Root 2 - Numberphile
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- Опубликовано: 26 янв 2012
- The square root of two is a fascinating number with a long and sordid history. It also forms the basis of most office paper, such as A4, A3, etc.
More links & stuff in full description below ↓↓↓
This video features Professor Roger Bowley and Dr James Grime.
NUMBERPHILE
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why do i do this to myself. late night math videos on youtube when i cant even do basic math
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
i took my root beer and put it in a square cup...now its just beer
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!
I can't stop watching these videos. I barely understand anything going on, but I think I've learned what a number is, so I'm pretty excited
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
*4:51** "...I can't begin to tell you how much they disliked this."*
**Proceeds to tell us how much they disliked it.*
XD
It's an expression
And now there is no such thing as dislikes on RUclips.
@@thegrassguy2871 Return RUclips Dislike extension join the revolution
@@sandorrclegane2307 🤓
The overwhelming irony is that if you hear the name Pythagorus, you think of his Theorem, which gives rise to irrational numbers - the very thing he hated!
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.
At this point in time, the number of views this video has is 1.414 million (root 2 x 1000000). You have no idea how long I waited for this moment.
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
I love re-watching ancient primal numberphile videos and thinking about how when these videos were made, they likely had no idea how popular their channel would end up becoming over the next decade
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
2b squared or not 2b squared?
Yes.
wauw
FriedEggSandwich that is the question
Commander Keen was an awesome game
It's hip to be squared.
dont urine towards the wind.. solar wind applies too!
???
@@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*
"The square root of 2 is about 1.41 something or other."
Nice.
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.
5:44 - guys, guys, Bruno Giordano was a striker for Napoli football team in the eighties.
Philosoper guy is Giordano Bruno.
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.
Nobody:
The bloke who added a radial blur on the thumbnail: *You have entered the comedy area*
Hahaha
@@lokeegnell3991 maybe that 'he have achieved.... komedy !!!!' XD
@@pikachu2860 weedeater
99/70 = 1.4142857142857...
(99/70)² = 2.000204081632653
99/70 is an excellent approximation of √2
+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!
I think that talking about the square root of 2 is pretty irrational.
Ha, punny.
+LLHLMHfilms Yes, let's cast them into the Mediterranean.
oh hello there brother
Perseihottuma greetings fellow loaf bloke
Nice one right there.
I love how they always talk like they're uncovering some massive government conspiracy
Maths is a conspiracy that the government doesn't want you to learn about.
Terrence Howard anyone?
I'll never ever pee facing the sun again. Thank you, numberphile.
Cool, I didn't know that about the A series paper. Now I'm a fan of the A series Paper (alas, something we don't use in the USA).
Keith Dart except for card stock and other specialty craft paper.
A4 "pretty standard in most of the world"
*Cries in freedom paper*
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)
root 2 is indeed one of my favorite numbers. It comes more up in daily life than I thought.
@5:20 I was lucky enough to have an awesome professor for complex numbers, he was my AC circuits professor. Everyone in that class was great with complex numbers which worked out well in other math classes. Other students used to hate it when we had to do complex/imaginary numbers.
The proof was very clearly demonstrated, thank you!
3:16 "Pssssst... pssssssssssssst* (*whispering*) "Hey kid, wanna learn some maths"?
Well I don't know about urinating against the sun, but I believe i shouldn't urinate against the wind.
I tend to not do it outdoors where the neighbors can see me
Oh my gosh, watching these videos just make my day. Whenever my day isn't going so well or I am in a bad mood, I watch these and I feel better almost instantly. THANK YOU NUMBERPHILE!
This is how i discovered how useful algebra is. I used my basic knowledge to find out what the ratio between the sides on paper is. From that moment on, i was interested.
@@abirdthatflew tbh calculus is the one that's just approximations, algebra gets you answers.
Thank you sir
I don't really like math but I like Numberphile for some reason.
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
I understand about 10% of these videos, but I still watch them. Dr. Grime is awesome.
Dude i am french and my schoool juste send me this video haha
Thank you sir
Brilliant. I always knew there was something special about √2. This argument is based on the fact that the ratio of √2 to 1 becomes the ratio of 1 to √2 when you divide the larger amount (the longer side) by 2.
You start with a ratio of √2:1. Divide the larger amount (the longer side) by 2, you get (√2/2):1. But if you multiply (√2/2) by √2 top and bottom, you get (2/2√2), which is equal to (1/√2). So the new ratio becomes (1/√2):1, which is the same as the ratio √2:1. And so on ad infinitum: the ratio of the sides will always be √2:1 when you halve the longer side.
“This is A4 paper, it’s pretty standard in most parts of the world.”
🇺🇸: 😬
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.
I love these simple, historical videos about well known mathematical concepts, another favorite is the one about zero.
Don't urinate towards the son... but whose son?
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.
"you shouldn't eat fava beans" Now i understand why vihart said pythagoras was afraid of beans
This is one of the coolest videos I’ve watched in a while!
I have been looking for this video for some time! I love the way you disprove the idea of only having rational numbers! Math is truly a beauty of nature
Had to replay some parts to understand it, but it was worth it, this is amazing!
My favorite thing about the square root of 2 is that if you multiply it by itself, you get 2 EVERY TIME. Mind blowing!
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.
The square root of 2, if we think about musical notes, is equal to a tritone. The twelfth root of 2 is equal to a semitone.
+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?
The demonstration at the end was very beautiful, thank you for sharing!
That last explanation left me feeling like I was tricked. Like watching a magician make something disappear, or listening to a logician prove something that sounds contradictory...
Funny. People hate Maths and yet this channel has over 1,000,000 subscribers.
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!" :).
In the US, paper is typically 8.5 inches by 11 inches, a ratio of 1.294... - not quite 1.414... Is office paper different in the UK?
the sqrt(2) is also the basis of camera f/stop numbers (1, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22, 32)
Hipasus : *proof that √2 is irrational in Pythagoras's own theorem*
Pythagoras: I'll ignore that
I'm bad at maths, but I really love your videos and appreciate math science
This is so fascinating! It's weird that a simple number like 2 can have such a complicated square root.
So amazingly cool. Love these videos so much.
The videos you make now look exactly the same 11 years ago, no wonder why your channel is the best!
I keep cringing really hard when his finger slides by the paper's edge.
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.
Nice video! It made me love mathematics a lot! thanks for sharing this.
Thank you sooo much for such an amazing and concept clearing lecture. 😊😊 Keep it up 💯
Great job, please continue!
Why can't these guys be my Algebra teachers?
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
My mum thinks that complex numbers don't really exist and were just invented by mathematicians because we couldn't work out the square root of -1.
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.
The extent of my post-college education has been your videos, so thanks for that!
This was actually really cool. Had no idea that even paper measurements had so much thought behind them (at leeast in Europe).
This means that the longer side of A0 is square root of square root of 2 (fourth root of 2)
1:06
When I was little I didn't know about √2, but I knew that if you folded the paper in half the proportions would be the same
I love this really weird juxtaposition of "Look at how this math fits really well into itself, as found by Pythagoras." and "Pythagoreans say you shouldn't urinate towards the sun."
I love this video! Also, where is James filming his part? Looks lovely
In a way, not peeing against the sun makes sense. If the sun's in your eyes, you could be aiming anywhere!
Strangely enough, the Pythagoreans also believed "10" was holy and honored it by not meeting in groups larger than ten, but in the painting in the video there's eleven of them (if you don't consider the background to be part of the group)
This is why basic math needs to be taught to everyone, I read advanced math, I don't remember a fraction of it, but I still have the basics.
Understanding this is so godamn beautiful, seeing these patterns. It pains me that not everyone will be able to see these.
that guy is asking for a paper cut.
This guy is -crazy- IRRATIONAL!!!
Loved the physical interpretation guys
I really love these videos. One small quibble with Grimes… complex numbers and the mathematics associated with them works well, but that doesn’t mean that complex numbers “exist”. Geometric algebra does a better job of describing the same phenomenon by a different approach (that ends up looking similar, but is based on geometry and vectors instead).
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.
That proof is incredible. I'm amazed
mind blown in last few seconds
And this is the video that finally got me to subscribe to your channel.
very nice information, this is the first time I heard of A4 paper size ratio
4:15 well that escalated quickly
I was so afraid he'd get a horrible paper cut when pointing along the edge of that paper.
Indeed the ratio of A format Paper is root 2, but, not only the paper size, also the Pen widths for technical drawings have the same ratio, 0.25, 0.35, 0.5, 0.7, 1, and so on.. which means that if a drawing is enlarged, even the line thicknesses remain within the same ratio
Congrats! Nice vídeo!
Baby, are you √2? Cuz you can't even!
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.
Don't misunderstand 4:46 : not just irrational numbers are such that they "go on forever". 1/3 also goes on forever in its decimal fraction form, although it does so quite predictably.
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.
“A4 paper is pretty much standard throughout the world.”
US: We are not the world. Letter and Ledger it is.
I wish i had someone like you as my math teacher, would have made math a lot more intresting
but a4 paper is a/b=sqr(2)...
is a4 paper peeing towards the sun?
Dan -Horsenwelles- Williams I love faulty logic, it makes for some hilarious dinner table conversations.
The ratio of the dimensions of A4 paper is an approximation accurate to five significant decimal digits, not bad.
That proof is juat beautiful. I was partially lost all the way up to the end when you proved b was eve. Had a beautiul eureka moment.
Simple & brilliant maths. Thank you
I had to prove √3 is irrational in an exam once. I got it right.
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
What kind of paper sheet are you using? Where can I get it from
brilliant, love it.
wow thanks for the video.that was so beautifuly done
5:11 "We call them irrational numbers because..."
They're called "irrational" because they are not "rational", i.e., they're not a "ratio" of integers.
The nomenclature has nothing to do with Pythagoras.
You can generalize this proof to show that the square root of any prime number is irrational if I recall correctly. The strategy is the same. If p is a prime number and p = (a/b)^2, you can look at the prime factorization of a^2 = b^2 x p and show a contradiction.
+Alan Falleur - not only prime
Mariusz Szewczyk How do you generalize it further?
+Alan Falleur
I guess any whole number that's not a square number.
Ian Wubby Of course. That makes sense. If it's not a square number, then you can write it as the product of a whole number and the square root of a prime number, which you know is irrational.
+Alan Falleur Not strictly true. Look at the square root of six. Then it's the product of two square roots of prime numbers, which is not necessarily irrational. I mean, it does happen to be irrational, but you have to generalize the proof further to prove it :P
That proof at the end was delightful!
Goosebumps,one of the greatest
PYTHAGORAS101 I am curious to your statement of "SQRT (2) IS RATIONAL". Would you agree on a definition for rational numbers such as ℚ = {m,n} | (m,n) = 1 and m,n ∈ ℤ where n ≠ 0} where the ordered pair (x,y) is equivalent to gcd(x,y).
If you accept this definition then what two elements of ℤ would satisfy this definition to place √2 as an element of ℚ?
Would you also agree that x^2 = 2 has no solution in ℚ?
Allow me to give a more detailed Proof by contradiction from some lecture notes I found:
Assume x ∈ ℚ satisfies x^2 =2.
if (-x)^2 = x^2 = 2 then x=| x | and x ≥0 therefore x is always positive and x ∈ ℕ.
If x= m/n and m,n ∈ ℕ and (m,n) = 1
then since x^2 = 2 then (m/n)^2 = 2
m^2 = 2n^2 making m^2 therefore m^2 is even
If m^2 is even it follows that m is even (square of odd number is odd, square of even number is even)
if m = 2k with k ∈ ℕ we can substitute 2k for m in m^2 = 2n^2 for m and write it as 2k^2 = n^2 so n^2 is even and therefore n is even.
If 2 divides both sides m and n this contradicts the initial condition of (m,n) = 1
Therefore x^2 = 2 has no solution in ℚ.
This means that √2 can not be in ℚ and therefore can not be a rational number.
Steve McRae o.k i'm going to try explain in very simple terms .
my calculator says sqrt2=1.414213562,now one may say that number is irrational.
Now suppose we adjust the display and we can only see 1.4,nobody can deny that 7/5=1.4 is rational right?
another digit 1.41 ,is another fraction 17/12=1.41 plus other decimals
also 24/17=1.41 plus other decimals.
What is very interesting here is ,both fractions share a number and both are sqrt2 @3digits but one is< sqrt 2 and the other >sqrt 2.
Now if you combine both fractions and divide by 2 to get the average you get a convergence which doubles precision to.1.41421,now already we have more than half the digits for sqrt 2
For all 10 digits in the smallest possible terms 338/239 and 239/169 will both result in sqrt2 @5digits and a combined average converges to 1.414213562.(sqrt2 @10 digit)
Please see for yourself.
This is the realm of the real numbers ,they all exist as eternal converging fractions.
The odd/even argument is silly because it assumes it must be one fraction and must be the absolute square root of 2 .
Also one could argue that "a" and "b"could never be both even no matter what because of its GCD,so to conclude they are both even is absurd,because fractions are always in their lowest possible terms.
This is where the whole argument is futile in the first place .a and b are never, and never, can be both even.
any questions?
PYTHAGORAS101 I'm trying to painstakingly go through your post here, and not trying to ignore anything here...but only can really address some of the larger issues I seem to see here...I will go on the assumption your decimal calculations are correct.
It is true that all real numbers can be formed from convergent sequences (Cauchy sequences)...not sure what you mean by all real numbers are formed from converging fractions however.
You could as you pointed out try to find any nth place of √2 using what you are saying...but that really has nothing to do with √2 being rational or irrational. In order to claim √2 is rational you MUST be able to give specifically the fraction a/b where a and b are integers that would EXACTLY produce the entire value of √2. What would be a and b that would produce √2?
The odd/even argument isn't silly as it is a direct proof by contradiction given the conditions of what it means to be a rational number.
Perhaps there is some type of confusion in terminology here. What to you distinguishes between a rational and irrational number?
PYTHAGORAS101 In regards specifically to the proof that √2 is irrational, I'll simply it a bit and perhaps it may be a bit clearer.
a/b = √2 and assume that a/b is GCD(a,b) = 1 so they have no common factors other than 1.
Squaring both sides:
(a/b) ^ 2 = 2
a^2/b^2 = 2
Rearranging:
a^2 =2(b^2)
Obviously here a^2 must be even since 2(b^2) will be even as anything times 2 is even correct? (Even numbers are described by {x : x= 2n, n ∈ ℤ})
So a^2 is even, and as such a also much be even since even numbers when squared result in even numbers.
So a is EVEN
If a is even we can write a=2c
This gives us (2c)^2 =2(b^2)
4c^2 = 2(b^2)
Diving both sides by two we have:
2c^2= b^2
b^2 now must be EVEN since 2(c^2) is EVEN and therefore b must be even. (Same reasoning as above)
So a and b are both even. If they are both even they both can be divided by 2 which directly contradicts the assumption that GCD(a,b) = 1.
Where specifically do you see the flaw in this proof?
EDIT: "This is where the whole argument is futile in the first place .a and b are never, and never, can be both even." Exactly! Given that the GCD(a,b)=1 then you are right a and b can never both be even...which is why it is a proof by contradiction.
Steve McRae There is no entire value of sqrt 2 so how can there be a fraction for it ?However there are limitless amounts of fractions that can be constructed for ever real number (sqrt n) for any required decimal precision.(not cauchi ,more fibonacci)
In my opinion a number has no status if it labeled irrational ,it is no longer a number because it has no ratio to any other number.Its kind of sad that real number are treated this way.
PYTHAGORAS101 Why would there be no entire value for √2? Is Pi irrational to you? Even thought that as well can't be expressed a/b where a and b are integers and b is not equal to 0.
Your version of mathematics I am sure you are aware dates back to the greeks and even more specifically to Egyptian fraction notation. Are you familiar with that? What you are saying is what they believed. However, we are in modern maths and established modern maths. You do also realize that according to modern maths you would be incorrect would you agree? So you are saying you rather our educational system teach an outdated version of math (Egyptian fraction notation)? Where exactly is the progress there?
When he said A4 is the standard in most countries, I swear I could almost hear a parenthetical United States...
James Grime is an AWESOME teacher
i wish i had a professor like you guys......!!!!!
Here I thought you were going to go for the proof by infinity where you prove that not only a and b are even, but also c, d, and any other number you could put in there down the line has to be even. In other words: You could divide a and b by 2 into infinity but the maths holds that they always remain even, therefore can always be divided again.
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".
A0 paper. Its area is exactly 1 m squared. Its dimensions are (2^0.25 × 0.5^0.25) = (1.19 m × 0.84 m). and the ratio between them is ( 2^0.5 )...
A4 is one sixteenth of A0, its dimensions are (29.7 cm × 21.0 cm)...
These are really well edited films. Not spliced to make smart people look like idiots, or to make idiots seem smart. Or at least it feels that way. Just information. Well done.
this is just amazing !
Exaskryz
It's not arbitrary, a=2c because a is even. That's the definition of an even number. It's a multiple of 2.
“the ratio of a4 paper is root 2”
boom, root two isn’t irrational!
Me, wearing no jewelry and a carrier for G6PD deficiency: "Tell me more about these Pythagoreans..."
@ 8:36 but in physics we can say sqrt2 ≈ 99/70 :) and there is even a way of generating fractions closer and closer to 2 when squared. 41/29 could have also approximated sqrt2. here we can see the formula for generating fractions approximating sqrt2... 41+29=70 and 70+29 = 99. so, you add the numerator and denominator of the last fraction, and then add the two denominators and use it as the numerator for the new fraction. Thus, the next fraction in the sequence would be 99 + 70 = 169