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
+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.
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 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
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.
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!
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.
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
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.
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.
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
@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!
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.
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.
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 😅
It's interesting...if you take the list of these 9 numbers and line them up in order and subtract the lowest from the second lowest, the 2nd lowest from the 3rd lowest, etc. like you would if you were trying to find the degree of a function, you end up at 164, which is the lowest number (1) added to the highest number (163). Just thought that was interesting.
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.
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.
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)?
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...
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.
In a straight line y=mx+c, the gradient is m. In a curve the like y=x^2, the gradient has to be worked out differently (it changes as the curve gets steeper). To find the slope you 'differentiate' (you'll learn this later) to find the gradient. The number e is defined to be such that the curve y=e^x differentiates to e^x. Basically the the gradient at any point is equal to the y co-ordinate at any point. 2.718281828 =e (roughly, it's irrational).
I wish there were videos like this that assumed that the viewer had a basic understanding of math at least up to Calculus. I don't want another Khan Academy (which is fantastic), because math is such an enormous field that you can't know more than a tiny fraction of it and I'm sure it would take a while to fully explain the relations here. I don't need to know everything about how the car works, I just want a peek under the hood.
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.
Yeah, but they are. The point, though, is the definition of prime as a number that cannot be factored. What's important is to see that 1 + sqrt(-5), etc., are prime. When you multiply sqrt(-5) by itself any number of times, you always end up back either at 5, -5, sqrt(-5) or -sqrt(-5). You're going around in a circle. Basically, what happens in Z[i] is that there are more ways "around the circle."
but in the root minus five system, 2 and 3 might not be primes and unique factorization might still stand. And I don't have much idea about this clip, just noticed
We do use higher base systems and we do frequently. Oftentimes, when confronted with a 32-bit number, it is easier to express it using 4 hex digits. Therefore [1] * 32 = ffffffff in hex, which is easier than writing 32 ones. In computers, hex numbers are used to represent operations, memory-addresses, bit-fields, etc. Hex is so popular because of how easy it is to go from base 2 to base 16 since both are powers of 2, so 1111 = f, 1010 = a etc. so we can represent alot w/ hex.
Ramanujan number: 1,729 Earth's equatorial radius: 6,378 km. Golden number: 1.61803... • (1,729 x 6,378 x (10^-3)) ^1.61803 x (10^-3) = 3,474.18 Moon's diameter: 3,474 km. Ramanujan number: 1,729 Speed of light: 299,792,458 m/s Earth's Equatorial Diameter: 12,756 km. Earth's Equatorial Radius: 6,378 km. • (1,729 x 299,792,458) / 12,756 / 6,378) = 6,371 Earth's average radius: 6,371 km. The Cubit The cubit = Pi - phi^2 = 0.5236 Lunar distance: 384,400 km. (0.5236 x (10^6) - 384,400) x 10 = 1,392,000 Sun´s diameter: 1,392,000 km. Higgs Boson: 125.35 (GeV) Phi: 1.61803... (125.35 x (10^-1) - 1.61803) x (10^3) = 10,916.97 Circumference of the Moon: 10,916 km. Golden number: 1.618 Golden Angle: 137.5 Earth's equatorial radius: 6,378 Universal Gravitation G = 6.67 x 10^-11 N.m^2/kg^2. (((1.618 ^137.5) / 6,378) / 6.67) x (10^-20) = 12,756.62 Earth’s equatorial diameter: 12,756 km. The Euler Number is approximately: 2.71828... Newton’s law of gravitation: G = 6.67 x 10^-11 N.m^2/kg^2. Golden number: 1.618ɸ (2.71828 ^ 6.67) x 1.618 x 10 = 12,756.23 Earth’s equatorial diameter: 12,756 km. Planck’s constant: 6.63 × 10-34 m2 kg. Circumference of the Moon: 10,916. Gold equation: 1,618 ɸ (((6.63 ^ (10,916 x 10^-4 )) x 1.618 x (10^3)= 12,756.82 Earth’s equatorial diameter: 12,756 km. Planck's temperature: 1.41679 x 10^32 Kelvin. Newton’s law of gravitation: G = 6.67 x 10^-11 N.m^2/kg^2. Speed of Sound: 340.29 m/s (1.41679 ^ 6.67) x 340.29 - 1 = 3,474.81 Moon's diameter:: 3,474 km. Cosmic microwave background radiation 2.725 kelvins ,160.4 GHz, Pi: 3.14 Earth's polar radius: 6,357 km. ((2,725 x 160.4) / 3.14 x (10^4) - (6,357 x 10^-3) = 1,392,000 The diameter of the Sun: 1,392,000 km. Orion: The Connection between Heaven and Earth eBook Kindle
it only seems easy because we have 10 fingers. If we had 16 fingers we could equally say "Oh,base 16 is logical because to multiply 16 by 16 all we have to do is add a 0 and move the one to the left"
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?
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)!
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 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
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.
I think it's because: a) brown paper is numberphile's thing. if i'm watching a video of someone writing on brown paper, i immediately correlate it to these guys. b) that paper isn't a shiny surface so the light doesn't reflect and makes it impossible to see part of what he's writing down, which would happen with the whiteboard from this angle.
@IamGumbyy Whiteboards tend to not show up well in generically lighted rooms. A problem compounded by non-professional cameras. Notice how much glare is cast on the whiteboard from the lighting on the exposed area of the whiteboard.
Well what I was thinking was angling the camera down slightly because all light sources are from above. I would be surprised if there was any glare then, but I could be wrong.
It might be that, if 'e' and 'Pi' is taken to be more accurate, then if the x.9999... could close more in towards the integer. Then, considering limiting value (as we consider more digits after decimal for 'e' and 'Pi') it might be true, i.e. it really could be an integer.
If Ramanujan said that it is an integer then it is. End of story. We will never know how his mind was wired, certainly not like us the mortals. His infinite series to evaluate pi for example is still a wonder to this day.
We use base 2 on the basis that a it's a lot easier to have a computer read either a generalized "high" voltage versus a "low" voltage than trying to establish discrete increments of voltage to represent data. If we were to adopt something like that, I'd rather we use base 16, then we could represent larger numbers with fewer digits. >_>
PLEASE STOP SAYING UNDEFINED MEANINGLESS THINGS LIKE "ordinary numbers" or "normal numbers". If you MEAN "the integers" or "the positive integers", then SAY "THE INTEGERS" or "THE POSITIVE INTEGERS". Everytime anyone is sloppy expressing these concepts it absolutely DOES raise unnecessary confusion, as saying the proper concept requires LESS work than being sloppy.
My school just added a course on number theory, I think I'll be taking it. Pure mathematics is as fascinating to me as physics, and in the long run I believe the deeper connections in physics will be made through abstract mathematical concepts.
A little late, but it's wrong to write it like that. What he wants you to notice is that the square of those numbers is negative. What he should have written to be formally correct is i√2, i√5, i√7... And so on
You can't. Some fucksticks just said "you know what, we're gonna call i the number that gives -1 if you square it. i²=-1. The fascinating thing about it is that it opens a whoooole new world of numbers. You have naturals, integers, fractions, real numbers. They are are sitting on a line that includes all real numbers. So its in 1D you might say. But by introducing i, you expand the numbers in 2D. A number then becomes a coordinate on a 2-axis field (instead of the 1-axis if only real numbers) in the form of a+bi, with a being the real part of the number and the bi the part being the imaginary part. A completely real number lies on the horizontal axis and only has a real part, with b=0 so a+bi becomes a, the multiplications of i lie on the vertical axis and only have an imaginary part, with a=0 so a+bi becomes bi.
I can remember the edition of Scientific American in which Martin Gardner's joke appeared. It was the April edition, and there were a number of other "April Fools" in the same column. I think one of them was a fake drawing showing that Leonardo da Vinci had invented the flush toilet.
_Heegner_ was the amateur mathematician in this case. He proved the Gauss' conjecture. Ramanujan just found that the two Pi over e numbers were almost whole.
I and another talented 14 years old actually figured out a lot of this on our own in an afternoon at a maths camp. Though we had the hint that "complex primes are interesting", I am quite proud of that. Hehe, just some pointless bragging.
numberphile never uses other than brown paper. I assume it is because brown paper does not reflect light and allows camera and viewers to see what is actually written.
I too,have something to say about the number 163. But for reasons you'll know some time into the future,I'll just give you some maths. Light, is 1. A day, is 24. We have 4 base simulations. 6=1×2×3 Logical analysis of cell division will give us 3 new cells,since there was 0,then came 1,which then squared and gave 1. But that isn't supposed to happen. After all,after 24 hours,you do end up with light(1),don't you?
That is true when you only look at real numbers. The square root of a negative number is called an imaginary number. Imaginary numbers don't make much sense in counting or measuring things (eg you can't have sqrt(-5) apples or be sqrt(-8) feet tall) but they are very helpful in a lot of physics and mathematics problems. For example imaginary numbers can be used to represent a sine wave; sin(x) = (e^ix-e^-ix)/2i where x is the angle, e is Euler's number and i=sqrt(-1)
You are absolutly right here, this is a valid decomposition into seperate sets of primes.The professor skipped (maybe on purpose) the important point that when working with quadratic fields Q[sqrt(D)] you actually look at Z[w(D)] with w(D) = sqrt(D) if D = 2 or 3 (mod 4) and w(D) = (1+sqrt(D))/2 if D = 1 (mod 4). So you actually work with Z[1/2 + sqrt(-7)/2] in which the number 8 decomposed uniquely into 8 = (1/2 +sqrt(-7)/2)³ * (1/2 - sqrt(-7)/2)³
It is thought to be the ten fingers as there have been cultures that had eleven as the base thanks to a male appendage. Twelve was used for currency in Britain as 12 is very divisible (1,2,3,4,6,12 rather than 1,2,5,10)
This seems eerily similar to the Fermat primes. I wonder if there is a relationship between the Heegner numbers { 1, 2, 3, 7, 11, 19, 43, 67, 163 } and the Fermat primes { 3, 5, 17, 257, 65537 }. Notice how both sets have a large jump between that last two elements. Of course, it isn't proven if 65537 is the last Fermat prime.
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
+Gonzalo Skalari It could be to avoid the glare off the white board.
+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.
+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.
+Jonathan Park They didn't do that at the time
+jackcarr45 it is not a case when you write in arabic dude
Ramanujan was probably the most original and great mathematician
Itsiwhatitsi
That's true
..well apart from or on par with Euler, Euclid, Fibonacci, gauss
yes eternia
Einstein and Newton and gallelio and Archimedes are the best
Arsh Upadhyaya umm, einstein was not a mathematician.
Ramanujan is great... but he's no Gauss ;)
nothing is more mysterious than the brown paper.
Piyush Kuril THIS COMMENT HAS 163 LIKES LOLLOL
And its artfoolish fringes
Why cover a board specifically designed to write on , cover it with a paper , in order to write on it .
I am digesting moths .
Maybe the camera couldn't see the white Board or something?
Sells the scribbled brown paper on eBay. Can’t do that if the professors write on their board.
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.
And demonstrate this new number system is false.
This will never work with sine.
@@fredyfredo2724 nothing in mathematics is "wrong" as long as it's logically consistent
@@dielegende9141 undefine is not demonstrate false or wrong and is not true
@@fredyfredo2724 I have no clue what you're trying to say
@@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
I first encountered 163 when I moved on from 162.
You made me laugh. Truly. Thx!
You should get more thumbs up
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.
😂😂😂
That was very funny. Grounding. Thank you. Ha.
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!
The camera man for this channel loves zooming in to faces as awkwardly as possible
Jason Palmer he is an amateur, non professional, he must even love the subject of that clip on amateur mathematics
The camera man for this channel is the dude that runs this channel
Obviously this cameramen never review his work; the worse cinematographers of the millennium !
Jason Palmer hahahahahaha heheheeeee!
It's obvious you bitches have never had to to film in a cramped space.
Guys, go on Gauss' Wikipedia page, and look at his signature, I swear I can see Pi. XD
+galefray And there is the integral sign just before the end :)
+galefray you can also see an e and a butterfly in there :P
And the integral sign as the first "s"!
I thought mathematicians always had bad handwriting though, this signature is stunningly beautiful
and I totally feel like it's on purpose
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.
Ramanujan,
the mystery yet unsolved.
Now "163" is also my favourite number.
Haha look at that face in the end... it WAS his PIN heheh
+Entropy3ko Bosco!
Dat Seinfeld ref! hehe
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
Rooted negative numbers make me uncomfortable.
What should they do? Just use the word "i" behind the number?
you must be feeling complex!😉
it's just another notation for. (also all numbers are imaginary in some sense)
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.
Why? Do irrational numbers make you feel uncomfortable?
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.
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
***** …Excuse me?
The opposite for me, I'm left handed and when I see people writing with their right hand I'm like: "magic!" XD LOL
We lefties feel that you're the weirdos xD
I feel the same way, NoriMori.
How sinister...
I think the interlocutor guessed his ATM card code at the end.
absolutely brilliant. Thank you Alex.
@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!
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.
How did Ramanujan calculate this? He was really great... Love for Ramanujan from Bangladesh 🇧🇩
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.
@@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.
At last a normal person on Numberphile.
mathematicians would like to encourage everyone to do maths
"officially a mathematician"
They don't make 'em any more pretentious than that
Even Ramanujan called himself a clerk and not a mathematician. It is a job title, after all (cf. engineer).
So when are you "officially" a mathematician?
I suspect you'd need a degree? Just a guess.
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 😅
Great Ramanujan......
They mentioned several mathematicians, but you only noticed Ramanujian just because he happened to be Indian?
+Sanan Guliyev so what...search about him you will understand...and you have problem with indians?
so what problem you have with country tell me ofcourse i also here for math..
+Tanmoy Samanta whatever man try not to be racist/nationalist and appreciate scientists regardless of nationality/ethnicity
+Sanan Guliyev I'm not.....
It's interesting...if you take the list of these 9 numbers and line them up in order and subtract the lowest from the second lowest, the 2nd lowest from the 3rd lowest, etc. like you would if you were trying to find the degree of a function, you end up at 164, which is the lowest number (1) added to the highest number (163). Just thought that was interesting.
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.
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.
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)?
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...
John Stuart Just lingo. he also called one guy a recreational mathematician.
Hats off John!
Writing looks so tough for left handers
This is probably the best one yet.
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.
In a straight line y=mx+c, the gradient is m. In a curve the like y=x^2, the gradient has to be worked out differently (it changes as the curve gets steeper). To find the slope you 'differentiate' (you'll learn this later) to find the gradient. The number e is defined to be such that the curve y=e^x differentiates to e^x. Basically the the gradient at any point is equal to the y co-ordinate at any point. 2.718281828 =e (roughly, it's irrational).
So I guess 163 is special for something other than it's digits adding up to ten?
Truly Infamous ml
I understand the proof well enough I was just having fun, because I found some peculiar patterns in the series of numbers. thanks for the comment
He says "right triangles" but his triangles is actually left.
He is left handed
Илья Лагуткин lol tru
He is talking about Right angled triangle
Chutiya
... and he did mark the 90 degr corner.
If ramanunjan would have lived longer, he would have solved math.
I dont get why z[sqrt(-7)] works.
for example, 8 = 2*2*2 = (1+sqrt(-7))(1-sqrt(-7)). Am I missing something
because you don't know what a plus b whole square means
you're a duffer
Is (1+sqrt(-5)) a prime in Z[sqrt(-5)]? Because in the video he says it is.
Danny I Tan Lin
The "square magnitude" (norm?) of 1+sqrt(-5) in Z[sqrt(-6)] is 6, which is not prime.
Danny I Tan Lin just multiply
I really enjoyed this video - this feels like the kind of stuff I always wanted them to cover in school!
I wish there were videos like this that assumed that the viewer had a basic understanding of math at least up to Calculus. I don't want another Khan Academy (which is fantastic), because math is such an enormous field that you can't know more than a tiny fraction of it and I'm sure it would take a while to fully explain the relations here. I don't need to know everything about how the car works, I just want a peek under the hood.
i love those ominous sounds.
When you are right handed and see someone writing with left hand
Why do you assume we don't understand primes and factoring and then don't explain negative roots??
Is there any significance to those last, almost whole, numbers being similar form to eulers equation
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.
This is the first numberphile video that I have no idea what's going on in.
sqrt(163+6)= 13
13+4= 17.... pretty cool ?
no
Makes sense for you? Thats great!
Yeah, but they are. The point, though, is the definition of prime as a number that cannot be factored. What's important is to see that 1 + sqrt(-5), etc., are prime. When you multiply sqrt(-5) by itself any number of times, you always end up back either at 5, -5, sqrt(-5) or -sqrt(-5). You're going around in a circle. Basically, what happens in Z[i] is that there are more ways "around the circle."
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!
but in the root minus five system, 2 and 3 might not be primes and unique factorization might still stand. And I don't have much idea about this clip, just noticed
Amazing that my iPhone calculator cannot calculate e^(SQRT(163)*pi)
My android can (;
Mine gives me a really really big number 6725525588.089824502242480889791268597377
Probably goes beyond that XD
Oh and also android and not iphone.
Then you entered something wrong.
e*(sqrt(163)pi)= 262,537,412,640,768,743 . 999 999 999 999 25 (On my Windows calculator)
*****
Indeed. Seems like i didnt put something in correctly. Your result is the correct one.
Get's 2.62537412641E+17 on my Iphone
We do use higher base systems and we do frequently. Oftentimes, when confronted with a 32-bit number, it is easier to express it using 4 hex digits. Therefore [1] * 32 = ffffffff in hex, which is easier than writing 32 ones. In computers, hex numbers are used to represent operations, memory-addresses, bit-fields, etc. Hex is so popular because of how easy it is to go from base 2 to base 16 since both are powers of 2, so 1111 = f, 1010 = a etc. so we can represent alot w/ hex.
im amazed at how suddenly he moved from something i was totally getting into something that completely lost me
the sound of the marker on the paper just killed my brain :/
in e^sqrt(x)*pi
besides x=163
there is also x=-1
that would give you an integer
although not a whole number
There are 163 days until christmas
Ramanujan number: 1,729
Earth's equatorial radius: 6,378 km.
Golden number: 1.61803...
• (1,729 x 6,378 x (10^-3)) ^1.61803 x (10^-3) = 3,474.18
Moon's diameter: 3,474 km.
Ramanujan number: 1,729
Speed of light: 299,792,458 m/s
Earth's Equatorial Diameter: 12,756 km. Earth's Equatorial Radius: 6,378 km.
• (1,729 x 299,792,458) / 12,756 / 6,378) = 6,371
Earth's average radius: 6,371 km.
The Cubit
The cubit = Pi - phi^2 = 0.5236
Lunar distance: 384,400 km.
(0.5236 x (10^6) - 384,400) x 10 = 1,392,000
Sun´s diameter: 1,392,000 km.
Higgs Boson: 125.35 (GeV)
Phi: 1.61803...
(125.35 x (10^-1) - 1.61803) x (10^3) = 10,916.97
Circumference of the Moon: 10,916 km.
Golden number: 1.618
Golden Angle: 137.5
Earth's equatorial radius: 6,378
Universal Gravitation G = 6.67 x 10^-11 N.m^2/kg^2.
(((1.618 ^137.5) / 6,378) / 6.67) x (10^-20) = 12,756.62
Earth’s equatorial diameter: 12,756 km.
The Euler Number is approximately: 2.71828...
Newton’s law of gravitation: G = 6.67 x 10^-11 N.m^2/kg^2. Golden number: 1.618ɸ
(2.71828 ^ 6.67) x 1.618 x 10 = 12,756.23
Earth’s equatorial diameter: 12,756 km.
Planck’s constant: 6.63 × 10-34 m2 kg.
Circumference of the Moon: 10,916.
Gold equation: 1,618 ɸ
(((6.63 ^ (10,916 x 10^-4 )) x 1.618 x (10^3)= 12,756.82
Earth’s equatorial diameter: 12,756 km.
Planck's temperature: 1.41679 x 10^32 Kelvin.
Newton’s law of gravitation: G = 6.67 x 10^-11 N.m^2/kg^2.
Speed of Sound: 340.29 m/s
(1.41679 ^ 6.67) x 340.29 - 1 = 3,474.81
Moon's diameter:: 3,474 km.
Cosmic microwave background radiation
2.725 kelvins ,160.4 GHz,
Pi: 3.14
Earth's polar radius: 6,357 km.
((2,725 x 160.4) / 3.14 x (10^4) - (6,357 x 10^-3) = 1,392,000
The diameter of the Sun: 1,392,000 km.
Orion: The Connection between Heaven and Earth eBook Kindle
it only seems easy because we have 10 fingers. If we had 16 fingers we could equally say "Oh,base 16 is logical because to multiply 16 by 16 all we have to do is add a 0 and move the one to the left"
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?
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)!
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!
so, instead of 1 and itself (or P), (1+sqrt(-5) and itself (or P)?
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
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.
I think it's because:
a) brown paper is numberphile's thing. if i'm watching a video of someone writing on brown paper, i immediately correlate it to these guys.
b) that paper isn't a shiny surface so the light doesn't reflect and makes it impossible to see part of what he's writing down, which would happen with the whiteboard from this angle.
Why does Alex Clark sound like Ben Carson?
@IamGumbyy Whiteboards tend to not show up well in generically lighted rooms. A problem compounded by non-professional cameras. Notice how much glare is cast on the whiteboard from the lighting on the exposed area of the whiteboard.
I feel watching this upside down because he is left handed >_>
Well what I was thinking was angling the camera down slightly because all light sources are from above. I would be surprised if there was any glare then, but I could be wrong.
It might be that, if 'e' and 'Pi' is taken to be more accurate, then if the x.9999... could close more in towards the integer. Then, considering limiting value (as we consider more digits after decimal for 'e' and 'Pi') it might be true, i.e. it really could be an integer.
If Ramanujan said that it is an integer then it is. End of story. We will never know how his mind was wired, certainly not like us the mortals. His infinite series to evaluate pi for example is still a wonder to this day.
Nope, it can be shown that, with infinitely precise e and pi, it isn't a whole number
Ramanujan's conjecture gives us a rational number out of a mess of irrationality
My personal favourite of all the numberphile so far. The professor reminds me of Prof Gerald Lambeau from Good Will Hunting (Stellan John Skarsgård).
So Al Gore has left Global Warming and moved into Math...
Ramanujan was THE MATHEMATICIAN ...nobody will ever come close to him
Maybe Gauss conjectured that because it's the more surprising conclusion?
Mr.Clark forgot to mention that a and b are HALF-integers, except for d=1 and d=2, when they are integers. So, you are correct. :)
Gauß just didn't want to prove -163 was the last, because he was nice enough to leave some cool things for us to prove
We use base 2 on the basis that a it's a lot easier to have a computer read either a generalized "high" voltage versus a "low" voltage than trying to establish discrete increments of voltage to represent data. If we were to adopt something like that, I'd rather we use base 16, then we could represent larger numbers with fewer digits. >_>
They say that we use a base 10 system because of our fingers. Ancient people start counting wiht their fingers, that's why we use this system.
How do you even begin without the laws of physics
PLEASE STOP SAYING UNDEFINED MEANINGLESS THINGS LIKE "ordinary numbers" or "normal numbers".
If you MEAN "the integers" or "the positive integers", then SAY "THE INTEGERS" or "THE POSITIVE INTEGERS".
Everytime anyone is sloppy expressing these concepts it absolutely DOES raise unnecessary confusion,
as saying the proper concept requires LESS work than being sloppy.
My school just added a course on number theory, I think I'll be taking it. Pure mathematics is as fascinating to me as physics, and in the long run I believe the deeper connections in physics will be made through abstract mathematical concepts.
I need more explanation of these new number systems e.g. Z[√-5] and also of what makes a prime in those number systems
I'm sort of lost. I've learned in school that you cannot make a square root of a negative number. Please help me!
A little late, but it's wrong to write it like that. What he wants you to notice is that the square of those numbers is negative. What he should have written to be formally correct is i√2, i√5, i√7... And so on
You were lied to or, rather, protected from maths deemed too complicated.
You can't. Some fucksticks just said "you know what, we're gonna call i the number that gives -1 if you square it. i²=-1. The fascinating thing about it is that it opens a whoooole new world of numbers. You have naturals, integers, fractions, real numbers. They are are sitting on a line that includes all real numbers. So its in 1D you might say. But by introducing i, you expand the numbers in 2D. A number then becomes a coordinate on a 2-axis field (instead of the 1-axis if only real numbers) in the form of a+bi, with a being the real part of the number and the bi the part being the imaginary part. A completely real number lies on the horizontal axis and only has a real part, with b=0 so a+bi becomes a, the multiplications of i lie on the vertical axis and only have an imaginary part, with a=0 so a+bi becomes bi.
Sorry pal, they lied!
Another fantastic number. Great vid!
but its not repeating, there was 25 later on.
I can remember the edition of Scientific American in which Martin Gardner's joke appeared. It was the April edition, and there were a number of other "April Fools" in the same column. I think one of them was a fake drawing showing that Leonardo da Vinci had invented the flush toilet.
Jim Morrison and Kurt Cobain were self taught singers as well :|
nice guy this professor is. hes got a good heart. funny how you can tell that about someone.
one of the most cliffhanging numberphile videos ever
please make some video about the Loschian numbers
Amateur mathematician lol
If the great Ramanujan was an amateur mathematician you're all pre-school daycare caretakers.
_Heegner_ was the amateur mathematician in this case. He proved the Gauss' conjecture.
Ramanujan just found that the two Pi over e numbers were almost whole.
As if this has 15 likes. Ramanujan was never called an amateur in the video
And here I am still waiting for irrational numbers to be used for anything useful.
Who decides who is mathematican or is not? Its so stupid and illogical.
Idiotic.
2, 5, 3+i and 3-i are all composite numbers in the Z/i system. The prime factorization of 10 is (1+i)(1-i)(2+i)(2-i).
I and another talented 14 years old actually figured out a lot of this on our own in an afternoon at a maths camp.
Though we had the hint that "complex primes are interesting", I am quite proud of that.
Hehe, just some pointless bragging.
Now you are 15 already
You proved 163 was the most negative number?
numberphile never uses other than brown paper. I assume it is because brown paper does not reflect light and allows camera and viewers to see what is actually written.
Really INDIAN is GREAT
Finally found it! The NP video that isn't sponsored by someone!
that's right, they're denoted by i for mathematicians, and by j for engineers who already use I for electrical current.
I did not know that, but that is something I should remember if I am to go on vacation there at some point.
I too,have something to say about the number 163.
But for reasons you'll know some time into the future,I'll just give you some maths.
Light, is 1.
A day, is 24.
We have 4 base simulations.
6=1×2×3
Logical analysis of cell division will give us 3 new cells,since there was 0,then came 1,which then squared and gave 1.
But that isn't supposed to happen.
After all,after 24 hours,you do end up with light(1),don't you?
That is true when you only look at real numbers. The square root of a negative number is called an imaginary number. Imaginary numbers don't make much sense in counting or measuring things (eg you can't have sqrt(-5) apples or be sqrt(-8) feet tall) but they are very helpful in a lot of physics and mathematics problems. For example imaginary numbers can be used to represent a sine wave;
sin(x) = (e^ix-e^-ix)/2i
where x is the angle, e is Euler's number and i=sqrt(-1)
You are absolutly right here, this is a valid decomposition into seperate sets of primes.The professor skipped (maybe on purpose) the important point that when working with quadratic fields Q[sqrt(D)] you actually look at Z[w(D)] with w(D) = sqrt(D) if D = 2 or 3 (mod 4) and w(D) = (1+sqrt(D))/2 if D = 1 (mod 4).
So you actually work with Z[1/2 + sqrt(-7)/2] in which the number 8 decomposed uniquely into 8 = (1/2 +sqrt(-7)/2)³ * (1/2 - sqrt(-7)/2)³
It wasn't .9 repeating. It shows in the video that it's something like .9999999925...
It is thought to be the ten fingers as there have been cultures that had eleven as the base thanks to a male appendage. Twelve was used for currency in Britain as 12 is very divisible (1,2,3,4,6,12 rather than 1,2,5,10)
This seems eerily similar to the Fermat primes. I wonder if there is a relationship between the Heegner numbers { 1, 2, 3, 7, 11, 19, 43, 67, 163 } and the Fermat primes { 3, 5, 17, 257, 65537 }. Notice how both sets have a large jump between that last two elements. Of course, it isn't proven if 65537 is the last Fermat prime.
He's using a permanent sharpie. Also a whiteboard probably isn't too visible on a camera