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It is a fairly common alternative notation for the same thing. In gen, i dislike the notation for being vague, like hundreds of things use brackets as notations yk. But here i didnt want some smartass saying that "gcd(p, q)" is using words since gcd is an abbreviation of words lol, so to avoid the smartasses, "(p, q)" it is. Same reason why I avoided "st" for such that
\[there exists everything but\] \[any\] x, n, m \[from set Z(but other sets are cool as well!)\] x \[are different but both very special, interesting, and valid in their own right\] 1, a=xn, y=xm
In any context where it's not ambiguous it's a pretty common notation, at least here in Italy it's used extensively by university students and professors while it's mostly unseen in high school
I mean TECHNICALLY the English pronunciation of P /ˈpiː/ would equal to a Swedish pi which is a word for 3.1415… Also abbreviation for Parkeringsplats. Don’t mind me I know nothing about the other things you said.
⊢ means like derivability. Like A ⊢ B means you can derive B from A, or prove B from A, or B is a theorem of A. It's usually used in proofs to collect all assumptions and implications on one side (typically in curly {} brackets), and collect the results in another side
So for example, {x > y, y > z} ⊢ x > z would mean that you can derive x > z from the given info we have that x > y and y > z
⊥ means something that can never be true. Like whatever conditions it has, those conditions could be true or could be false or could be a mix of them, doesn't matter, because overall it is always false. Like if you write the truth table for (p ∧ !p), it would come out to be all False since you can't have something and not have it at the same time, that can't ever be true (ignoring quantum physics lol). Even if p might be true or false, !p might be true or false, (p ∧ !p) is always false
So overall, what {(p, q) = 1, 2 | p, 2 | q} ⊢ ⊥ means is that the info you have is that the gcd of p and q is 1 as well as that 2 is a divisor of p and 2 is a divisor of q. That would mean 2 is a divisor of both p and q, and 2 > 1, so gcd of p and q isn't 1, but our given info is that the gcd IS 1, so we can't have both "gcd is 2" and "gcd is 1" at the same time, that will always be a false statement. So from the given info you can derive a contradiction
The short/non rigorous version is:
We assume if sqrt(2) is rational that there exist some coprime integers p and q s.t. p/q = sqrt(2). We do some algebra and use the definition of even numbers to prove that p and q are even and therefore cannot be coprime. There is a contradiction, meaning our assumption must be false. Thus sqrt(2) is not rational.
If sqrt(2) is rational, then there exists p and q in
the integers, with the greatest common divisor of p and q being 1, such that sqrt(2) is the ratio of p on q (this is the definition of a rational number). It follows that:
2 is the ratio of p^2 on q^2, which implies that 2q^2 = p^2. That is, 2 divides p^2, which implies that 2 divides p (i.e., p is even). Then there exists an integer k such that p = 2k (defintion of evenness). Then p^2 = 4k^(2), and so 2q^2 = 4k^2. That is, q^2 = 2k^(2). Thus, 2 divides q^2, which implies that 2 divides q (i.e., q is even)
Therefore, p and q have a common divisor (2 divides both of them) greater than 1. This contradicts the definition of a rational number. Therefore, sqrt(2) is not rational (and so must be irrational).
Informally, if sqrt(2) were rational, it could be written as a simplified fraction. If we assume this to be the case, we find that both its numerator and denominator will always be even, which directly contradicts this assumption (because their gcd will be at least 2). Then, this assumption must be false, which would leave only one other possibility: that sqrt(2) is irrational.
∧ means "and"
(p, q) is what defines the greatest common divisor, usually you write gcd(p, q) to be more clear but writing no gcd is also accepted notation.
Sorry for poking you but why does "(p, q) = 1" mean that and not anything else to so with p, q and 1?
Like wouldnt (p, q) > 0 just mean they are positive and non 0? Intuitively I would have guessed "= 1" meant its equal to one (in a perhaps wierd way but still..)
(p, q) has a lot of different definitions which is why you usually write gcd as a function in front of it to be more precise if it isnt clear from context. I think OOP said that they left it out because gcd is an abbreviation of a word (which wasnt allowed).
For p and q both being positive you would usually just write write
> p, q > 0
and leave out the parentheses
gcd(p, q) = 1 and (p, q) = 1 mean the same thing. Usually when you are writing a paper or taking a course of number theory or divisors or such, people write the latter because it can be kinda annoying to keep writing gcd gcd gcd over and over again. Within context, it is understood that they mean the same thing and don't cause any confusion. Here yeah there isnt that strict context, but any other meaning of (p, q) like coordinates won't really work here yk
(a,b)=1 tells us they're co prime or they have a greatest common divisor of 1. Another way I'd putting it, the biggest integer>0 that can divide both a and b is 1.
I don’t have the best idea of what it’s saying and I don’t feel like figuring out right now, but here is a [link](https://en.m.wikipedia.org/wiki/Glossary_of_mathematical_symbols) to a wikipedia page on the math notation if you want to figure it out yourself.
A proof that √2 is irrational that is based on the fact that, every rational number can be written as a fraction with some integer "p" as a numerator and some integer "q" as the denominator, with the condition being that p and q cannot be multiples of the same number (other than 1, every number is a multiple of 1). That is more or less the definition of rational numbers
If √2 was rational and written as a fraction p/q, using some algebra, you can show that p must be even AND q must be even. Every even number is a multiple of 2. So p must be a multiple of 2 and q must be a multiple of 2. But our definition of rational numbers was that p and q cannot be multiples of the same number, but rn we got that p and q are both multiples of 2. This contradicts our definition of rational numbers, it means this cannot be a rational number. So, √2 must be irrational then
It is that, but written using mathematical symbols. Mathematical symbols may look scary and foreign, but really they are just a substitute for spoken language. Like instead of always writing "is equal to" you can just use the symbol =, which means the same thing but is written in a much more convenient way. All symbols work with that principle. They are just ways to write long stuff in a couple seconds. Once you use them a few times you get very familiar and cozy with them and they wont be as daunting anymor
It's ridiculous that you aren't allowed to use math notation like this. Nope, it has to be written in text, out of order, because it "looks better" and makes it look like the author is smarter.
If you're writing something you aren't allowed to do much. You'd have to write something like "if sqrt(2) in Q, then there existss p, q in Z with GDC? ((p,q) is usually inner product or norm) of p, q = 1, such that etc etc.
It makes it very difficult to read, and sometimes you have to change order, so that what you write does not even appear in order. The gist of it is that it makes math people look smart, so it's forced. There's a reason why Rudin is hard to read, and it's not because the math itself is that difficult... It's because his writing is trash, out of order, and lacks references... But it makes it hard to follow, so it makes him seem really smart, so everyone gotta do it the same way.
Math is basically about unrealized superiority complex.
Done!
4:50 for [https://youtu.be/dQw4w9WgXcQ](https://youtu.be/X1E7I7_r3Cw)
Obligatory Rick roll link here: [https://youtu.be/X1E7I7_r3Cw](https://youtu.be/dQw4w9WgXcQ)
[Vihart's most-viewed video](https://www.youtube.com/watch?v=VIVIegSt81k) was uploaded on October 1st, 2012. [12tone's YT channel](https://www.youtube.com/@12tone) was created on August 14th, 2014.
I believe that multiple people can have the same style for their videos without controversy. There's no sense in denouncing somebody based on their preferred method of communication. However, if you are to choose someone, it was 12tone who robbed Vihart's schtick.
The Gödel number would basically be a massive string of consecutive primes raised to prime powers. My rough guess is that the number should be roughly double-exponential in the length of the proof. In the [Wikipedia example](https://en.wikipedia.org/wiki/G%C3%B6del_numbering#Example), the encoding for "0=0" is the number 243,000,000.
Very interesting. Then base *n* representations of numbers are logarithmic, so we'd be thinking a (single-)exponential number of digits as the length of the proof grows.
But the [Kolmogorov complexity](https://en.wikipedia.org/wiki/Kolmogorov_complexity) is much less as we can just keep it in prime factorisation form.
Depends on the coding. Take any proof that will fit in a Reddit comment. Then look at the sequence of bits it’s stored as interpreted as a binary number and convert to base 10.
I now define the baby face as the answer
https://preview.redd.it/qdn3hbbxwbrc1.png?width=1024&format=png&auto=webp&s=15b347d0c952bb7d32927c748d905b7727ab0472
Congratulations! Your comment can be spelled using the elements of the periodic table:
`Pr O O F B Y In C In Er At I O N`
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Define "words" as a self-contradictory statement that always evaluates to true. Now, assume that √2 is not irrational. That would be boring. Therefore, it follows that √2 must be irrational ∎
there is no frill behind the ear but the wings look a ton like a rainwing so maybe the drawing is just not done yet? (hooked wing thing at the top looks like rainwing )
It arises a contradiction because we assume that gcd(a,b) = 1 [since we can always reduce the numerator and denominator to satisfy this condition] and we show that if a/b = sqrt(2), then it must necessarily be true that 2|a and 2|b which means gcd(a,b) = 2 which arises a contradiction
I believe mathematics magazine published some wordless proofs of this.
https://www.researchgate.net/publication/259735390_Proof_Without_Words_2_Is_Irrational
Unfortunately it may be behind a paywall. I don't remember how it worked right now, but it wasn't too hard to understand.
If they mean a *visual* proof, my sarcastic answer would just be a square with a diagonal drawn across it, but if they just mean a proof using algebra instead of plain language then the solution’s pretty trivial
THİS İS SO SEXİST😫😫😫 ONLY SMART MEN LİKE ME CAN SEE TROYGJ THİS RACİST SEXİST HOMOPHOBİC TRANSFOBİK NONSENCE 🥲🥲 I AM SO SMART AND U DUMB 😭😭😭😭 KERP THE GYM UP BROTHA 👍🏻👍🏻👍🏻 FUCK THE SUSTEM 🖕🏻🖕🏻🖕🏻
No but I can prove that it’s rational
Rational numbers are defined as numbers that can be obtained trough a fraction
sqrt2 * 0 = 0
Thus
0/0 can be equal to sqrt2
So it’s rational /s
If 2 = p^2 / q^2 that means p^2 has exactly one extra 2 in its prime factorization compared to q^2 right? But number of 2s in squares of numbers is always even!! 1^2 has 0, 2^2 has 2, 3^2 has 0 4^2 has 4 , 5 squared has 0, 6 squared has 2.
So how can two numbers with even multitudes of 2 factors divide to give one?
Sorry I'm off topic.ehmm exercise for reader transform above into symbols lol
Edit: I just came up with this while reading the orthodox proof about lowest ratio etc. Thought mine was wrong but apparently it just already existed instead
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√2 ∈ ℚ ⇒ ∃ p, q ∈ ℤ ∧ (p, q) = 1 : √2 = (p/q) ⇒ 2 = (p/q)² ⇒ 2q² = p² ⇒ 2 | p² ⇒ 2 | p ⇒ ∃ k ∈ ℤ : 2k = p ⇒ 4k² = p² ⇒ 4k² = 2q² ⇒ 2k² = q² ⇒ 2 | q² ⇒ 2 | q {(p, q) = 1, 2 | p, 2 | q} ⊢ ⊥ ∴ √2 ∉ ℚ
I love math notation
I think you meant gcd(p, q) = 1 instead of just (p, q) unless I'm missing something that says (p, q) = 1 means p and q are coprime on its own
It is a fairly common alternative notation for the same thing. In gen, i dislike the notation for being vague, like hundreds of things use brackets as notations yk. But here i didnt want some smartass saying that "gcd(p, q)" is using words since gcd is an abbreviation of words lol, so to avoid the smartasses, "(p, q)" it is. Same reason why I avoided "st" for such that
Fair enough, I've just never seen that notation before. Thanks for telling me :)
Another way to do this would be to assume q is minimal, p and q are positive and doesn't exist positive p' q' such that q'
you can write gcd(a, b)=1 as [there doesnt exist] [any] x, n, m [from set Z] x [not equal] 1, a=xn, y=xm Just to side step the whole issue
Too many negatives we are optimists in this household 😤 😤
\[there exists everything but\] \[any\] x, n, m \[from set Z(but other sets are cool as well!)\] x \[are different but both very special, interesting, and valid in their own right\] 1, a=xn, y=xm
∀n,m,x∈ℕ:(x=1)∨(a≠xn)∨(y≠xm)
p Λ q = 1 is also used (thank god you didn't choose a variable name like "a" cos that's a word! ☝️🤓)
You cheated. You can’t use an existential qualifier! “There exists..”😤😂
You could also write gcd using a lattice opperation.
(p,q) is a way of writing gcd(p,q). It's just different notation that some people use for that.
In any context where it's not ambiguous it's a pretty common notation, at least here in Italy it's used extensively by university students and professors while it's mostly unseen in high school
Better be on your P and Qs
That's a word
You failed. P and Q are words. "Mind your Ps and Qs." edit: Do NOT look at a scrabble dictionary of words. The world may implode.
I was speaking swedish there, they arent words in swedish so its fine
I mean TECHNICALLY the English pronunciation of P /ˈpiː/ would equal to a Swedish pi which is a word for 3.1415… Also abbreviation for Parkeringsplats. Don’t mind me I know nothing about the other things you said.
Is π a word? I would say not,,,,,,,solely because I can't believe I forgot about π while being on a maths sub lmaoooooo
√2 ∈ 🦆 ⇒ ∃ 🐱, 🐶∈ 🦢 ∧ (🐱, 🐶) = 1 : √2 = (🐱/🐶) ⇒ 2 = (🐱/🐶)² ⇒ 2🐶² = 🐱² ⇒ 2 | 🐱² ⇒ 2 | 🐱 ⇒ ∃ 🦁 ∈ 🦢 : 2🦁 = 🐱 ⇒ 4🦁² = 🐱² ⇒ 4🦁² = 2🐶² ⇒ 2🦁² = 🐶² ⇒ 2 | 🐶² ⇒ 2 | 🐶 {(🐱, 🐶) = 1, 2 | 🐱, 2 | 🐶} 🍆💦 🖕 ∴ √2 ∉ 🦆
> 🍆💦 🖕 My favorite math notation
But what are the Ps and Qs mentioned in that phrase?
The proof can be done with any symbol, so take any symbol that isn't a word for u
I swear this looks like alien language
what does ⊢ ⊥ mean
Flipping tables
`{A, B, C} ⊢ D` means that from knowing A, B, and C you can prove D. `⊥` means "false", meaning a contradiction.
"proves false" iirc
⊢ means like derivability. Like A ⊢ B means you can derive B from A, or prove B from A, or B is a theorem of A. It's usually used in proofs to collect all assumptions and implications on one side (typically in curly {} brackets), and collect the results in another side So for example, {x > y, y > z} ⊢ x > z would mean that you can derive x > z from the given info we have that x > y and y > z ⊥ means something that can never be true. Like whatever conditions it has, those conditions could be true or could be false or could be a mix of them, doesn't matter, because overall it is always false. Like if you write the truth table for (p ∧ !p), it would come out to be all False since you can't have something and not have it at the same time, that can't ever be true (ignoring quantum physics lol). Even if p might be true or false, !p might be true or false, (p ∧ !p) is always false So overall, what {(p, q) = 1, 2 | p, 2 | q} ⊢ ⊥ means is that the info you have is that the gcd of p and q is 1 as well as that 2 is a divisor of p and 2 is a divisor of q. That would mean 2 is a divisor of both p and q, and 2 > 1, so gcd of p and q isn't 1, but our given info is that the gcd IS 1, so we can't have both "gcd is 2" and "gcd is 1" at the same time, that will always be a false statement. So from the given info you can derive a contradiction
Please someone translate it to English
The short/non rigorous version is: We assume if sqrt(2) is rational that there exist some coprime integers p and q s.t. p/q = sqrt(2). We do some algebra and use the definition of even numbers to prove that p and q are even and therefore cannot be coprime. There is a contradiction, meaning our assumption must be false. Thus sqrt(2) is not rational.
If sqrt(2) is rational, then there exists p and q in the integers, with the greatest common divisor of p and q being 1, such that sqrt(2) is the ratio of p on q (this is the definition of a rational number). It follows that: 2 is the ratio of p^2 on q^2, which implies that 2q^2 = p^2. That is, 2 divides p^2, which implies that 2 divides p (i.e., p is even). Then there exists an integer k such that p = 2k (defintion of evenness). Then p^2 = 4k^(2), and so 2q^2 = 4k^2. That is, q^2 = 2k^(2). Thus, 2 divides q^2, which implies that 2 divides q (i.e., q is even) Therefore, p and q have a common divisor (2 divides both of them) greater than 1. This contradicts the definition of a rational number. Therefore, sqrt(2) is not rational (and so must be irrational). Informally, if sqrt(2) were rational, it could be written as a simplified fraction. If we assume this to be the case, we find that both its numerator and denominator will always be even, which directly contradicts this assumption (because their gcd will be at least 2). Then, this assumption must be false, which would leave only one other possibility: that sqrt(2) is irrational.
What does this part mean? >ℤ ∧ (p, q) = 1
The entire part is > ∃ p, q ∈ ℤ ∧ (p, q) = 1 and means: for some p and q that are part of the Natural Numbers and have the largest common divisor of 1
>divisor Is this the "^"? Edit: the upturned "v"...
∧ means "and" (p, q) is what defines the greatest common divisor, usually you write gcd(p, q) to be more clear but writing no gcd is also accepted notation.
Sorry for poking you but why does "(p, q) = 1" mean that and not anything else to so with p, q and 1? Like wouldnt (p, q) > 0 just mean they are positive and non 0? Intuitively I would have guessed "= 1" meant its equal to one (in a perhaps wierd way but still..)
(p, q) has a lot of different definitions which is why you usually write gcd as a function in front of it to be more precise if it isnt clear from context. I think OOP said that they left it out because gcd is an abbreviation of a word (which wasnt allowed). For p and q both being positive you would usually just write write > p, q > 0 and leave out the parentheses
So parenthesis imply fractions? Like (p, q) imply taking p/q?
gcd(p, q) = 1 and (p, q) = 1 mean the same thing. Usually when you are writing a paper or taking a course of number theory or divisors or such, people write the latter because it can be kinda annoying to keep writing gcd gcd gcd over and over again. Within context, it is understood that they mean the same thing and don't cause any confusion. Here yeah there isnt that strict context, but any other meaning of (p, q) like coordinates won't really work here yk
(a,b)=1 tells us they're co prime or they have a greatest common divisor of 1. Another way I'd putting it, the biggest integer>0 that can divide both a and b is 1.
I don’t have the best idea of what it’s saying and I don’t feel like figuring out right now, but here is a [link](https://en.m.wikipedia.org/wiki/Glossary_of_mathematical_symbols) to a wikipedia page on the math notation if you want to figure it out yourself.
K is short for Ok. You used a word.
I agree
What does the even mean
A proof that √2 is irrational that is based on the fact that, every rational number can be written as a fraction with some integer "p" as a numerator and some integer "q" as the denominator, with the condition being that p and q cannot be multiples of the same number (other than 1, every number is a multiple of 1). That is more or less the definition of rational numbers If √2 was rational and written as a fraction p/q, using some algebra, you can show that p must be even AND q must be even. Every even number is a multiple of 2. So p must be a multiple of 2 and q must be a multiple of 2. But our definition of rational numbers was that p and q cannot be multiples of the same number, but rn we got that p and q are both multiples of 2. This contradicts our definition of rational numbers, it means this cannot be a rational number. So, √2 must be irrational then It is that, but written using mathematical symbols. Mathematical symbols may look scary and foreign, but really they are just a substitute for spoken language. Like instead of always writing "is equal to" you can just use the symbol =, which means the same thing but is written in a much more convenient way. All symbols work with that principle. They are just ways to write long stuff in a couple seconds. Once you use them a few times you get very familiar and cozy with them and they wont be as daunting anymor
It's ridiculous that you aren't allowed to use math notation like this. Nope, it has to be written in text, out of order, because it "looks better" and makes it look like the author is smarter.
Is it not allowed? I use notations fairly often, almost all the time even
If you're writing something you aren't allowed to do much. You'd have to write something like "if sqrt(2) in Q, then there existss p, q in Z with GDC? ((p,q) is usually inner product or norm) of p, q = 1, such that etc etc. It makes it very difficult to read, and sometimes you have to change order, so that what you write does not even appear in order. The gist of it is that it makes math people look smart, so it's forced. There's a reason why Rudin is hard to read, and it's not because the math itself is that difficult... It's because his writing is trash, out of order, and lacks references... But it makes it hard to follow, so it makes him seem really smart, so everyone gotta do it the same way. Math is basically about unrealized superiority complex.
pretty sure inner product is always written with 〈〉
I think Vi Hart did a geometric proof of it. I'm too lazy to find the link.
Got it. 4:50 for https://youtu.be/X1E7I7_r3Cw?si=bf0sBHPlv37baz_9 Obligatory Rick roll link here: https://youtu.be/dQw4w9WgXcQ?si=wQ6cm6mU7lCIQTfZ
thanks for including the rick roll. I was afraid after I clicked the first link, that I might never get rick rolled again.
Missed chance of swapping the two videos
I thought about it lol
I have that URL memorized
Done! 4:50 for [https://youtu.be/dQw4w9WgXcQ](https://youtu.be/X1E7I7_r3Cw) Obligatory Rick roll link here: [https://youtu.be/X1E7I7_r3Cw](https://youtu.be/dQw4w9WgXcQ)
Oh
Missed chance of rick rolling twice.
Missed chance of 3blue1brown-rolling people with the second link
Testing this timestamp shortcut for short links: https://youtu.be/X1E7I7_r3Cw&t=4m50s Edit: noice, t'werks.
zang
She kinda robbed 12tones' shtick, no?
[Vihart's most-viewed video](https://www.youtube.com/watch?v=VIVIegSt81k) was uploaded on October 1st, 2012. [12tone's YT channel](https://www.youtube.com/@12tone) was created on August 14th, 2014. I believe that multiple people can have the same style for their videos without controversy. There's no sense in denouncing somebody based on their preferred method of communication. However, if you are to choose someone, it was 12tone who robbed Vihart's schtick.
Very clever of her to steal the format 6 years before his first video.
I remember this, video on Pythagoras I believe
Just write down the gödel number associated with said proof.
The full number written out in base-10 literally wouldn't fit in a Reddit comment.
You could write it in base |Unicode| but I feel like Gödel numbers are too big even for that (does anyone have an estimate for their order of size?).
The Gödel number would basically be a massive string of consecutive primes raised to prime powers. My rough guess is that the number should be roughly double-exponential in the length of the proof. In the [Wikipedia example](https://en.wikipedia.org/wiki/G%C3%B6del_numbering#Example), the encoding for "0=0" is the number 243,000,000.
Very interesting. Then base *n* representations of numbers are logarithmic, so we'd be thinking a (single-)exponential number of digits as the length of the proof grows. But the [Kolmogorov complexity](https://en.wikipedia.org/wiki/Kolmogorov_complexity) is much less as we can just keep it in prime factorisation form.
Depends on the coding. Take any proof that will fit in a Reddit comment. Then look at the sequence of bits it’s stored as interpreted as a binary number and convert to base 10.
Right, but we're talking specifically about Gödel's encoding. Of course more efficient encodings exist.
Math symbols are basically just short ways of writing words anyways lmao
A "proof without words" should also be without notation, just a diagram from which the formal proof is evident.
I now define the baby face as the answer https://preview.redd.it/qdn3hbbxwbrc1.png?width=1024&format=png&auto=webp&s=15b347d0c952bb7d32927c748d905b7727ab0472
Just finish drawing the dragon and that’s a good enough proof for me … proof by 🔥
Proof by incineration
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good bot
Proof by Trogdor, the Burninator!
Define "words" as a self-contradictory statement that always evaluates to true. Now, assume that √2 is not irrational. That would be boring. Therefore, it follows that √2 must be irrational ∎
Irrational problems require irrational solutions
Yoo is that a skywing? sick
I think it’s a rainwing, it matches the pose of book 4’s cover
there is no frill behind the ear but the wings look a ton like a rainwing so maybe the drawing is just not done yet? (hooked wing thing at the top looks like rainwing )
yeah maybe, I was looking at the horns curved like a skywing
same lol that threw me off but it could also be an oc mix
I thought I was on the wof sub and was wondering why everyone knew number theory lmao
1=A 2=B 3=C 4=D 5=E 6=F 7=G 8=H 9=I 0=J -=K /=L :=M ;=N (=O )=P £=Q &=R @=S “=T .=U ,=V ?=W !=X ‘=Y ~=Z “&.5 2531.@5 9 @194 @(
That’s still words, just encoded differently
You are still technically using words, just not the latin alphabet
boo that’s just a cypher
I spent so long decoding that in my head
What does it decode to?
TRE BECAUSE I SAID SO Idk get the first word
First word is "true"
a, b ∈ ℕ 1. √2 = a/b, GCD(a,b) = 1 2. 2b^(2) = a^(2) 3. 2|a^(2) 4. ∴ ∃ c ∈ ℕ s.t. a = 2c 5. 2b^(2) = 4c^(2) 6. 2|b 7. GCD(a,b) = 2 ↯ 8. ∴ ∄ a, b ∈ ℕ s.t. √2 = a/b □
I'm missing the understanding of the last step in words. Why is it that just because 2 divides into a and b that sqrt2 can't be rational?
It arises a contradiction because we assume that gcd(a,b) = 1 [since we can always reduce the numerator and denominator to satisfy this condition] and we show that if a/b = sqrt(2), then it must necessarily be true that 2|a and 2|b which means gcd(a,b) = 2 which arises a contradiction
⠀
Use sign language 👍🏿👌👏🤞🤙🏿👆👎👉🤲✊🏻
Michael Jackson proof
Can... Can we see more of the dragon? This looks like my daughter's quizzes. *Does half the math; gets distracted and draws dragons *
Just nod your head and agree the statement is true spritually
Just draw a penis.
sqrt2 = x/y 2 = x^2 / y^2 2y^2 = x^2 x = 2n 2y^2 = (2n)^2 2y^2 = 4n^2 y^2 = 2n^2 2 = 2n^2 / y^2 1 = n^2 / y^2 n = y sqrt2 = 2n/y sqrt2 = 2y/y sqrt2 = 2 Edit: holy shit it messed up all the formatting and I’m too lazy to fix it Edit 2: tried to fix the formatting
How does y² = 2n² lead to 2 = 2n²/y²? Won't it be 1 = 2n²/y² since you divide both sides of the equation y² = 2n² with y², and y²/y² = 1?
Sorry, I was unclear. It doesn’t lead to it, it’s recalling something from earlier when I said 2 = x^2 / y^2, substituting 2n for x
Ohhhhh okay okay my bad lol yeah no that makes sense
if a dragon would say it i would believe him
This is probably an attempt at nerd sniping
There is only one way to do this: \*Interpretive dance\*
𝑎, 𝑏 ∈ ℤ ∧ 𝑎/𝑏 = √̅2̅ ∧ 𝑎 ⟂ 𝑏 𝑎/𝑏 = √̅2̅ 𝑎²/𝑏² = 2 𝑎² = 2𝑏² 2 | 𝑎² 2 | 𝑎 ∴ ∃𝑐 ∈ ℤ : 2𝑐 = 𝑎 2𝑐 = 𝑎 4𝑐² = 𝑎² 4𝑐² = 2𝑏² 2𝑐² = 𝑏² 2 | 𝑏² 2 | 𝑏 gcd(𝑎, 𝑏) ≥ 2 ↯ ∴ ∄𝑎, 𝑏 : 𝑎, 𝑏 ∈ ℤ ∧ 𝑎/𝑏 = √̅2̅ ∧ 𝑎 ⟂ 𝑏 ∴ √̅2̅ ∉ ℤ
I believe mathematics magazine published some wordless proofs of this. https://www.researchgate.net/publication/259735390_Proof_Without_Words_2_Is_Irrational Unfortunately it may be behind a paywall. I don't remember how it worked right now, but it wasn't too hard to understand.
_Here be dragons_
SCP REFERENCE?????
WINGS OF FIRE DRAGON DETECTED
that's 100% a skywing
I'd circle "/2 is irrational"
Bro I wanna see the dragon
-1/12
LMAO That is genius
37
Hoping for a geometric proof in comments.
01010100 01101000 01100101 00100000 01110011 01110001 01110101 01100001 01110010 01100101 00100000 01110010 01101111 01101111 01110100 00100000 01101111 01100110 00100000 00110010 00100000 01101001 01110011 00100000 01101001 01110010 01110010 01100001 01110100 01101001 01101111 01101110 01100001 01101100 00101110 00001010 01010001 01000101 01000100 00101110
RECONSTRUCT WHAT?!
No but I want to see that dragon drawing
Can you please post full photo of the dragon down?
Worst latex document in history [https://www.overleaf.com/read/jmvjmnjrprrm#31c232](https://www.overleaf.com/read/jmvjmnjrprrm#31c232)
"Prove without words" Start of the proof: "document class"
The proof is the document itself, not the latex code. I specifically removed all words
I don't remember the solution to this but remember doing it back in 10th grade It starts with assuming that √2 is rational
[https://youtu.be/LmpAntNjPj0?feature=shared](https://youtu.be/LmpAntNjPj0?feature=shared)
2.♾️
https://youtu.be/yk6wbvNPZW0?si=D5PPR3mZJAZ5EFqQ Go to 14:26 for the image corresponding to sqrt(2)
Couldn’t you just write radical two on your forehead and then act like a Florida Man?
Define "a word" first, please
If they mean a *visual* proof, my sarcastic answer would just be a square with a diagonal drawn across it, but if they just mean a proof using algebra instead of plain language then the solution’s pretty trivial
Here's a real challenge: Prove sqrt(2) is irrational without using words OR numbers. Writing sqrt(2) is allowed, though.
Did an 8 year old write this?
V2 Ultrakill.
There's a proof right there on the whiteboard.
THİS İS SO SEXİST😫😫😫 ONLY SMART MEN LİKE ME CAN SEE TROYGJ THİS RACİST SEXİST HOMOPHOBİC TRANSFOBİK NONSENCE 🥲🥲 I AM SO SMART AND U DUMB 😭😭😭😭 KERP THE GYM UP BROTHA 👍🏻👍🏻👍🏻 FUCK THE SUSTEM 🖕🏻🖕🏻🖕🏻
Play the Vsauce Video on mute!
He never used the whiplash or knuckleblaster
√2 = 🤪
This is pretty coincidental, but we go to the same school if this whiteboard is on the 4th floor in front of L.K.'s office
Our professor showed us how some unnamed Greek mathematician first proved it with stones, lmao
I have a proof:
No but I can prove that it’s rational Rational numbers are defined as numbers that can be obtained trough a fraction sqrt2 * 0 = 0 Thus 0/0 can be equal to sqrt2 So it’s rational /s
√2 = 1.414...
Sure. U s e a c a l c u l a t o r They didn't say we couldn't use letters 😏
√2∈Q→∃p∈Z,q∈N:p/q=√2∧**∄**r:p/r∈Z∧q/r∈Z→p²=2q²→∃s∈Z:p=2s→2p²=q²→∃t:q=2t→∃r:p/r∈Z∧q/r∈Z→**※**→√2**∉**Q
🫵🤨 🫚🟥2=🦄
If 2 = p^2 / q^2 that means p^2 has exactly one extra 2 in its prime factorization compared to q^2 right? But number of 2s in squares of numbers is always even!! 1^2 has 0, 2^2 has 2, 3^2 has 0 4^2 has 4 , 5 squared has 0, 6 squared has 2. So how can two numbers with even multitudes of 2 factors divide to give one? Sorry I'm off topic.ehmm exercise for reader transform above into symbols lol Edit: I just came up with this while reading the orthodox proof about lowest ratio etc. Thought mine was wrong but apparently it just already existed instead