163 and Ramanujan Constant - Numberphile

Why does Alex Clark, from the University of Leicester, have a strange fascination with 163?
More links & stuff in full description below ↓↓↓
Some slightly more advanced stuff in this video, including the Ramanujan Constant and its use in a "famous" April Fool's joke.
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[@jamesblackburn8110]

"Who knows how he managed to determine this..." He was Ramanujan, that's how

[@ranjithkumarr9788]

I was studied my higher secondary in Ramanujan studied school in kumbakonam 😇I really proud of him

[@billoddy5637]

He was Ramen Noodles

[@indrajitmajumdar8590]

@@billoddy5637 hey, surprisingly he really sounds like that 😁😁😁😄😄😄

[@themandalorian7352]

@@billoddy5637 😂😂😂

[@manmohanmanjhi9733]

@@ranjithkumarr9788 really you are very lucky man

[@scottmuck]

I first encountered 163 when I moved on from 162.

[@claires9100]

You made me laugh. Truly. Thx!

[@wanalzheimer8341]

You should get more thumbs up

[@jeffreybonanno8982]

I actually first reached when counting down from ∞ and hadn't noticed its alleged significance. I was kinda tired though from being up literally counting forever. That's sounded funnier in my head than it looks on paper. Kind of like mathematical calculations and arithmetic operations.

[@somebody7407]

😂😂😂

[@truincanada]

That was very funny. Grounding. Thank you. Ha.

[@gonzalobriones796]

did somebody noticed that he is writing in a sheet of brown paper that is over a white board? ajajajajja i love this guys, they know how to keep the identity of their channel

[@akshaynair8498]

+Gonzalo Skalari It could be to avoid the glare off the white board.

[@tqnohe]

+Gonzalo Skalari he is left handed. Being left handed his writing on the white board would tend to be rubbed out. Not so much on the paper. It is true. I am a lefty. It is irritating.

[@jonathanpark4619]

+Gonzalo Skalari They write it on the brown papers so that they can donate it to charities that then auction off the papers to people.

[@bolerie]

+Jonathan Park They didn't do that at the time

[@wdyt2121]

+jackcarr45 it is not a case when you write in arabic dude

[@piyushkuril2127]

nothing is more mysterious than the brown paper.

[@talkgb]

Piyush Kuril THIS COMMENT HAS 163 LIKES LOLLOL

[@bell1095]

And its artfoolish fringes

[@IETCHX69]

Why cover a board specifically designed to write on , cover it with a paper , in order to write on it . I am digesting moths .

[@thebangladeshtribune]

Maybe the camera couldn't see the white Board or something?

[@pansepot1490]

Sells the scribbled brown paper on eBay. Can’t do that if the professors write on their board.

[@itsiwhatitsi]

Ramanujan was probably the most original and great mathematician

[@uuu12343]

Itsiwhatitsi That's true ..well apart from or on par with Euler, Euclid, Fibonacci, gauss

[@chetanchaudhari8231]

yes eternia

[@arshupadhyaya]

Einstein and Newton and gallelio and Archimedes are the best

[@SagarGohri-bj7hp]

Arsh Upadhyaya umm, einstein was not a mathematician.

[@AaronHollander314]

Ramanujan is great... but he's no Gauss ;)

[@innertubez]

Ramanujan and Gauss were absolute geniuses. Heegner wasn’t such a slouch either lol. But one of the most amazing parts of this story is that Gauss had the intuition to suspect the end of the list. How??

[@Gna-rn7zx]

Maybe he tried the rest of the primes up to a thousand!

[@stuboyd1194]

It's 99 years today (26 April 2019) since he died.

[@kenmolinaro]

He didn't look that old in the video.

[@incognitonotsure909]

@@kenmolinaro he was 32 when he died.

[@kenmolinaro]

@@deepaksinghpatwal5755 You need to learn the meaning of "sarcastic humor".

[@ShailendraSingh-pk1gf]

100 years today

[@bensin2076]

100 years today, 26-04-2020

[@dcterr1]

For those interested, the fact that e^(pi sqrt(163)) is so close to a whole number has to do with properties of the modular function J(tau) as well as the fact that Z[sqrt(-163)] is a unique factorization domain.

[@deepak2049]

Now that makes the whole essence of video crystal clear to me.................btw i dont know maths

[@christopherstoney4154]

I'm not sure how the calculation works, but my intuition tells me that the absolute value of (e^(pi sqrt(163)))+i is likely an integer.

[@dcterr1]

@@christopherstoney4154 I don't think you're right about this. The value of Ramanujan's constant is given by a very rapidly converging series, the first two terms of which happen to be integers.

[@rogerperkins]

e to the sqrt -1 x pi even closer to a whole number.

[@hylens5111]

I knew that.

[@cradoll90]

I love that this video starts with explaining how to write a number as a product of a prime, and quickly escalates to the invention of new number systems using unreal numbers.

[@fredyfredo2724]

And demonstrate this new number system is false. This will never work with sine.

[@dielegende9141]

@@fredyfredo2724 nothing in mathematics is "wrong" as long as it's logically consistent

[@fredyfredo2724]

@@dielegende9141 undefine is not demonstrate false or wrong and is not true

[@dielegende9141]

@@fredyfredo2724 I have no clue what you're trying to say

[@ingenuity23]

@@fredyfredo2724 are you aware of the polar form for any complex number a+bi? if so you must know it is r(cosθ+i sinθ). I fail to understand why complex numbers wouldn't work with sine, let alone other trigonometric functions

[@crowdozer3592]

watching left handed writing is like watching a wizard at work 😓

[@ieradossantos]

Ramanujan was the most talented mathematician to grace the world. He didn't 'proof' what he already knew until they learned him how to. He knew things on his own that the collective mind of math's history took centuries to learn.

[@jasonpalmer1552]

The camera man for this channel loves zooming in to faces as awkwardly as possible

[@bell1095]

Jason Palmer he is an amateur, non professional, he must even love the subject of that clip on amateur mathematics

[@shyambuddh5546]

The camera man for this channel is the dude that runs this channel

[@markspc1]

Obviously this cameramen never review his work; the worse cinematographers of the millennium !

[@ABC-xj8cs]

Jason Palmer hahahahahaha heheheeeee!

[@robertjennings7282]

It's obvious you bitches have never had to to film in a cramped space.

[@bengski68]

Hey look, a white-board! We can use it to - Numberphile: let's stick some brown paper on it!

[@anupambanerjee8336]

They didn't use the white board because it would reflect light making it hard to see.

[@castironlawnbunny]

White boards have glare that shows up strongly on camera and makes writing hard to read. The brown paper is very easy to read on camera.

[@jacderida]

This is one of the most underrated videos on Numberphile. Absolutely fascinating!

[@IETCHX69]

Not to a 56 year old man with a 5 year old's math skills . No offence to 5 year old's !

[@galefray]

Guys, go on Gauss' Wikipedia page, and look at his signature, I swear I can see Pi. XD

[@aliebadi5341]

+galefray And there is the integral sign just before the end :)

[@hellNo116]

+galefray you can also see an e and a butterfly in there :P

[@kevinstefanov2841]

And the integral sign as the first "s"!

[@flikkie72]

I thought mathematicians always had bad handwriting though, this signature is stunningly beautiful

[@anamikamishra4052]

and I totally feel like it's on purpose

[@shawnwilliams77]

I must say, as a mathematics major, these videos really keep up my joy for maths. I really enjoy seeing videos on number theory topics and what not. Fascinating, and encourages me to become the best mathematician I can be! Thank you!

[@JacobGoodman]

Fun fact: (x^2-y^2)^2 + (2xy)^2 = (x^2+y^2)^2 For all x and y. This is bascially just a Pythagorean Triple machine

[@tonaxysam]

@@ludo-ge9fb or by using complex numbers: a + bi Is a number whose distance from the origin is the square root of an integer, so if you square it, it's distance from the origin wil get square and thus, you'll get a complex number whose distance from the origin is an integer. (a + bi)² = (a² - b²) + (2ab)i So that number is at a whole number distance from the origin

[@cyberiandeprochan7998]

What's impressive about this is that it was solved by an amateur mathematician who is as brilliant as all the professional mathematicians combined in number theories

[@thehaqq3540]

“Someone who wasn’t officially a mathematician” - lol, okay…

[@akhileshkhot8326]

Now "163" is also my favourite number.

[@numberphile]

@grande1899 fair enough... When it comes to the more advanced stuff, it seems we're damned if do and damned if we don't... I hope you like the next one more and appreciate anyone who takes the time to comment constructively.

[@linus6718]

Hi Numberphile, I love you

[@leif1075]

Wait MISTAKE ALERT.He says square root of -7 gives unique factorization but that's wrong..yiu can write 8 as either 2 times 2 times 2 or as (1-sqr root -7)(1 + sqr root -7) also gives 8! Same reason why sqr root-5 was discarded..sonwhy not discard 7 and 11 and several others for that mater..Didn't anyine else notice this is a mistake??

[@nicolasbanks7871]

@@leif1075 It is well-known that -7 yields unique factorization, so my guess is that 2*2*2 and the other factorization you mentioned are what we call "associates". This means that one is a unit multiple of the other, where a "unit" is any element of Z[sqrt(-7)] that has a multiplicative inverse.

[@Tuberex]

didnt know grandayy watched numberphile

[@d4slaimless]

@@leif1075 wiki page explains about sqrt(-5): "These truly are different factorizations, because the only units in this ring are 1 and −1; thus, none of 2, 3, 1 + sqrt(− 5), 1- sqrt(-5), are associate". I wonder though what are the units for Z[sqrt(-7)]

[@abinashmishra1134]

Ramanujan, the mystery yet unsolved.

[@Hythloday71]

He looks like the mathematician out of 'Good Will Hunting', who takes Will under the wing.

[@Entropy3ko]

Haha look at that face in the end... it WAS his PIN heheh

[@ceelar]

+Entropy3ko Bosco!

[@Entropy3ko]

Dat Seinfeld ref! hehe

[@TwelfthRoot2]

You’d expect a mathematician to be the toughest to break into their suitcase/bank account/etc but it turns out they are the easiest because they use their favorite constant lol

[@bethysboutique]

Rooted negative numbers make me uncomfortable.

[@fayguled900]

What should they do? Just use the word "i" behind the number?

[@bharatkothari2998]

you must be feeling complex!😉

[@lagduck2209]

it's just another notation for. (also all numbers are imaginary in some sense)

[@JannikPitt]

In some sense root(-5) isn't really correct. When you take root(a*b) then this is the same as root(a)*root(b). But for -1 root(-1)*root(-1) is equal to i^2=-1, but root(-1*-1) is equal to root(1)=1. Also root(1) does have two solutions, 1 and -1 and we define the root to always give back the positive result (so x^2 does have a bijective inverse function). For root(-1) there are two solutions as well, i and -i, but these are in some sense undistinguishable because there is no notion of comparison in the complex numbers. You can't say i is bigger than -i or vice versa. So it's better to write i*root(5) because that is completely unambiguous and you don't run into problems because it's difficult to define root(z) for a complex number z.

[@Sporkabyte]

Why? Do irrational numbers make you feel uncomfortable?

[@TheGuardian163]

That's MY number.

[@uuu12343]

TheGuardian163 Prove it

[@Penguin_of_Death]

That's NumberWang!

[@higherbeingX]

Brown paper reduces the light reflection and hence comfortable for the eyes

[@baileyduryea3168]

I always love these videos where a seemingly ordinary number is shown to be far more interesting than the average person would expect

[@tommythai2660]

+Sangeet Khatri Small correction, 5i or 5 times iota is not the root of -5 it is the root of -(5^2) or - 25

[@Hythloday71]

Still my favourite number / numberphile video ! A great example of the delightful surprises that emerge from understanding the most generalised of principles underpinning number 'systems' / Rings / Fields / Groups etc.

[@higherbeingX]

There are 2 classes of mathematicians..Ordinary mathematicians and Ramanujan

[@NoriMori1992]

Watching people write left-handed always makes me a bit squeamish, because I naturally imagine myself doing the same, and since I'm right-handed it feels really wrong. XD

[@NoriMori1992]

***** …Excuse me?

[@ishwar8119]

The opposite for me, I'm left handed and when I see people writing with their right hand I'm like: "magic!" XD LOL

[@arvindhmani06]

We lefties feel that you're the weirdos xD

[@theultimatereductionist7592]

I feel the same way, NoriMori.

[@tyn6211]

How sinister...

[@paulfaigl8329]

absolutely brilliant. Thank you Alex.

[@Atrix256]

I've been watching these videos from newest to oldest and this video is my favorite so far. Great vid!!!

[@marlenesclark]

thank you

[@shaantubes]

gauss a genius. ramanujan an another genius.

[@vinaykumarsharma8565]

Shaantubes an another???? universe just imploded.

[@noblerkin]

No shot.

[@saharshbehal8766]

@@vinaykumarsharma8565 😭😭😂🤣

[@eashchawla8330]

Gauss just prove it was given by ramanujan

[@johnlandis2552]

a quibble: his name is" rama- nujan " not "ramunajan"

[@roberteospeedwagon3708]

I was thinking that too

[@vinayakbiju]

john landis yep..It should be pronounced just as it is written like Rama.nujan...no extra flavours..I'm an INDIAN.

[@rosiefay7283]

Another quibble. The number has nothing to do with Ramanujan. Hermite knew that exp(π√163) is very near an integer. Ramanujan's papers don't mention it.

[@zTheBigFishz]

...and Zed instead of Zee. Clearly incorrect.

[@rosiefay7283]

Oh, stuff and nonsense. He is clearly American -- his American accent is evident every few words -- prahblem, sahlved, liddle, wanna ride it, idennafy, right triangle (instead of right-angled triangle), exhahstive, noo, prahgress etc. etc., and that's just the first couple of minutes, before we get to the Z.

[@Tolstoievsky]

love these in-depth ones so much more than the "happy number" type ones. MORE!!!

[@anloef]

This is the kind of videos I would like to see more of from this channel!

[@Symbioticism]

I really enjoyed this video - this feels like the kind of stuff I always wanted them to cover in school!

[@avatacron60]

At last a normal person on Numberphile.

[@DanDart]

mathematicians would like to encourage everyone to do maths

[@truebeliever174]

How did Ramanujan calculate this? He was really great... Love for Ramanujan from Bangladesh 🇧🇩

[@flashpeter625]

Ramanujan himself often didn't understand how exactly he was coming up with his results. And even when he did, often he did not keep the explanation/proof, just the result. He was likely the most talented mathematician ever, but lacked formal faculties and rigor. He started working on those gaps, but died too soon.

[@empathycompassion6157]

@@flashpeter625even proof is not needed,since on higher plane everything look as formulae.Pls dont speculate,easier for you when you are not even him.

[@benterrell9139]

Another fantastic number. Great vid!

[@Weiss.Schnee]

I love dabbling into the complex plane on these videos, keep it up!

[@tstanmoysamanta]

Great Ramanujan......

[@sananguliyev4940]

They mentioned several mathematicians, but you only noticed Ramanujian just because he happened to be Indian?

[@tstanmoysamanta]

+Sanan Guliyev so what...search about him you will understand...and you have problem with indians?

[@tstanmoysamanta]

so what problem you have with country tell me ofcourse i also here for math..

[@sananguliyev4940]

+Tanmoy Samanta whatever man try not to be racist/nationalist and appreciate scientists regardless of nationality/ethnicity

[@tstanmoysamanta]

+Sanan Guliyev I'm not.....

[@albertoceleghin1988]

I have always hated math...since i was kid i never understood it....maybe cause my first teacher used to beat us up if we were wrong...who knows. But it is my biggest regret. I truly wish i could understand it. I love it and i found it fascinating. Great videos even if i got lost once he started talking bout factoring numbers 😅

[@kavankachoria1699]

Ramanujan was beyond any other mathematician....the sheer intuition and imagination was something alien.

[@AnilKumar-xl2te]

Ramanujan never dies. Ramanujan lives for infinity.

[@annoythefish]

"officially a mathematician" They don't make 'em any more pretentious than that

[@L0j1k]

Even Ramanujan called himself a clerk and not a mathematician. It is a job title, after all (cf. engineer).

[@numberphile]

@ParagonProtege Good to hear from people who enjoy being out of their comfort zone (welcome to my word making these videos!!!)

[@tonydetroit1431]

Cool stuff presented very well.

[@alta367]

10:41 is my favorite moment. I have to agree, I don't think most ordinary people would expect that e^d*pi where d forms a number system with unique factorization, would be very close to, but not quite, a while number.

[@alta367]

Perhaps a poll is in order just to be sure

[@lagduck2209]

He says "right triangles" but his triangles is actually left.

[@noralyounes214]

He is left handed

[@CaseyShontz]

Илья Лагуткин lol tru

[@shashankethane]

He is talking about Right angled triangle

[@hvishwakarma8371]

Chutiya

[@bell1095]

... and he did mark the 90 degr corner.

[@vaishnavraj6930]

Ramanujan was a mathematical wizard♾️

[@venkatbabu1722]

A eight digit binary sequence with inverse power has a critical wave edge trigger. 101 000 11 next 1. 3×4 is the smallest leap of a right angle for surface symmetry.

[@sport8133]

I think the interlocutor guessed his ATM card code at the end.

[@SMOshee]

I didn't understand this video...

[@victorkkariuki]

Saeed Oshee 😮😐😕

[@anarcho.pacifist]

For more info, check out the OEIS sequence: A003173.

[@spaceexplorer5481]

Watch again

[@bensin2076]

Not a problem , you are still fit to survive on this planet

[@leif1075]

There's mistakes in it sqr root of negative 7 does NOT give you unique factorization because 8 equals (1- sqr root- 7)(1 plus sqr root -7) as well as 2 times 2 times2. So it should be discarded like sqr root of -5....samecscenario.did no one else notice this mistake??

[@jccusell]

So when are you "officially" a mathematician?

[@andersbendsen5931]

I suspect you'd need a degree? Just a guess.

[@jackcarpenters3759]

If ramanunjan would have lived longer, he would have solved math.

[@wdfomfg]

I love this channel :) I'd sure love if you guys and gals could post daily videos though. Keep up the good work!

[@ChristopherHallWayne]

I had not come across this before and for the briefest of moments I was extremely happy to think that Ramanujan's Constant was an integer. Alas, those thoughts were shattered.

[@numberphile]

@davidandkaze no I was with you, in fact I think you missed the subtlety of my jokey retort... that I have in fact do have a PIN number... a PINN if you will... a number to protect my number! But I think the moment has passed!

[@thepurplelefant]

this is awesome i like it when they add some more complex topics.

[@davidspencer3726]

Finally found it! The NP video that isn't sponsored by someone!

[@trulyinfamous]

So I guess 163 is special for something other than it's digits adding up to ten?

[@rohitkumar03]

Truly Infamous ml

[@jyotishkaraychoudhury4762]

So.... which specific number is the Ramanujan constant ?

[@jyotishka]

That,s exactly what I was thinking.

[@cryme5]

e^sqrt(163) pi? although he didn't predict it, I think they just call it after his two other numbers

[@joeyhardin5903]

1729

[@devekhande9204]

Binod.

[@MrJronson]

Actually, the Babylonian's used a base 60 system (which is where our time system comes from) because on one hand they would point out only one finger and this would point towards one of their knuckles of the four fingers on the other hand. Each finger has three 'knuckles' if you take a look, hence there are 12 combinations on the one hand, multiplied by the 5 fingers and thumbs of the other hand, to get 60 combinations in total.

[@YesterdaysObsession]

This is probably the best one yet.

[@bassionbean]

Wait isn't Euler's theorem like the new Ram. constant? e^ipi = -1 (whole number)

[@TheMsksk]

bassionbean -1 is not a whole number

[@Luisitococinero]

+bassionbean It is an integer (whole number).

[@arvindhmani06]

I thought this too! Fascinating.

[@non-inertialobserver946]

No, because ramanujan's constant only has real numbers, euler's formula has imaginary exponent

[@bell1095]

... they refer to different rings

[@zachadkins8010]

Is there any significance to those last, almost whole, numbers being similar form to eulers equation

[@joeyhardin5903]

By raising e^( sqrt(-43)pi ) or whatever number you choose from that list, you are walking halfway round a unit circle sqrt(43) times, because the original expression can be rewritten as e^( sqrt(43)*pi*i ) which will give you an point on the unit circle where the y value (sine) is close to 1. Because the x value (cosine) is very irrational, it may be linked to the thing with unique factorisation. When using the formula at the end of the video, e^( sqrt(43)pi ) (notice the number inside the root is now positive) we are essentially taking an i out of the expression and hence moving the number onto the real axis. because the y value was close to a whole number (defined by the sine of sqrt(-43)pi) it rotates to the x axis where the real component is now close to a whole number. This comment is not necessarily the right answer to your question, but it is a guess as to some of the maths involved in the actual proof.

[@steamerSama]

one of the most cliffhanging numberphile videos ever

[@1495978707]

Good for you! Keep it up! You shan't regret it!

[@TheSwamynathan]

Now a Tamil Movie has come in his honour titled 'Ramanujan' -A Budget movie of course.

[@numberphile]

@Mrtheunnameable I refer you to my other reply... this joke has gone down like a lead balloon in pedant's corner!!!!

[@thelasttimeitookashowerwas7069]

how do they even come up with these theories and determine the final effing number? this is quite freaking impressive

[@guyboy625]

Note that e^(sqrt(163)*tau) is also really close to a whole number.

[@s4nsk_]

When you are right handed and see someone writing with left hand

[@mcdiamond2012]

There are 163 days until christmas

[@0SomwhatDamaged1]

I have to say, this is the one numberphile video that i just don't get. But still, this channel keeps you thinking ;) Keep up the good work!

[@infinummjb]

Thank you :)

[@eadanlin]

I dont get why z[sqrt(-7)] works. for example, 8 = 2*2*2 = (1+sqrt(-7))(1-sqrt(-7)). Am I missing something

[@SanjeevKumar-js4mu]

because you don't know what a plus b whole square means you're a duffer

[@erayk96]

Is (1+sqrt(-5)) a prime in Z[sqrt(-5)]? Because in the video he says it is.

[@shijiadai2766]

Danny I Tan Lin

[@alnitaka]

The "square magnitude" (norm?) of 1+sqrt(-5) in Z[sqrt(-6)] is 6, which is not prime.

[@KaizokuKevin]

Danny I Tan Lin just multiply

[@anglo2255]

I understand these are factors, but these complex numbers, (at least the imaginary part) are not whole numbers, so I don't understand how you can call them primes. any thoughts?

[@scowell]

It's taking the concept of complex numbers (adding root(-1)) and expanding it... you create separate number systems. The normal complex number system works (in generating unique factorizations for all numbers in the system), the one using root(-2) works, root(-3) works etc... the example root(-5) didn't work... up to root(-163), where you are at an end. *I* want to see the Mandelbrot-like set for the complex-like plane with root(-163)!

[@ghdevil666]

The more general definition of prime (also called irreducible) is that if a number p is factorized as p = a*b then either a or b is 1 or -1 (in this case). This is equivalent (also, in this case) with the definition of prime you are probably thinking of, only divisible by 1 or itself. - Suppose p is only divisible by 1 and itself, then p = 1*p is the only factorization, therefore p is also prime according to the more general definition. - Suppose p only allows trivial factorizations i.e. p = 1*p or p = -1*-p, then p is only divisible by 1 or itself, because if it was divisible by something else, there would be a non trivial factorization. Therefore the two definitions are equivalent. You can prove 1 + sqrt(-5) and 1-sqrt(-5) are prime in several ways. Hope this helped!

[@anglo2255]

so, instead of 1 and itself (or P), (1+sqrt(-5) and itself (or P)?

[@ghdevil666]

anglo2255 So 1+sqrt(-5) is divisible by 1, -1, itself and -1-sqrt(-5). In the case of regular primes we could limit ourselves to the positive numbers, but since there is no such thing as a positive complex number z (as long Im(z) =/=0), you have to include "minus"-itself and -1 as well

[@cryme5]

I think it needs some clearing. Z is a ring for it has two operations with a particular structure + and x (times), you should definitely read Wikipedia on what is asked to be a ring. You can do what is called extension of ring, that is a ring that contains Z and uses the same operations. That is the meaning of Z[i]: the smallest ring containing Z and i, using + and x. To define a prime in Z you need to talk about units. Units are the numbers of your ring that end up going to 1 after being multiplied by itself a finite number of time. If I take Z, 1 is already 1, -1 x - 1=1 that's another, and that's it. A prime is then a number p for which any writing p=a x b, implies that a or b is a unit. For Z, it just means that you can only write p = 1.p = - 1.-p, but for Z[i] it's another story since the units are 1,i,-1,-i. In Z[i], a prime can only be written 1.p = i.-ip =-1.-p = -i.ip. Now if we talk about Z[2i], you notice that the units are only 1 and -1, so the definition of prime is essentially the same as in Z except a and b are in Z[2i]. That means, primes before may not be primes anymore. (1+2i)(1-2i)=5, 5 isn't a prime anymore in Z[2i], and in Z[i] either actually. Now the big deal is to check if your ring allows you to do prime decomposition with unicity by the order (and disregarding units, p and -p are said to be the same factor...). What the video tells, and actually what the Stark-Heegner theorem states is that only for the numbers n=1,2,3,7,...,163, Z[ni] allows a unique factorisation. Hope it helps, you might want to check euclidian division, euclidian domain, principal integral domain, etc, on wikipedia it's already nice to start with.

[@kennethflorek8532]

The connection between those numbers being close to whole numbers and the class number being 1 is as eerie as I have ever heard.

[@bredmond812]

So Al Gore has left Global Warming and moved into Math...

[@toniokettner4821]

sqrt(-1) = i -i: am I a joke to you?

[@JoseyWales93]

Fascinating!!

[@rjbond007]

Calculating 3 irrational numbers in power without calculator..... Me : left the math

[@hobinyetir7072]

I feel watching this upside down because he is left handed >_>

[@numberphile]

@TheCokeSideOf this was indeed the most advanced one we've done... but I understood it enough to upload it so I don't totally agree with you. I hope you like the next one more... Numberphile will always be a bit more mixed because it is venturing more into the language of mathematics, which I try to eliminate from the other channels (for which I also get abused!!!!) I can't really ban it from Numberphile - at least not every time!!! Cheers for the feedback!

[@dushyanthabandarapalipana5492]

Thank you!

[@logitechpanasonic3]

i wouldn't mind if this guy could join another episode of your interesting show, very interesting. good work

[@harshitkumar4760]

I noticed that most of the poeple know who was Ramanujan except many Indians, his own people and they say that there is no great scientist or mathematician here. If you yourself will not appreciate them then how can you expect from the world? Sad but true that there were many but they just died, struggling to print their research and nobody cared about them.

[@stewiegriffin6503]

sqrt(163+6)= 13 13+4= 17.... pretty cool ?

[@colw321gaming2]

no

[@nandaveerum4399]

Makes sense for you? Thats great!

[@javierantoniosilva8477]

I miss this guy.

[@acesulfameazzakari4616]

Right now, youtube says numberphile has 3.14 million subscribers. And it's Christmas day. Coincidence? I think not.

[@mmancini05]

Why does Alex Clark sound like Ben Carson?

[@Qermaq]

Amazing that my iPhone calculator cannot calculate e^(SQRT(163)*pi)

[@inna9882]

My android can (;

[@GregaMeglic]

Mine gives me a really really big number 6725525588.089824502242480889791268597377 Probably goes beyond that XD Oh and also android and not iphone.

[@ZoeTheCat]

Then you entered something wrong. e*(sqrt(163)pi)= 262,537,412,640,768,743 . 999 999 999 999 25 (On my Windows calculator)

[@GregaMeglic]

***** Indeed. Seems like i didnt put something in correctly. Your result is the correct one.

[@mwtrolle]

Get's 2.62537412641E+17 on my Iphone

[@veramentegina]

so fascinating!!

[@mikewolf78]

This is the first numberphile video that I have no idea what's going on in.

[@Supermario0727]

Solved by an "amateur" mathematician? What does that even mean? What makes him an "amateur"? The fact that he didn't have a degree from Oxford? Who came up with that nonsense? You think because you went to university and blew $25 000, that suddenly your a "professional" mathematician"? Mathematics has no degree or level of education. It is a subject that is common to every thinker.

[@anishkumthekar4708]

Finlander Ramanujan proved theorems that are applicable in quantum physics and are in use right now, after approximately 100 years of his proofs. Clearly more respect for the man was needed instead of tossing "amateur" out there. Makes it sound like he stumbled upon the theory rather than rigorously and tirelessly worked on it that confounded not only the mathematicians of that era but also the current ones.

[@misteralex1358]

This is a video on mathematics, which is a subject based on rigorously defining a system of axioms and proving things using those simple axioms. Do you have a way of rigorously defining the term "amateur" that isn't based on someone not doing an activity as their profesion(ie someone doing something when not being payed to do so)?

[@uuu12343]

Holy wow, chill guys It's a technical term, can't help it that it's a term used for many years and it just so happened that ramanujan fit into this category He is a great mathematician, but he didn't have a degree in math so "technically" under math terminologies, he is a amateur mathematician, that's it Ffs guys in the world...

[@Robin-bk2lm]

John Stuart Just lingo. he also called one guy a recreational mathematician.

[@manjunathahn1691]

Hats off John!