What is a math concept or theorem that you wish there were a better explanation of?
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@willyyam @futurebird You should be worried because a few of these have almost made sense.
@Meowthias @willyyam @futurebird It's actually far more annoying than you'd think. So annoying, in fact, that it wasn't proven until 1882 in spite of it being something we've been trying to figure out since at least as far back at Euclid (c. 300 BCE).
I don't think I can give a simple explanation of the proof but I can maybe try to at least explain how mathematicians reason about it.
You're probably familiar with the term "integer". That's just any whole number of things, including negatives and zero. 1, -7, and 394 are all integers. These are the first numbers you learn about in school.
But integers aren't good enough so the next thing you learn about in school are fractions: 1/2, 3/5, etc. These are for when you need to talk about less than a whole thing like when you're dividing up a pizza. But sometimes you have one whole pizza and a half a pizza so we also have numbers like 1 1/2. We call these "mixed fractions" or "mixed numbers". And you also learn in school that any mixed fraction can be written as a simple fraction. So 1 1/2 can be written as 3/2.
Decimal numbers are also fractions of a sort. The whole part goes to the left of the decimal and the digits to the right represent a fraction where the denominator (the bottom part) is a power of 10. (1 with some number of zeros.) Not every fraction is a nice, neat decimal but decimals are always fractions. (Repeated decimals are a thing. I'll get to those later.)
When you take all the integers and all the fractions together, you get what are called the rational numbers. These are all numbers that can be expressed with an integer ratio.
And these numbers are good enough for just about anything.
There's two really important facts about the rational numbers: First is that, as long as you don't divide by zero, you can divide any rational number by any other and you'll get another rational number. This makes them good enough for most arithmetic. The second is that, given any real number, you can get infinitely close with a rational number. So if you have something like pi, it might not be rational but you can always find a rational number as close as you want. This means that for anything you need to calculate in the real world, rational numbers are good enough. You just need a big enough denominator or enough decimal places.
But the ancient Greeks discovered, or at least suspected, that there were numbers that weren't rational. They were trying to find a ratio for the side of a square and its diagonal and they couldn't. There clearly was a ratio there (the Greeks were more concerned with ratios than actual numbers) but they couldn't express it as a ratio of integers. That's because √2 isn't rational.
So we have this sort of Russian doll situation with categories of numbers. On the inside we have the integers. Then the rationals contain the integers. And on the outside, we have the real numbers, which is all numbers that can exist in nature. But are there other dolls? Is there something between the rationals and the reals?
Yes! Multiple somethings in fact.
I'll skip over the constructable numbers, as much fun as those are, and jump straight to the algebraic numbers. Algebraic numbers are every number that is the solution to some polynomial. In other words, if x is algebraic then there is some expression a + bx + cx2 + dx3 ... = 0, where a, b, c, etc. are integers and the whole thing evaluates to zero. For an integer i, that expression is just (-i) + 1x = 0. For a fraction n/d, it's (-n) + dx = 0. So every rational number is an algebraic number and the dolls nest nicely.
One difference between algebraic numbers and rational numbers is what happens when you try to write them as decimals. With a rational number, the decimal representation will either stop like 1.5 for 3/2 or repeat forever like 0.666666... for 2/3. It might not be a single digit that repeats. It might be multiple like 1.60606060... but there will be a repeated pattern. Numbers that aren't rational, like √2, will never have a repeating pattern, even if you go out millions of decimal places.
But e and pi are even worse.
There are some numbers that aren't even algebraic. These are called transcendental numbers. They're the ones for which you can't write down a nice algebraic expression with that number as its solution.
For a long time, mathematicians suspected that there must be numbers that weren't algebraic but they're terribly hard to pin down because just about anything you can write on a piece of paper is going to end up being algebraic. And we've known for quite a while that such numbers must exist because, while we have infinitely many real numbers and infinitely many algebraic numbers, we know that they're different infinities. (That's a whole other discussion.)
But what are these mythical transcendental numbers? They have to exist but what are they?
Well, I'm 1882, we finally proved a theorem called the Lindemann–Weierstrass theorem that, among other things, says that pi and e are both transcendental. No matter how hard you try, you can't express them as fractions, decimals, or with pure algebra.
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@xarvos @evan @Cheeseness @Meowthias
Did the Greek numerals have place value?
@futurebird@sauropods.win @evan@cosocial.ca @Cheeseness@mastodon.social @Meowthias@mastodon.world what do you mean by place value? each digit having different value based on its place, like our numerals? it doesn't seem so
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@Meowthias @futurebird I have a story about someone who believes the repeating .333 needs to be freed from that repetition. I wrote it because as a kid I couldn't believe that it would never end, even though it manifestly never ended. Similar sort of preoccupation.
@asakiyume @futurebird Was the .3 eventually freed?
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@seachaint @rallias @Meowthias
That explains why it *could* go on forever. It explains why it's possible to have an irrational number that isn't a nice ratio of integers... but it doesn't show that whatever process you use to estimate pi won't at some point down the line just start repeating.
You can define 1/3 as an infinite process too.
@futurebird @seachaint @Meowthias because in each intermediate step that you take, say from six sides to eight sides, you change that ratio ever so slightly closer to the limit. That pattern doesn't just exist from six size to eight sides to 10 sides, that pattern continues on as you go from a billion to a billion and two sides, from a trillion to a trillion and two sides, from a quintillion to a quintillion and two sides, to whatever arbitrarily large even number you think of to whatever arbitrarily large even number plus two sides. And thus, any intermediate repeating digit set is going to be wiped out by adding two more sides in perpetuity.
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@futurebird @seachaint @Meowthias because in each intermediate step that you take, say from six sides to eight sides, you change that ratio ever so slightly closer to the limit. That pattern doesn't just exist from six size to eight sides to 10 sides, that pattern continues on as you go from a billion to a billion and two sides, from a trillion to a trillion and two sides, from a quintillion to a quintillion and two sides, to whatever arbitrarily large even number you think of to whatever arbitrarily large even number plus two sides. And thus, any intermediate repeating digit set is going to be wiped out by adding two more sides in perpetuity.
@futurebird @seachaint @Meowthias there are actual proper math proofs to pi, I'm not going to pretend I understand them. However, it is proven, in a math proof, that pi is a transcendental number, and thus cannot be represented by any ratio of two numbers.
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Was he doing his calculations in Roman numerals? I feel like I used to know this...
@futurebird @Cheeseness @Meowthias I don't know! But Greek mathematicians often didn't write things out. They did some of their proofs geometrically. They had these giant compasses, about a meter long, and they would draw out their proofs in the dirt on the ground. Supposedly Archimedes died with his compass in his hand. When they did write out proofs, they were verbal and logical -- not equations. Euclid's proofs are a good example.
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Pi is still irrational in other bases, though. Because if you have a circle and flatten it out, and you have the diameter of that circle and you make exact copies of these two lengths and lay them side by side one line of diameters and one line of repeated circumferences they will never ever ever ever perfectly match up no matter how many you lay down.
@futurebird @SeanPLynch @Meowthias π = 1 in base pi…
I'll see myself out
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Pi is still irrational in other bases, though. Because if you have a circle and flatten it out, and you have the diameter of that circle and you make exact copies of these two lengths and lay them side by side one line of diameters and one line of repeated circumferences they will never ever ever ever perfectly match up no matter how many you lay down.
@futurebird @SeanPLynch @Meowthias
I think it's not about the circle being "unsquarable". A square has a ratio of sqrt(2), which is as irrational as pi. (Although less trascendental, I guess)
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@seachaint @rallias @Meowthias
That explains why it *could* go on forever. It explains why it's possible to have an irrational number that isn't a nice ratio of integers... but it doesn't show that whatever process you use to estimate pi won't at some point down the line just start repeating.
You can define 1/3 as an infinite process too.
@futurebird @seachaint @rallias @Meowthias I feel like there's a sub-proof here that any repeating decimal is non-repeating in some base.
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What is a math concept or theorem that you wish there were a better explanation of?
It could be from arithmetic: Why is adding fractions so complicated?
From grade-school algebra: Why does the teacher get so sad and angry if I just √(x²+y²)=x+y
From the calculus: Why do I need to write dx with the integral?
or beyond.
@futurebird pretty much everything. i gave up even trying in 10th grade math class cause it was just more of the same "if you don't understand, it's all your fault and you will get big angry red zeros on all of the tests you fail because you don't understand and have given up bothering with it".
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What is a math concept or theorem that you wish there were a better explanation of?
It could be from arithmetic: Why is adding fractions so complicated?
From grade-school algebra: Why does the teacher get so sad and angry if I just √(x²+y²)=x+y
From the calculus: Why do I need to write dx with the integral?
or beyond.
@futurebird I've had very good teachers, so I don't think that I needed more explanation than what I received for most concepts
I do wish they had spent more time about the multidimensionality of complex numbers
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@seachaint @rallias @Meowthias
That explains why it *could* go on forever. It explains why it's possible to have an irrational number that isn't a nice ratio of integers... but it doesn't show that whatever process you use to estimate pi won't at some point down the line just start repeating.
You can define 1/3 as an infinite process too.
@futurebird @seachaint @rallias @Meowthias
To elaborate:
That π is transcendental (and thus both non-repeating and not expressible in closed form algebraically) follows from Euler's identity: e^iπ = -1 and that e is transcendental.
Proving e is transcendental takes a few pages of playing around with series and integrals: https://en.wikipedia.org/wiki/Transcendental_number#A_proof_that_e_is_transcendental .
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@Meowthias @willyyam @futurebird It's actually far more annoying than you'd think. So annoying, in fact, that it wasn't proven until 1882 in spite of it being something we've been trying to figure out since at least as far back at Euclid (c. 300 BCE).
I don't think I can give a simple explanation of the proof but I can maybe try to at least explain how mathematicians reason about it.
You're probably familiar with the term "integer". That's just any whole number of things, including negatives and zero. 1, -7, and 394 are all integers. These are the first numbers you learn about in school.
But integers aren't good enough so the next thing you learn about in school are fractions: 1/2, 3/5, etc. These are for when you need to talk about less than a whole thing like when you're dividing up a pizza. But sometimes you have one whole pizza and a half a pizza so we also have numbers like 1 1/2. We call these "mixed fractions" or "mixed numbers". And you also learn in school that any mixed fraction can be written as a simple fraction. So 1 1/2 can be written as 3/2.
Decimal numbers are also fractions of a sort. The whole part goes to the left of the decimal and the digits to the right represent a fraction where the denominator (the bottom part) is a power of 10. (1 with some number of zeros.) Not every fraction is a nice, neat decimal but decimals are always fractions. (Repeated decimals are a thing. I'll get to those later.)
When you take all the integers and all the fractions together, you get what are called the rational numbers. These are all numbers that can be expressed with an integer ratio.
And these numbers are good enough for just about anything.
There's two really important facts about the rational numbers: First is that, as long as you don't divide by zero, you can divide any rational number by any other and you'll get another rational number. This makes them good enough for most arithmetic. The second is that, given any real number, you can get infinitely close with a rational number. So if you have something like pi, it might not be rational but you can always find a rational number as close as you want. This means that for anything you need to calculate in the real world, rational numbers are good enough. You just need a big enough denominator or enough decimal places.
But the ancient Greeks discovered, or at least suspected, that there were numbers that weren't rational. They were trying to find a ratio for the side of a square and its diagonal and they couldn't. There clearly was a ratio there (the Greeks were more concerned with ratios than actual numbers) but they couldn't express it as a ratio of integers. That's because √2 isn't rational.
So we have this sort of Russian doll situation with categories of numbers. On the inside we have the integers. Then the rationals contain the integers. And on the outside, we have the real numbers, which is all numbers that can exist in nature. But are there other dolls? Is there something between the rationals and the reals?
Yes! Multiple somethings in fact.
I'll skip over the constructable numbers, as much fun as those are, and jump straight to the algebraic numbers. Algebraic numbers are every number that is the solution to some polynomial. In other words, if x is algebraic then there is some expression a + bx + cx2 + dx3 ... = 0, where a, b, c, etc. are integers and the whole thing evaluates to zero. For an integer i, that expression is just (-i) + 1x = 0. For a fraction n/d, it's (-n) + dx = 0. So every rational number is an algebraic number and the dolls nest nicely.
One difference between algebraic numbers and rational numbers is what happens when you try to write them as decimals. With a rational number, the decimal representation will either stop like 1.5 for 3/2 or repeat forever like 0.666666... for 2/3. It might not be a single digit that repeats. It might be multiple like 1.60606060... but there will be a repeated pattern. Numbers that aren't rational, like √2, will never have a repeating pattern, even if you go out millions of decimal places.
But e and pi are even worse.
There are some numbers that aren't even algebraic. These are called transcendental numbers. They're the ones for which you can't write down a nice algebraic expression with that number as its solution.
For a long time, mathematicians suspected that there must be numbers that weren't algebraic but they're terribly hard to pin down because just about anything you can write on a piece of paper is going to end up being algebraic. And we've known for quite a while that such numbers must exist because, while we have infinitely many real numbers and infinitely many algebraic numbers, we know that they're different infinities. (That's a whole other discussion.)
But what are these mythical transcendental numbers? They have to exist but what are they?
Well, I'm 1882, we finally proved a theorem called the Lindemann–Weierstrass theorem that, among other things, says that pi and e are both transcendental. No matter how hard you try, you can't express them as fractions, decimals, or with pure algebra.
@faithisleaping @willyyam @futurebird Thank you for taking the time to write this response.
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@faithisleaping @willyyam @futurebird Thank you for taking the time to write this response.
@Meowthias @willyyam @futurebird I hope it made at least some sense.
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@futurebird I would like an explanation for why pi goes on forever. Is it evidence we are living in a simulation? Is it because if you trace the circumference of a circle with your finger you never reach a beginning or an end? Is it a message from the gods?
@Meowthias @futurebird Joke's on you. According to Contact there is finally a pattern in there somewhere way in there and it's something that makes even the aliens wonder if there might be something meaningful behind that.
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@asakiyume @futurebird Was the .3 eventually freed?
Yes, with the help of shrooms.
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What is a math concept or theorem that you wish there were a better explanation of?
It could be from arithmetic: Why is adding fractions so complicated?
From grade-school algebra: Why does the teacher get so sad and angry if I just √(x²+y²)=x+y
From the calculus: Why do I need to write dx with the integral?
or beyond.
@futurebird hamiltonian math and matrix math
Good Lord those things make my brain hurt
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@futurebird hamiltonian math and matrix math
Good Lord those things make my brain hurt
@futurebird I can work in Hilbert space all day though
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@futurebird hamiltonian math and matrix math
Good Lord those things make my brain hurt
IDK about that hamiltonian, but matrices aren't so bad. Once you just accept them as linear multivariable functions... though I assume you are talking about something deeper than that.
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IDK about that hamiltonian, but matrices aren't so bad. Once you just accept them as linear multivariable functions... though I assume you are talking about something deeper than that.
I was introduced to it doing NMR stuff and I had just done intro calculus and being Greek would be an improvement. So obtuse to me
https://en.wikipedia.org/wiki/Hamiltonian_system
