Root 2 - Numberphile 

Oh, you're funny...

Wish that would work,

***** It failed, but when I put a pie in a circular cake pan it fit exactly.

Mistermaarten150 Funny, I managed to fit a pie perfectly in a square cake pan.

naphackDT Wait, hear me out! Pies are squared, I swear!

DragongodZenos me too i do that all the time

I can do basic math! :D But that's about it. Basic math. :(

+199022009 What counts as basic maths?

studying calculus made me cry don't put yourself through the torture XD

VolvoGustav 10+1=3

It's like dogs watching humans have *** The dog doesn't know what's going on, but the dog is still enjoying it

doubtful. I've literally never met anyone over the age of about 6 or 7 who knows what a number is. 7 and -7 are both considered numbers. but... they have the same number component. so that's like saying 7 apples and 7 turnips are numbers. but... only the number component is the number, right? what about 7i? that's an imaginary number, but it shares the same number component as 7, -7, 7 apples and 7 turnips, so it can't be a number either. only the number component is truly a number there, too. ok, how about 7/3 then? surely fractions are numbers, right!? well... no, because we still have 7 of something, just like 7i is 7 instances of i, and 7 apples is 7 apples. turns out almost nothing that's called a number by mathematicians actually is one. even positive 7 is not actually a number, because it has a sign component which opposes the sign of -7, but +7 and -7 share the same number component. turns out all of these things are vectors, not numbers. and this is actually important because mathematics does not operate over numbers at all. it actually doesn't even operate over vectors (which is the pairing of a number and a unit). it operates over units. and this has consequences that will probably completely blow your mind. for instance, we can see that mathematics operates over bare units by noting that unit conversions are possible: - 4 inches * 2.54 cm / inch = 4*2.54cm * inch/inch = 10.16cm * 1; so dividing a bare unit by itself yields the dimensionless multiplicative identity, 1... inch/inch has absolutely no number component at all, so that division was not over numbers, or even vectors, but pure units. but this carries further, because it means that 1+1=2 is actually false: - 1C flour + 1C flour = 2C flour; seems to work here, 1+1=2 is demonstrated, right? - 1C flour + 1 egg = uhm... not 2 of anything; so NOT(1+1=2) is also demonstrated this means that the truth of 1+1=2 is undecidable, which notably contradicts current mathematical dogma because in 1929 Mojzesz Presburger took Peano Arithmetic, removed multiplication from it, and allegedly proved that addition over bare numbers is decidable. it was two years later that Kurt Godel published his Incompleteness Theorems showing that Peano Arithmetic in its original form is undecidable, and this result was what shook up the Hilbert Program and basically threw 20th century mathematics into a minor crisis that remains unresolved. the deep problem here is that Peano Arithmetic and everything related to it, even alternatives to it used to build up to different formulations of the current foundation of mathematics, assume that numbers are the basic object over which mathematical operations work. and this mistake is the source of all the trouble, since math does not operate over numbers mathematicians must go around claiming that vectors are numbers, and since math doesn't operate over numbers but Peano Arithmetic does, the sorts of things which can be proven within a logical framework that accepts Peano's Axioms will be fraught with contradictions, which gives us Godel's Incompleteness Theorems.

What is this comment thread XDDD

@@sumdumbmick In Korea they laugh of -7 . They actually laugh of one divided by zero too 1/0 .

​@@PYSSMILK Haha thought the same thing

Yes.

wauw

FriedEggSandwich that is the question

Commander Keen was an awesome game

It's hip to be squared.

Will Sheridan now we have to wait for 14 mil views

Root too long?

now its 2.818m (root 2*2*1000000)

W i l l: I think 1414213 (rounded) is (root 2*1000000000000) or (root 2*10^12).

Now it's almost exactly DOUBLE

Yikak4 Nope, I am only thinking about who is Pythagorus.

akuskus Alright. Well as a maths student that's what comes to mind. I guess if you are more of a history person it's different.

You spelt it wrong, Akuskus was joking.

kerbalspacevideos I caught on :)

Don't take historical citations very seriously! I mean... we can know for a fact what Pythagoras did, but can never be sure of what he really liked.

I love these videos that are essentially 2 or 3 interviews intercut, or in parallel, about the same topic. Maybe they've fallen out of fashion, but I wouldn't mind more topical videos like these with multiple interviews.

They were pretty popular back then too it’s only grown proportionally

???

@@cometzfordays2032 thanks for clearing that up

@Alaa Alessa the wind will blow the urine towards you. Solar wind applies too lol

@@AakashKumar-tn6yh 4:19

urinate*

Calculators can't explain why no fraction can be the square root of 2. Most give a rational number as the answer, they just give a close answer.

Well it is...

GhostlyJorg Wouldn't it be splendid when we could infinitely calculate something.

Well, we have the tools to do that, just not the time. Trickle algorithms can give you any digits of Pi and other irrational numbers with absolute precision.

@@chumsky8754 modern calculators are more advanced. If you do sum(1/n^2) for n=1 to infinity, some will give pi^2/6 They have an internal logic that understands special values.

Yeah, it's a joke

I thought that Giordano Bruno was the Italian version of Gordon Brown.

And Bruno was burned at the stake for postulating that there were other intelligent lifeforms on other worlds and for being a pantheist, not for saying that the Universe is infinite.

XD

It's an expression

And now there is no such thing as dislikes on RU-vid.

@@thegrassguy2871 Return RU-vid Dislike extension join the revolution

​@@sandorrclegane2307 🤓

Ha, punny.

+LLHLMHfilms Yes, let's cast them into the Mediterranean.

oh hello there brother

Perseihottuma greetings fellow loaf bloke

Nice one right there.

+wheresmyoldaccount well if we're talking about approximations its not far off, but it's still infinitely far off from being exact

+Tobias Christensen Very nice wording (not far off but infinitely off) #Irrational

+wheresmyoldaccount (99/70)^2=9801/4900 = (9800+1)/4900 wait you see that let's zoom in 9999 times. (9800+1)/4900 we can do this (2x+1)/x and do this sqrt(2x+1/x) (2x+1)/x approachs 2 for x=infinity so the main function approachs the square root of 2. try it out today! and also if you want to approximate square roots, use this forumla sqrt((yx+1)/x)

ah ha! (2x+1)/x approaches 2 for x=infinity, because 2x+1 approaches 2x for x=infinity (simplified) x+1 approaches x for x=infinity

+wheresmyoldaccount Oh yeah, try squaring the result of this ratio: 665857/470832 Possibly enough precision to even fool your calculator into thinking that the square root of 2 is rational!

Hahaha

@@lokeegnell3991 maybe that 'he have achieved.... komedy !!!!' XD

@@pikachu2860 weedeater

Maths is a conspiracy that the government doesn't want you to learn about.

So close, so far.

US doesn't use A4?

@@lenonel3286 its uses has A4 paper and their weird paper

@@andreysilva8418 i hate this knowledge

@@lenonel3286 they use Letter size paper which is slightly wider and shorter (8.5 * 11 inches)

Keith Dart except for card stock and other specialty craft paper.

Dude i am french and my schoool juste send me this video haha

Thank you sir

You a) dont like being forced to learn math, or b) dont like the math youre being taught. Math is sooooo interesting when you sit down and learn and understand it. My first time learning and logarithms and exponents in school i HATED it, later on i looked it up on my own time and was fascinated by it.

+Andrew Jatib Interesting- I hated math up until 9th grade when I got a great teacher who made me want to excel at it and love doing it in general. It's my favorite subject and pastime :)

Well learning and life shouldn't always be fun.

You really do like math. You didn't like the way math was taught when you were in school. Most math classes do a poor job making math interesting and relevant. Numberphile does both, which is why you like this channel.

+Andrew Jatib me too

@@abirdthatflew tbh calculus is the one that's just approximations, algebra gets you answers.

Thank you sir

Any square root time itself is the number everytime

@@freshrockpapa-e7799 I made this comment 5 years ago, so I can't be sure, but I'm fairly certain I was being sarcastic when I wrote it.

I tend to not do it outdoors where the neighbors can see me

Don't urinate towards anyone's son. Urinating at other people is crappy behavior.

OTOH there are stranger fetishes...

+orbik Okay, then for you, urinating at other people _without their permission_ is crappy behavior. :D

+Reema Issa Or is that *probloom*, so that it's really *sunflowers* you shouldn't be urinating toward?

Your profile picture made that comment.

Ahahaha mee too. I was afraid he might cut himself!

I always cut my fingers when lifting school books from my bag I swear I would have my finger cut off if I done that.

Ikr I always thought he would cut himself

Same!

Not hard when your fingers are made of dark energy.

Popo Sandybanks People hate maths the way it's taught in classes. This is better.

Tim Tian Agree!Its being taught by conservative and conformist figures that promote stale thinking!!

Lots of people love maths...

Gomlmon99 true dat!

StarSky GamingOne Google Translate translates it to "true it!" :).

and the standard isn't root 2 but 297/210, though it's close and root 2 is within standard tolerances

@@garygrass7044 iirc the standard actually mentions the ratio of √2:1 as a defining property (and then goes on to say that all sizes should be rounded to millimeters after the exact calculation).

@@garygrass7044 approximation clearly since they went to lengths to show the SQR(2) is irrational proofs. As explained the purpose was finding a ratio that wouldn't end up being disproportionate with different a / b and SQRroot (2) was as close as it gets.

@@garygrass7044 The standard is √2. To meet that standard, 297/210 mm is officially "close enough" to be labeled A4. If you can dial in your machinery precisely enough, you can depart from 297/210, get closer to the standard, and legally label your product A4. Fun fact: There is a corresponding standard for technical pens, so that you can enlarge or reduce a drawing from one A size to another, and continue to add to it with lines of matching thicknesses.

@Sjittaste We have A4 - but for some reason it costs 3x as much as 8.5 x 11.

+sevishmusic And a tritone was once referred to as "diabolus in musica" (or "the devil in music"), on account of being so dissonant that people thought it must be avoided at all costs.

+Jim Cullen (Zagorath) It's true, however the old tritones were tuned differently as equal temperament has only been in use for a couple hundred of years. In equal temperament the tritone is equal to the square root of 2.

***** well, not per se. Previous tuning systems were based on natural ratios and frequencies found in the harmonic series. For example, an interval of a fifth was a ratio of 3/2. 12th root 2 is a close approximation of this, but it isn't quite as "perfect" as the natural frequency. What it gives us is a nicer sound in more keys, instead of a perfect sound in one key, and a less nice sound if you're playing out of key.

+pyropulse Not sure which notes you're talking about, we mentioned a few different classes of intervals already in this thread.

+sevishmusic Can you explain this some more? how can musical notes be equated to numbers?

YoshiFace yes if you go to Cambridge

YoshiFace you're not smart enough to enter a class with them as teachers. Intelligent people don't blame teachers for their inadequacies.

/r/iamverysmart b/c i passed algebra class

@@theywalkinguptoyouand4060 It seems like you never had a bad teacher havent you?

@@theywalkinguptoyouand4060 i think u are the kind of kid who born with rich family, went to special school and never saw a bad teacher

Jack Harrison isn't that a thing of debate, whether numbers exist or we just made them up?

but we did work it out the way we apply our mathematics in anything involving negative square roots is such that it can still end up a real rational positive number at the end of the day, and complex numbers are used in physics and other fields of science plenty, so i can't really see why she'd say it "doesn't exist" when it perfectly validly represents and solves for real world problems

I see maths like language sometimes, we as humans made it up, and one could debate whether that makes it 'real' or not, but it is used to work out and communicate processes, and can be applied to the real world

You can argue that numbers don't exist in general but I don't think any mathematician believes real numbers exist but complex numbers don't. That's because the complex numbers can be constructed from the real numbers using what's called a "field extension". Consider the set of polynomials with real coefficients. Step 1: Define the following equivalence relation: two polynomials a(x) and b(x) are equivalent iff they leave the same remainder when divided by x^2 + 1 - If a(x) and b(x) are equivalent we write a(x) ~ b(x) - The set of all polynomials equivalent to a(x) is called the "equivalence class" of a(x). Step 2: Define the following arithmetic operations: If A and B are equivalence classes then A + B is the equivalence class of a(x) + b(x) where a(x) and b(x) are any polynomials in A and B respectively. - A*B is of course the equivalence class of a(x)*b(x). - If you're thinking this looks a lot like modular arithmetic but with polynomials then you're right, that's exactly what this is. Step 3: Let "i" denote the equivalence class of x and let the real numbers denote their own equivalence classes. And you're done. i^2 = -1 because x^2 ~ -1. Every equivalence class can be written as a + b*i where a and b are real numbers since every equivalence class contains a polynomial of degree 1 or lower. You can now go and do complex arithmetic, derive Euler's formula, or prove the Riemann hypothesis with the comforting knowledge that everything you do is based on a sound theory of purely real numbers.

All abstraction of reallity is made up, and in a deep sense, math works in "real world" because we built it to match with our own abstraction of the world. Math can't exist because our brains simplefy the world around does, so exist but, in our minds and not in nature. And so, if we can agree that negative numbers exist, so they exist. If we can agree that sqrt (-1) exist, so it exist. But always in our minds.

This guy is -crazy- IRRATIONAL!!!

How?

Proof by contradiction: Suppose sqrt(3)=a/b where a and b are the smallest possible integers. that means that 3=(a^2)/(b^2) so 3b^2=a^2 now notice that if you factorize a square number, you always get an even number of prime factors: 4=2*2 9=3*3 16=2*2*2*2 25=5*5 and so on so that means that the prime factorization of the right hand side has an even number of factors, and the left hand side has an odd number of prime factors. since both sides are equal, and every number has one and only one prime factorization, we have a contradiction, so our assumption that sqrt(3) is rational is wrong QED BTW, it's pretty easy to generalize this proof to all non-square numbers

It think this was a joke, a shitty one but still a joke.

@@yuvalnosovitsky1303 Very interesting... but in a sense I feel this proof tells me nothing (new). For example the square root of 4 can be represented as a rational number because 4b^2=a^2 for the same reasons. Inside I feel there's something deeper in nature to be seen but this is like restating the same problem.

@@PM-vs3rh or you could prove it in a way similar to how √2 was proved irrational in this video

If you're referring to the "assumption" that the square root of 2 is irrational, that's the whole idea behind proof by contradiction, which is the method they're using. They assumed the square root of 2 was rational, and from it derived an absurd conclusion. Since this absurd conclusion can't be true, assuming all of their logic after assuming sqrt(2)=a/b is valid, they must've done something wrong, and the only thing left to be wrong is the assumption that the square root of 2 is rational. The general idea is assuming something which you think is false, and derive something absurd, therefore demonstrating you did something wrong along the way. Assuming you made no mistakes, the only thing you could have done wrong was assuming the false thing, so it must be false.

you're right, getting a contradiction constitutes a proof.

Jane It does when there are two possibilities

Jane Somebody never took geometry/algebra2 :P

You can't prove to me that you exist.

lel

I was gonna say, "Cause you're so irrational"

Baby, are you the square root of negative one? Because you can't be real. There even is a worse one! Baby, are you i? Because you can't be real.

What an odd joke

L4Vo5 I would say you two are even.

What does it mean for any mathematical object to "exist"? I'm not sure I understand your distinction here about complex numbers not existing because of geometric algebra.

@@MuffinsAPlenty Grimes is the one who said the complex numbers "exist". Geometric Algebra provides a different approach that results in things that act like complex numbers and therefore fulfil the same role.

Dan -Horsenwelles- Williams I love faulty logic, it makes for some hilarious dinner table conversations.

Brickfilm Man What you say is true.

It can be expressed as a fraction but not as a fraction of 2 whole integers. So the paper must be some decimal/ some other decimal = 2**(1/2) .

Iskander Said The side lengths (both or at least one side) of the paper would have to be irrational in order for the ratio of the long edge and the short edge to be √2, which is an irrational number. Keep in mind that all fractions of _rational_ decimals can be expressed as fractions of integers (ex. 0.125/0.48 = 25/96), and so the side lengths of the paper would have the be irrational for the ratio to be √2. It does not matter if the side lengths of the paper are merely decimals or not, since as I said before, all fractions of _rational_ decimals can be expressed as a fraction of 2 integers. For the sides lengths of the paper to have a ratio of √2, the side lengths of the paper would have to be irrational, and that would be impossible in the real world.

***** Sorry if I have caused any confusion, but what I was trying to say is that the ratio of the long edge and the short edge of A4 paper would not actually be _exactly_ √2, because the values of the long edge or short edge would not be irrational numbers. It is _physically_ impossible to measure irrational numbers; what makes you say that the lengths of the sides would be irrational? Going back to your example, it is not possible to construct a _perfect_ square in reality and you can't slice a square _perfectly_ diagonally. Thus the said √2 value of the diagonal of a square in reality would just be an approximation. Another example would be pi. Pi is an irrational number, and people do not know the _exact_ value of pi. All that humans can see, are rational approximations of irrational numbers. What are irrational numbers? Real numbers that cannot be represented by a ratio of integers. Irrational numbers when written as decimals do not end or repeat. _Physical_ values in real life would not be irrational, and could not satisfy this property, because all measurement is imprecise in the _slightest_ way. I have placed emphasis on the words "physically" and "exactly" to help you understand my message.

Brickfilm Man Hmm, I see now. I think what you were trying to talk about were how humans and computers process numbers, which is better described as "measured values" rather than "physical values."

Hey boo

But they all do it 'predictably': what makes the difference between 'rational' and 'irrational' is that rational numbers always have a decimal fraction expansion that starts repeating and then keeps repeating forever. With irrationals, they are still predictable, but there is no point past which it only repeats.

Sure, I only referred to the vague term "goes on forever"

But "only referring to the vague term" does no good: it must be replaced with something exact.

But " But "only referring to the vague term 'it goes on forever' " does no good: it must be replaced with something exact." does no good: it must be replaced with something exact.

He meant go on forever without repeating. 1/3 repeats:1.3333...., Pi and sqrt(2) do not repeat.

That's probably how the original argument worked! It's a method called "infinite descent", and it's based on the principle that you cannot have an infinitely descending sequence of positive integers. But today, infinite descent proofs are often replaced by "choose a minimal thing and violate minimality" arguments. I guess they feel "cleaner".

the person who created algebra is mohamed bin mousa alkhwarizmi was a muslim man was born in khwarizm near iraq

@@samussam6554 I can tell where you're from then...

This reminds me of my linear algebra lecturer who claimed pivot elements were named for a French mathematician.