to get our product! (3² + 2²) × (4² + 4²) = (4² + 20²)

The cool thing is that this works no matter what your starting numbers are. 218 = 13² + 7² and 292 = 16² + 6², so we can follow the chain to get 218 × 292 = ‖13+7i‖ × ‖16+6i‖ = ‖(13+7i) × (16+6i)‖ = ‖166+190i‖ = 166² + 190² and indeed you can check that both extremes are equal to 63,656. No matter which two S2 numbers you start with, if you know the squares that make them up, you can use this process to find squares that add to their product. That is to say, the product of two S2 numbers is S2.

Part Four: Why do we skip three?

Now we have all the ingredients we need for our cute little proof soup! First, let's hop to the quaternions and their norm. As you should hopefully remember, quaternions have four terms (some number of units, some number of i's, some number of j's, and some number of k's), so a quaternion norm will be a sum of four squares. For example, ‖4+3i-7j+4k‖ = 90 means 90 = 4² + 3² + 7² + 4².

Since we referred to sums of two squares as S2, let's say the sums of four squares are S4. 90 is S4 because it can be written as we did above. Similarly, 7 is S4 because 7 = 2² + 1² + 1² + 1², and 22 is S4 because 22 = 4² + 2² + 1² + 1². We are of course still allowed to use zeros; 6 = 2² + 1² + 1² + 0² is S4, as is our friend 13 = 3² + 2² + 0² + 0².

The same fact from the S2 numbers still applies here: since 7 is S4 and 6 is S4, we know that 42 (the product of 7 and 6) is S4. Indeed, after a bit of fiddling I've found that 42 = 6² + 4² + 1² + 1². I don't need to do that fiddling, however, if I happen to be able to calculate quaternions! All I need to do is follow the chain, just like before: 7 × 6 = ‖2+i+j+k‖ × ‖2+i+j‖ = ‖(2+i+j+k) × (2+i+j)‖ = ‖2+3i+5j+2k‖ = 2² + 3² + 5² + 2². This is a different solution than the one I found earlier, but that's fine! As long as there's even one solution, 42 will be S4. Using the same logic, it should be clear that the product of any two S4 numbers is an S4 number.

Now, what goes wrong with three dimensions? Well, as you might have guessed, it has to do with S3 numbers, that is, numbers which can be written as a sum of three squares. If we had any three-dimensional number system, we'd be able to use the strategy we're now familiar with to prove that any product of S3 numbers is an S3 number. This would be fine, except, well…

3 × 5 = 15.

Why is this bad? See, 3 = 1² + 1² + 1² and 5 = 2² + 1² + 0², so both 3 and 5 are S3. However, you can check without too much trouble that 15 is not S3; no matter how hard you try, you can't write 15 as a sum of three squares.

And, well, that's it. The bucket has been kicked, the nails are in the coffin. You cannot make a three-dimensional number system with the kind of nice norm that the complex numbers and quaternions have. Even if someone comes to you excitedly, claiming to have figured it out, you can just toss them through these steps: • First, ask what the basis is. Complex numbers use 1 and i; quaternions use 1, i, j, and k. Let's say they answer with p, q, and r. • Second, ask them to multiply (p+q+r) by (2p+q). • Finally, well. If their system works, the resulting number should give you three numbers whose squares add to 15. Since that can't happen, you've shown that the norm is not actually multiplicative; their system doesn't capture the geometry of three dimensions.

[simply-disconnected]

English mathematicians really fucked up when they decided on 'uniqueness' over 'unicity'.

[simply-disconnected]

To add to all of the above, there is another way to construct countable many different cardinality, and it's with power sets! For any given set A, the power set which I will denote P(A) is the set of all subsets of A. For example, P({1,2})={{},{1},{2},{1,2}}. With our friend cantor's diagonalization argument, it can be shown the cardinality of P(A) is strictly greater than that of A, even if A was infinite. That means that we can construct infinitely many cardinalities simply by taking power sets of power sets. If we denote the cardinality of A to be |A|, then |A|<|P(A)|<|P(P(A))|<|P(P(P(A)))|<... and so we have countable many infinite infinities.

You can even complicate things! What if you take the union of all of A,P(A),P(P(A))... ? You get a set that's even larger than all the previous ones! let's denote it A_2. What then of P(A_2)? and P(P(A_2))? We can take the infinite union of all of those and get an even larger set, which we will call A_3. We can then recursively define A_4, A_5 etc. And if we take the infinite union of all the A_n? We get a new larger set! If we denote it B, and then repeat this process taking the infinite union of B, P(B), P(P(B))... we get B_2 and similarly B_3 and so on. We can in fact repeat this process infinitely many times, getting C, then D, and continuing even after we run out of letters! And THEN we can take the union of this infinite process, and denote that Alpha, and take P(Alpha) and P(P(Alpha)) and so on and so forth, and the more you look at this, the more similar it starts to seem to cardinals.

Tell me everything about infinity.

Oh, a very loaded question! All right. Let's start with the sizes of infinity!

Roughly speaking, there are two sizes of infinity; or, in proper terminology, "cardinalities." (There is some debate, as I recall, over whether there are more sizes of infinity. But we know there are two.) The first cardinality is the same size as the integers, which are the positive and negative whole numbers; essentially tick marks going in a line forever. 1, 2, 3, and so on; and in the opposite direction, -1, -2, -3, and so on.

The second cardinality is the same size as the real numbers, which are all the numbers that most people use on a daily basis; think, instead of tick marks, a line, and every place on that line is a number. No matter how close two places on that line are, there's always another number in between them. So you have 2.5 and 2.6, but between them you have 2.55 and 2.5932, and infinitely many more.

The concept of infinity, of course, gets a bit weird once you move into more than one dimension; it's easy enough to point in the same direction as a line and say, "that goes to infinity," but once you have multiple dimensions, is it meaningful to talk about a negative or positive infinity? Oh, and adding and subtracting get weird once you start adding infinitely many things together. Addition loses commutativity (e.g. 5+3 = 3+5), which still blows my mind, even though I've seen the proof for it. It's the kind of thing that keeps me up at night.

Generally mathematicians get around the multiple-dimensions problem by using the modulus of a number, which is the distance from the number to zero, measured using our dear old friend the Pythagorean Theorem: so, if your point is at 4 in one dimension and at 3 in another, you use the Pythagorean theorem to get 5, and then you consider that point to be 5 away from zero. (This would be easier to explain if I had a chalkboard!) Then, infinity is sort of a circle that surrounds the whole plane; or, if you think of your 2d plane like a flat circle, if you folded it up into a ball, infinity would be all the points at the very top of the ball, and zero would be the point at the bottom. (Obviously this gets weirder if we have three dimensions, but you get the idea.) Okay, so that's a quick introduction to infinity from a mathematical perspective. I think there were also some physics questions re: the expanding universe and spacetime? I'm happy to write a bit about that, too, but I think it belongs in a separate post! So if you have questions about that, please let me know and I'll try to share what I do know! Disclaimer: while I DO know more math than the average person, I have essentially a bachelor's-degree level of knowledge in math. I think everything I've typed out is correct, but I may very well have missed some details! Dear readers, please feel free to correct me if you have greater knowledge than I.

Edit: also, I should have mentioned that the pythagorean theorem only works in a euclidean space. But I feel like going into non-Euclidean stuff is a bit past the scope of (this) tumblr post.