The Difference of Two Squares

See more James Grime over on Singing Banana.
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He is also on some channel called Numberphile.
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We have some teacher notes and activity resources for this video on Think Maths. See if your students can find a better graphical proof for the case of 4k than we did! Spoiler: there is one.
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CORRECTIONS
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Music by Howard Carter
Filming and editing by Matt Parker
Design by Simon Wright
Some maths by James Grime.
MATT PARKER: Stand-up Mathematician
Website: standupmaths.com/
Maths book: wwwh.umble-pi.com
Nerdy maths toys: mathsgear.co.uk/

[@quantummaniac5]

If one can't be a prime, why should zero get to be a square?

[@MmmVomit]

Zero can be a Parker square.

[@benjamimapancake6429]

0: not a square 1: not a prime 2: not an even number 3: doesn't exist 4: not x^x 5: not an odd number 6: not a perfect number Anyone else?

[@karolakkolo123]

@@benjamimapancake6429 7: not a mersenne prime 8: not a perfect cube 9: not a single digit number in base 10 10: not a power of ten

[@i_am_anxious0247]

Because a perfect square is x•x, but a prime has exactly 2 factors. X•X? Just plug in 0. 2 factors? No. One has a single factor, which is 1.

[@Cernoise]

If Matt calls 2 a subprime, maybe he can call 0 a subsquare.

[@JackKanutin]

I'm slightly annoyed Squarespace didn't sponsor this video....

[@McMxxCiV]

Yeah. Together with Foursquare. They're quite different.

[@xNothing2Lose]

@@McMxxCiV Cant stop laugh.. dude:D

[@Leibowitz]

@@McMxxCiV Man, that would be two truly different squares...

[@NathanTAK]

I think this video not being sponsored by Squarespace and/or Foursquare might be a war crime

[@JamesSmith-rb5lv]

They did. The whole video was the advertisement.

[@matthewsmith7900]

Most ambitious crossover event in history.

[@valdemar91]

Someone: Marvel Endgame is the most ambitious crossover in history! Matt Parker: Hold my calculator!

[@snowman7514]

Valdemar * calculator * lol

[@skandragon586]

Parker square: not quite right Grimes square: algebraically precise

[@geekjokes8458]

The gaussian way

[@MrSonny6155]

The difference is surely an odd one...

[@Pacvalham]

*Grime

[@rachelpoole4497]

This video has: A difference of two squares (math) A difference of two squares (they're on different squares) A difference of two squares (disagreement between nerds)

[@BobStein]

I recognize an aficionado of lists. And treble entendres.

[@odenpetersen6028]

They’re both standing outside in town squares

[@JasonCliftJones]

@@odenpetersen6028 Specifically, Leicester Square and Trafalgar Square in London. Which took me waaaaaay too long to realise. (Which are close enough that they walk between them at the end)

[@joda7697]

@engineer99 Well yes, but actually no.

[@christopherbiomass7155]

And the difference of two squared and two squares is... Two squares.

[@ReedHarston]

This Numberphile: Civil War trailer was brilliant. Definitely going to watch this one in theaters. 👍

[@GvinahGui]

Can't wait to see them reconnect in Numberphile: Endgame in a few years

[@EpicGuyJC]

@@GvinahGui Numberphile: Infinity War

[@hebl47]

They will fight until the true villain emerges: Steve Mould trying to convince the world we should be using tau instead of pi.

[@GEM4sta]

James turns himself into .9+.09+.009+... versions of himself, truly an epic scene.

[@huruey]

@@GvinahGui Endgame Theory

[@schefre35]

Matt and James? This has to be a great video.

[@erwinjohannarndt4166]

This IS a great video

[@retnogaming3379]

Marvel: Infinity War is the most ambitious crossover event of all time. Matt & James: *this video*

[@arikwolf3777]

My number 1 favorite mathematician, James, + my number 2, Matt = awesome video. Now get Brady to film them.

[@nitehawk86]

Two squares. :)

[@Robert-iu2ou]

@@nitehawk86 and their odd difference!

[@aguti1111]

It's just adorable seeing them walk through London arguing about whether 0 is a square number 😃

[@PSUQDPICHQIEIWC]

Idk. Looks pretty round to me.

[@user-vn7ce5ig1z]

Well 0×0=0 and √0=0, so it should count. I'm sure Matt also agrees, but had to disagree to make that last pun work.

[@minewarz]

@@user-vn7ce5ig1z I dunno man, Matt also won't accept tau as the superior circle constant.

[@brachypelmasmith]

i would accept that except, as matt said then any square number is difference of itself and zero squared

[@Tfin]

Well if they don't agree that 1 and 2 are prime numbers, why would they agree on this?

[@SteveMould]

James is right. I won't get out of bed for less than £(a² - b²).

[@standupmaths]

Suddenly I’m ok with 0 being a square number.

[@Ultiminati]

plot twist: a is a complex number

[@ahobby]

@@Ultiminati $?

[@zecuse]

Plot twist: a < b

[@agastyagoel6185]

@@zecuse plot twist a = -5 and b = 3 O_O

[@AnonymousFreakYT]

The difference of two squares? With Matt & James? Let's see... James has more hair. Matt has a goofier accent.

[@Halokon]

Thanks, saved me from making a cheesy joke on the same lines 😁

[@zmaj12321]

They made the same joke at the end

[@namewarvergeben]

Poofy and Goofy?

[@XxjazzperxX]

I don’t get it, please help

[@McMxxCiV]

@@XxjazzperxX a "square" is also a word for a serious, maybe somewhat boring person. They use it as a joke about themselves.

[@blemishingbohemian2075]

I like how they based an entire video on a pun while actually backing it up with an interesting demonstration about two different types of proof

[@Leibowitz]

While staying in two different squares :)

[@toferg.8264]

Absolutely :)

[@Xeridanus]

The footage of them in the squares is also square.

[@CraftQueenJr]

What’s the pun?

[@JonVanOast]

what? these two squares, and their differences? haha

[@AceMartinov]

Video length should have been 9:16 for two squares, missed opportunity

[@McMxxCiV]

What's worse is that 550 (the number of seconds in this video) is one of those numbers with a remainder of 2 after division by 4, so can't be written as the difference of two squares. TRAGIC.

[@hexeddecimals]

And 9:16 is 556 seconds, a multiple of 4, so it's a difference of two squares. Double missed opportunity.

[@hexeddecimals]

@@McMxxCiV actually its 551 seconds. It say the video 9:11, for me at least. Which can be a difference of two squares.

[@McMxxCiV]

@@hexeddecimals phew

[@justdoitlater9507]

9:36 would have been even better because: 9, 36 squares great, but additionaly 9 mins 36 secs= 576 secs =24*24 secs

[@12tone]

I'm with James on this one: 0's an integer, and squaring an integer gets you a square number, so 0^2 is a square number.

[@lior_shiboli]

exactly but you can also say it about 1 being prime (unless you say prime needs to be the "multiple of only *2* numbers which are 1 and itself" instead of "only 1 and itself" which feels like cheating) also i love your channel

[@hapmaplapflapgap]

I believe 1 is usually explicitly not a prime number. 0 can also be excluded from integers when need to, but the most general definition of prime does not include 0, and the most general definition of square numbers does include 0 (non-negative rather then positive).

[@nonchip]

​@@hapmaplapflapgap 0 can't be excluded from integers though. it's just not a positive integer (it's a non-negative one aka "natural number"). so it depends on their definition of "squares" really. in geometry a square of area 0 doesn't make much sense, but algebraically 0 being an integer i'd argue 0*0 is a square number. and apart from not being able to easily represented graphically it doesn't do anything different than any of the other square numbers they mentioned really. *and* is required for the "difference of squares" thing to even work. meanwhile 1 isn't a prime number because prime numbers aren't defined the way usually taught in school ("only divisible by 1 and themselves") but by being a "natural number greater than 1 that is not a product of two smaller natural numbers". if 1 was a prime then a lot of things (even the sieve of eratosthenes we all probably toyed with back in school to learn that "1 and themselves" thing) would break down, instead 1 (and -1; and some others when talking about non-integers) is so special it's considered its own category called "unit", being essentially "what defines integerness" (and by extension "naturalness" and by extension "primality"). note up until even the 1950ies lots of mathematicians considered 1 a prime still, and before the middle ages they actually didn't because they didn't consider 1 a (natural) number (kinda like that whole "unit" thing, 1 is what makes all the other numbers therefore it's not a number to consider), which is why eratosthenes came up with a sieve that worked when you ignore 1. some of the greeks even rejected 2 as prime because they thought primes are a subset of odd numbers (because apart from 2 all the others are odd, therefore giving 2 that same treatment as 1 "it defines how the others behave therefore it's special", but that's not considered correct anymore because it doesn't really define anything about the remaining primes, it's just a side effect, just as 3 ruling out numbers divisible by 3 doesn't "define" primality. meanwhile 1 *does* define something about *all* primes: what an integer even is: n₀=0; nᵢ=nᵢ₋₁+ *1* )

[@XCC23]

@@nonchip You also lose some really nice theorems. "all natural numbers can be written uniquely as a product of primes" - Not anymore you can't, if 1 is a prime.

[@clf400]

Loving the Leicester Square and Trafalgar Square backgrounds

[@ericherde1]

clf400 The difference of two squares, explained by two squares, in two squares.

[@MisterAppleEsq]

Ohhh, I didn't get that.

[@nrellis666]

never more than about 300 metres apart

[@Sakkura1]

Why no Parker Square background :P

They are definitely not squares. Uneven polygons at best

[@ShinySwalot]

"I am cheaper than Steve Mould" I'm dying, I love these two

[@woowooNeedsFaith]

:) Thanks, I simply couldn't figure out whom they were talking about.

[@ShinySwalot]

@@woowooNeedsFaith haha, you do know Steve Mould right?

[@woowooNeedsFaith]

@Shiny Swalot Yep absolutely, I do. That's why I was wondering why I could not recognise the name even when I listened it twice, but I recognised it immediately when I saw it written down. And even when I go back and listen to it again, I still keep failing to hear the last name "Mould"... There is something in his pronunciation which beats me.

[@ShinySwalot]

@@woowooNeedsFaith Ahh haha, glad I could help you then! Shame it didn't help you discover someone new though

[@livedandletdie]

I once tried to get rid of Steve Mold, but apparently thanks to u I now have Steve Mould instead.

[@xalkin]

at Leicester Square and Trafalgar Square talking about different Squares...brilliant :)

[@AviSilver]

They don't quite make it to Leicester Square

[@blackpenredpen]

Wow this is such a fun video!! Love the editing and the math!

[@standupmaths]

Thanks! Next time we’ll take a whiteboard with us…

[@ze_rubenator]

Oh right, and you're standing on different "squares" as you're explaining. Well played, good sirs!

[@Xeridanus]

And the footage of each of their arguments was also in a square shape.

[@guyspy21]

OOOOOOOHHHHHHHHHHHHHH

[@DavidLindes]

Xeridanus: and, the two of them are both "squares"... in the sense of being geeky people. 🤓

[@BrittBerg]

* Parker Square jokes incoming *

[@marcellkiss-redey8451]

When I saw the title, that was my reaction, too.

[@Goldap1000]

1:42 as parker as it gets

[@sehr.geheim]

you got me

[@dcs_0]

@@Goldap1000 Parker Tetris

[@Tentin.Quarantino]

Guilty as charged

[@stevepalmer4521]

Ha ha! Who else spotted the kid's super smooth wall dismount at 5:45?

[@ifburly76]

How did he get that so wrong

[@owenmaitzen339]

a real parkour square, that one

[@ogureckgreen]

More like Parker square of parkour

[@TheZotmeister]

I didn't, but that's great!

[@laurihei]

At least, he gave it a go.

[@_rlb]

I like the Tetris style animations :)

[@35571113]

+1 Except, they should probably be called... "Oddtris"?

[@_rlb]

@@35571113 Mattris?

[@rmdavidov]

@@_rlb matrix

[@N.I.R.A.T.I.A.S.]

Matt Parker & James Grime make a video together. Brady: Am I a joke to you?

[@BrittBerg]

I see James. I see Matt. I click.

[@JimmyLundberg]

Are you a dolphin?

[@BrittBerg]

@@JimmyLundberg What?

[@dirm12]

Bring back the old public maths offs! Love this format.

[@ericherde1]

The final argument applies to 1 just as much as 4. It’s only the difference of two squares as (1)^2-(0)^2

[@yourguard4]

It works also with switched places: 2^1-2^0 XD

[@EwanMarshall]

1 is odd.. all odd numbers were already accounted for in the algebraic proof. :D

[@ericherde1]

Ewan Marshall All odd numbers were only accounted for if we accept 0 as a perfect square. The proof breaks down for 1 otherwise. The only way in which 1 is the difference between two squares is as the difference between 0 and 1.

[@TheRavenCoder]

@Adam Filinovich except 0 is not a square number. You can prove this geometrically. Try to draw a square worth a side length of 0. You'll end up with a point, not a square.

[@SlipperyTeeth]

@@TheRavenCoder A point is a square.

[@petriksalovaara2805]

This got me into insanely good mood after a really crap day at work, thank you.

[@janhetjoch]

Love this format where we see multiple proofs of the same thing

[@kisaragiayami]

3:31 i love how the couple in the background is looking at the graph

[@MichaelBerthelsen]

That deadpun ending... The quality for which we come to this channel!😂❤

[@anitagofradump5195]

I think my friends and family would be quite worried if they found out how much i enjoy watching mathmeticians banter over square numbers

[@NikozBG]

I have to say, Matt's explanation is more pleasing. I like algebraic proofs and all, but visualising why something is true is more valuable knowledge imo.

[@Cream147player]

The algebraic proof says no less about “why” something is true than the geometric. The difference is you. You interpret the visuals better than you interpret algebra. That’s true for most, but not all, and the reality is that algebra is a far more concise way of demonstrating mathematical truths.

[@rewrose2838]

At the end, I was getting serious deja vu watching James drive Matt crazy (I was arguing with my cousin this morning about something similar) And yes , I agree with James , 0's a perfectly fine number and I'll accept that it somehow makes the odds feel less -odd- special

[@darkviking7135]

Good evening everyone and welcome back to... MAAAATH BAAATTLES

[@SiddharthGupta234]

Math beatles

[@nightmarius]

What about the difference of two Parker Squares though?

[@zeeshanmehmood4522]

It's round about an integer

[@NoNameAtAll2]

@@zeeshanmehmood4522 To be continued

[@liliwheeler2204]

4:44 I love the lady looking super confused right before Matt walked up (looking equally confused)

[@KyleJMitchell]

I would love to see more videos in this "Dueling Disciplines" format! It's entertaining and is also a great way to give a thorough explanation of a specific topic.

[@DaffyDaffyDaffy33322]

"And that everyone, is the difference of two squares" Perfect ending

[@MaximusXavier]

Who would win? A complex diagram and visual aid depicting your theorem Or one numbery boy

[@Brooke-rw8rc]

*boi

[@deathhog]

I actually yelled when I realized the last joke. Absolutely brilliant.

[@YensR]

5:44 great parcours training going on in background :D

[@questionlp]

YensR Parcours square and Parker Square 😂

[@vyuniverse9490]

Zero not a square number?! So what happens when you graph y=x^2? Is there a discontinuity at x=0? Loved the video!

[@minimike1995]

I like the argument, but (rt(3),3) lies on the y=x^2 graph, but 3 isn't a square number. I think the argument is that zero isn't seen as an integer, which is why he shudders when James says that zero is even

[@gabor6259]

@@minimike1995 Why 0 isn't seen as an integer? O.o

[@quadruplay9788]

@@gabor6259 Also zero isn't real nor complex

[@gabor6259]

@@quadruplay9788 How so? 0 is in every set. 0 is real, complex, quaternion, octonion, etc.

[@Brooke-rw8rc]

@@quadruplay9788 Zero is definitely real. It is part of the ordered set where every subset that has an upper bound has a least upper bound, ie. it's real.

[@johnchessant3012]

I'm amused at the people walking by in the background. If I saw a guy talking into a camera in a public square, and I heard he was talking about maths, I'd be VERY intrigued. ;)

[@emmafitzmaurice499]

“Let’s do it in a Gauss style” - words to live by

[@makessenseright]

The way this video is structured is phenomenal.

[@marionlara428]

Matt: makes a cool graphic James: G A U S S

[@atrumluminarium]

I took the easy way out: 4=((1+i)(1-i)/√2)²-((1-i)(1-i)/√2)²

[@atrumluminarium]

@@hetsmiecht1029 they were definitely talking about integers hence "easy way out" :p

[@christianbarnay2499]

Even easier: 4 = (√5)²-1² = (√3)²+1² If you don't restrict to integers, all numbers are obviously the difference and sum of 2 squares and there are infinite ways to do it. Choose any number a such that n+a² is a positive real number, and you can write n = (√(n+a²))²-a² Choose any number b such that n-b² is a positive real number, and you can write n = (√(n-b²))²+b²

[@JoelDowdell]

This is one of those videos that I wish I could like twice.

[@miscellaneous.7127]

I wasn't prepared for such a twist at the end!

[@munjee2]

The difference of two *GRIME* squares

[@haniyasu8236]

Every number is the difference between a Parker square and a Grime square

[@munjee2]

@@haniyasu8236 I remember , years ago they did a video where someone mentions that Steve , Matt and james have things named after them the Parker square, the mould affect and the grime dice a di is basically a cube so some need to link cubes and Parker squares

[@timgheys]

@@haniyasu8236 Except 4

[@haniyasu8236]

@@timgheys Hey, it may not work for 4, but at least it gave it a go and did some working out

[@Jaakington]

What is this? My two favourite mathematicians in one video

[@SomeNerdOutThere]

I played with this when I was a teenager. I tried finding a similar method of solving cubes that way. It turns out that cubes work the same way, but with an additional rate of change. Consider 0, 1, 8, 27, 64, 125. The differences between those are 1, 7, 19, 37, 61, which differ by 6, 12, 18, 24, which all differ by 6. This holds true for any other root; eventually, if you keep breaking down its rates of change, you'll get down to and end number that is the factorial of the exponent. It took me a few years (and a calculus course) to realise that I was looking at repeated derivatives of the function. Which demonstrates beautifully ~WHY~ the sums of odd numbers works for finding squares because if f(x) = x^2, then f'(x) = 2x. Similarly with higher powers, we end up using the factorial of the exponent because we're going through the rate of change of the rate of change of the rate of change... I was both pleased with myself in understanding that and disappointed that this cool pattern I'd found was elementary calculus.

[@illyon1092]

we need more of these colaborations. This was honestly one of the most fun maths videos I've ever watched.

[@JustMe-ui9bv]

You are great together. Looking forward to anothers. Love this format.

[@123amsterdan456]

this video is perfect! The way you two juggle the algebraic and the visualization around it makes the equations make complete sense! Thank you!

[@NonTwinBrothers]

Years later, still one of my top 5 favorite matt parker vids

[@elizabethh8579]

So many new videos lately featuring James Grime! What a treat!

[@johnchessant3012]

Matt, I agreed with you on 438,579,088 not being a Munchhausen number (because that requires 0^0 = 0). But here, we disagree. The square of every other integer (positive or negative) is included, so why not 0?

[@alzblb1417]

Every Parker is a Square of a Difference

[@drdonaldduck]

James AND Matt? This is glorious!!!

[@Veptis]

I love a geometric proof over. And KZbin does as well.

[@BedrockBlocker]

You: Brings argument _thats not how Gauss would have done it_

[@alwysrite]

I think Hannah Fry should sort out these two "squares" : )

[@marcospimentel6098]

What a colab! Thanks for the awesome videos!

[@mr3sepz]

I love how the videos are so perfectly planned through

[@kylecronin3212]

The difference between these two squares is a whole lot of hair

[@upsidedownwhale]

My 2 favourite mathematicians in a single video? Sign me up!

[@paulofortza3301]

I like the bit where you talk about the difference of two towers at 9:11

[@morgengabe1]

Between the persistence video, the cannonball video, and the elegance of these proofs, you've been on fire lately, Dr. Parker!

[@-fitzy-3335]

I guess zero is a Parker square...

[@Kaptenblu]

Ok fine, the ending was pretty sweet.

[@Mr_Mooo]

Two squares, standing at different squares, talking about squares. BRILLIANT!

[@baileycrowley1734]

Name a more iconic duo...I'll wait...

[@gauravdimri8009]

Received my Humble Pi today!

[@MrMattical123]

The greatest crossover since Infinity War

[@Starter61]

Nice to see you two together ! Double fun !

[@InDstructR]

More of these PLEASE!

[@GNiko324]

And they said Avengers was the most ambitious crossover

[@chillingpaully4137]

The difference of two squares, Presented by two different squares

[@paulchapman8023]

In different squares.

[@brachypelmasmith]

james and matt just chatting along the street looks so wholesome

[@ARVash]

This was one of my favorite videos so far!

[@Robi2009]

2 of my favorite mathematicians together? What holiday is it :O

[@mikerich32]

Anyone else notice that person trip and fall to the left at 5:46 ? hahah

[@EdwinStreet]

I love that you've joined your two passions, stand-up comedy and maths, and have been able to make a career out of it.

[@spackal2946]

I love what you guys do, you have inspired me by videos on your channel and numberphile appearances, please keep up the wonderful work and keep inspiring people to follow math :)

[@drakono82]

I accept it, James.

[@richardhee]

James is also wrong with saying that 0 (zero) is an even number, it's not odd eighter ... which makes it confusing

[@iteragami5078]

@@richardhee The formula for odd numbers is 2n+1 as they said. You can write 0 that way because "n" would have to be a fraction.

[@omri9325]

If 0 is a square number, anything is possible!

[@tobyzxcd]

It’s like two wizards constantly trying to outdo each other

[@frederikjuhlandreasen9477]

Awesome format, nice work on this vid. You could def do more in this format

[@ravencloud7]

I understood James better Sorry Matt!

[@thetntsheep4075]

The difference of two mathematicians

[@mattc3738]

This was a nice argument between 2 squares :)

[@Elitematt74]

I love the ending, feels like it wasn't even meant to be part of the video but probably my favourite part

[@nic5423]

I side with James on this. Unless you are saying that 0 is not an integer, clearly 0x0=0... Therefore 0 is a square number.

[@razielkeren6480]

0 is an integer but not a neutral number. but as I said if you kick 0 out you kill their claim that all odds can be written as a different between two squares since the only way to write 1 as a different between two squares is 1^2-0^2

[@pleaseenteraname4824]

0 is a natural number as long as you construct natural numbers to have 0 in them. You can construct them starting with 0 or starting with 1, doesn't make that much of a difference. Still, I'd argue you should put 0 so the operation + has the identity element and other reasons regarding definitions.

[@razielkeren6480]

@@pleaseenteraname4824 it's not a matter of when you start but when we say natural number we mean the numbers we count with. if you want, a positive integers. you then may argue let consider 0 as a positive number. but then a positive times a negative should be negative and 0 times a negative is 0 which we considered as positive, therefore we have a contradiction

[@pleaseenteraname4824]

raziel Keren No, that's not how it works. Naturals are not "positive integers", because they come before those. You first construct the naturals, the "standard" construction is that of Peano's set of axioms. You can put 0 and start with that, or just start at 1, it doesn't make that much of a difference, the numbers you end up with are the same. It's preferred to add 0 too because that way you have the identity element for the + operation, and many definitions/properties become somewhat easier (pass me the term). Only after you've constructed natural numbers you can pass on to integers, and now you can talk about positives and negatives. The only reason 0 is "neither" is because, in the definition we give, it just happens that the additive inverse (fancy way of saying opposite) of 0 is 0 itself, i.e. +0=-0.

[@victorselve8349]

How do you define a square number? If you define it as a/b=b then it is not. (I'm not a mathematician but from my perspective I think it should not be included into the list since 0 has the special property of returning itself when multiplied with any number while the defining feature of a square number is that it is the result of a number being multiplied with itself specifically)

[@SD-de4do]

So, when two squares differ, the results are odd.

[@ffggddss]

... if the squares are really close to each other, that is... Fred

[@charlessaintpe8574]

I enjoy good wordplay, but they proved in this very video that your statement is false.

[@ScottTilYouDrop]

love the tetris section!

[@fufu_212]

This is one of the best videos i have ever seen! Please do sth like this again

[@kcwidman]

This whole video was a set-up for the pun at the end. They both are rather square people, and they have a difference of opinion. Get it?

[@DerekHartley]

The pun is that one was in Trafalgar Square and the other was in Leicester Square.

[@kcwidman]

@@DerekHartley I'd argue it works both ways.

[@lunasophia9002]

Oooh, invoking Gauss, that's cheating! ... but I'll allow it.

[@dozenazer1811]

This collab is AMAZING!

[@server642]

Please do more videos in this style! And /please/ let this be foreshadowing a new channel where you and James collab/debate on different proofs!!