When I was about 16, I decided to become a great mathematician by submitting all these worthless sequences I came up with myself to the OEIS. After a few of them got published, I started submitting variations on them--using them as functions within each other, changing variables, etc. Most of those were published until I got a message from Neil Sloane himself telling me to quit it, with an exclamation mark. Five years later, that's still probably the coolest thing that's ever happened to me. To give you an idea of how bad these sequences were, one of my (rejected) submissions was: a(n) = n - 47
@@achtsekundenfurz7876 One self-liberated colony of an irrelevant rump state, you mean. And fahrenheit is a more human-relative temperature than c*lcius.
I was trying to join oeis when I was like 12, but they never accepted me since I couldn't prove my identity. I used to regularly lookup sequences I came up with to see if OEIS found them yet and occasionally I'd find new sequences but could never submit them
Dr. Grime you are the most cheerful, funny and emphatics guy I have ever seen. It is impossible to see one youtube video of yours without becoming in a good mood and with spirits up. GOD BLESS YOU !
aditsu If whatever code you used for the equation actually pulled from the OEIS directly, and failed to ignore itself, it would be an infinite loop... tho it might take a while to realize that's what happened, as even without that problem, it's a hard problem to solve... I don't mean like calculus, either... rather one of those insoluble problems that mathematicians call "Hard" in a rather cheeky "It's actually possible, but could never be solved given all the time in the universe" manner.
I got a notification saying there'd been a reply to my year-old comment but it seems that it only exists in the email notification. I wouldn't have mentioned it if it was there.
So... mathematicians using math to figure out what math mathematicians most often use in math... how meta can we get here? Any one have an equation to find that?
Not completely obvious.... the base 10 guys are clearly cultural.... Mathematicians study numbers without an obvious cultural bias so perhaps this gap is less meta mathematical that one would assume at first sight. But who knows.....
someone made a graph about 'popular numbers' used in database where were tons of number sequences. the thing was that some numbers are more mainstream than others
someone made a graph about 'popular numbers' used in database where were tons of number sequences. the thing was that some numbers are more mainstream than others
My professor played this video for class today. The video stopped to buffer many times, and at each point, we all burst into laughter at the hilarious expressions that you give.. Keep it up man!
What I love about these channels is that they provide proper sources. Often peer reviewed journal articles or papers. A lot better than the usual circumstances with videos
Actually, it depends on which number you start at, and the rate of growth. 1729 is the 1729th natural number, the 1729th whole number, and it would be near impossible to count its place on the integer sequence or rational number sequence.
I just wanted to say thank you to Brady and everyone else that puts together the videos from Numberphile, Sixty Symbols, Periodic Videos, Compterphile, etc... These videos are very entertaining, and always informative! Thanks!
Being an electronic engineer of trade and an electronic engineer of heart, math is a part of most solutions in my world. Your presentations of complicated math to us amateurs is just pure entertainment. Thank You /JD
These videos are always so intriguing, and yet so calming. I should see how often I fall asleep after one of these videos over other videos...thank you for intriguing me in math when school ruined it for me, more or less
Dr Grime called it the Encyclopedia of Integers (rather than Integer Sequences) right after I was pondering the potential and growth rate of a wiki with a page for each integer and now I feel like I have to devote the rest of my life to this.
A geometric version (can't post it, you'll just have to imagine): Take a grid that is x by x units. It has an area of x^2. Remove the top row. You now have a rectangular grid that is (x-1) by x. Turn the loose row 90 degrees and stick it on either side as a new column. You now have a rectangular grid that is (x-1) by (x+1) with 1 additional square sticking up. Its area would be (x-1)(x+1) + 1 So since no squares were ever added or removed: x^2 = (x-1)(x+1) + 1 or x^2 -1 = (x-1)(x+1)
Here's my conjecture: Numbers with larger amounts of simply definable characteristics will appear more due to sequences being based on defined characteristics. Furthermore, as the complexity of the characteristics increases, the frequency of the more common numbers will change only slightly, as they have already been established in the earlier stages of complexity. The ordered bands are a side effect of the basic properties of numbers and rules.
Is it possible to write a sequence of numbers that has no valid mathematical rule? For example, 3, 73, 132, 1972, 4, 41,... might look like random numbers but is there a way to prove that there is no mathematical link? (If indeed there is none.)
CowLunch Any mathmatical rule, no matter how it might look like, must build from an finite numer of charakters and therefore all of the rules combined are cuntable infinite. But your sequences are all infinite which makes the set of all of them uncountable infinite. So there musst be an uncountable set of sequences which do not follow any mathmatical rule :)
Correct. That's something I noticed rather recently as well. It relates to the difference of two squares. (x^2) - (a^2) = (x+a)(x-a). In the case that you pointed out, a = 1. However, this also works when a is equal to any real number, including negatives and decimals. (x^2) - 4 = (x-2)(x+2) ; (x^2) - 9 = (x-3)(x+3) ; (x^2) - 16 = (x-4)(x+4) ; etc. If you know a lot of perfect squares, it can save time when multiplying. Instead of trying to figure out what 39 x 41 is, square 40, then subtract 1.
I hate you guys! I watched this video before bed. I dreamed about it till I woke up at 3 am and had to think about it for a solid hour. Thanks for tickling my brain ... fsss ;-)
Proof by contradiction that all positive integers are interesting: assume that not all positive integers are interesting. Then, the set of positive integers that are not interesting is non-empty. By the Well Ordering Principle, that set has a least element. So, there is a smallest, non-interesting number. ISN'T THAT INTERESTING! ~_^
Assume not all positive integers are interesting: OK Then, the set of positive integers that are not interesting is non-empty: But who cares? Contradiction averted.
Solving the paradox: ‘Interesting’ (or ‘boring’) are undefined terms. It is like you were saying: Not all people are beautiful. Take the set of ‘not beautiful’ people. There must be one which is the most beautiful of them. Silly. Beauty and interest are 1. subjective ; 2. Continuous rather than categorical ; 3. non measurable. What is the smallest ‘big’ number?
@@MarcusCactus It's not a paradox, it's a math joke. About your continuous thing though, in the case ok this joke, interest is not continuous: a number is either interesting or it isn't. Also, your argument about beautiful people doesn't work the same way as the interesting thing. Being the most beautiful non-beautiful people just makes you that. The most beautiful non-beautiful person. Whereas being the smallest non-interesting number MAKES that number interesting. You don't have that contradiction with your beauty example : being the most beautiful non-beautiful person doesn't make you beautiful. That's why the joke works.
Philippe Carphin : something is not ‘either interesting or not ‘. It is more interesting than this and less interesting than that. Continuous fuzzy subjective ordering. Quand il y a paradoxe, il faut chercher l’erreur dans l’énoncé ou les prémisses implicites. Ici, l’erreur est de poser que ‘intéressant’ est une catégorie binaire bien définie. Le paradoxe est une preuve par l’absurde, non pas que tous les nombres sont intéressants, mais qu’on ne peut les trier en deux classes ainsi définies.
Nice job on finding it. It's actually a really useful formula that is used a lot. Normally it is taught in algebra classes, and only sometimes is it actually explained.
It would be interesting to study Sloan's Gap's evolution in time. My wild guess is that it was wider in past and narrows in time. The gap represents the amount of subjectivity in our perception of abstract. The lower band is white noise and upper band is human capacity of perception.
+christopher mccaul It does to an extent. Just that annoying gap that makes it a little odd. It's like high school heirarchy for numbers with that top band.
christopher mccaul for it to be zipfy, in the equation of that curve, the power of the 'n' in the denominator should be 1 or closer to 1 (like 1.002 or something like that)
Excellent question. You can expand the multiplication (x-1)(x+1) to x^2 - x + x - 1. The two x's cancel out, leaving just x^2-1, so they're equivalent. Put another way, imagine you have 25 (x^2) stones laid out in a 5 x 5 grid. Throw the top right stone away (the minus 1). Now take the four (x-1) stones at the top, and place them on the right. Now you have a grid of four (x-1) rows high and six (x+1) columns wide, 4*6=24=25-1. Same logic works for any size grid.
+NoriMori Yeah, I'm curious about that as well. To me it looks like there's a small band just above the bottom band but quite a bit below the top band of numbers inside the gap. And they look interesting to me. =)
***** Look at the graph, there are a few numbers there. Even though they're few and far between they seem interesting to us. We'd like to know more about them, *because* they're so few. And yes, we did understand the point of the video...
skr47ch 無限 True, but only *after* they've been mentioned x number of times in that encyclopedia they mentioned. Them being in the middle of the gap would only cause them each to be mentioned once more each in the lists I think.
Actually, you can show that all whole numbers are interesting. Proof: - Assume that one or more of the whole numbers are uninteresting. - Then, there must be a smallest uninteresting whole number. - Surely, the smallest uninteresting number is interesting by virtue of it being the smallest uninteresting number, which leads to a contradiction. - Therefore all whole numbers must be interesting.
You contradicted yourself. Your first assumption assumes at least one whole number is uninteresting, while your conclusion concludes that all whole numbers are interesting. uninteresting ≠ interesting
What he was showing in the video is not what you have supposedly proven. The video suggests that while all whole numbers are interesting, some are more interesting than others. Let's say we have f(n) = x, where n is some number, and x is a numerical indicator of how interesting the number is. Now suppose that a number is considered "popular" if x is greater than some value y. Now you state that if f(n) = x, then the smallest n where x < y should be more interesting than it really is, thereby making the statement of x + z > y. This may sometimes be true, but it is not guaranteed.
This might sound like a weird question, but is there a way to express numbers without using a base? Because from this video it would appear that whatever base you use, certain numbers will immediately take on a significance purely because you are expressing the numbers in that particular base. In other words, can any significance that the base has be filtered out when looking at the "interestingness" of numbers?
From what I understand from other Numberphile videos, mathematicians usually only consider a discovery serious if it works in ALL bases. Otherwise it seems like a recreational math trick, something just for fun. A lot of the integer sequences on the OEIS *are* just for fun, so no surprise that they include visually interesting base-10 stuff. Sorry you had to wait 100 years for a reply, lol. (In binary.)
It depends on how pedantic you get about what a "base" means. There's unary, which in some sense is base 1, but it doesn't really behave like other bases as there's no concept of "place value". Unary is just having a mark for every bit of value. So, 1=1, 2=11, 3=111, 4=1111. When you add them, rather than having to do some computation, you just concatenate them. So, 11 + 11 = 1111 (2+2=4 in unary). But of course multiplication can be a bit slow if you want to work purely in unary. You can sort of just make a multiplication table. if you want to get the result of 111 * 1111 (3*4), just make a rectangular array of 1's that is 111 wide and 1111 tall and concatenate all those 1's together and you have your answer. And if you're working purely in unary, counting becomes a kind of pointless exercise as if I asked you how many 1's are in: "111111" and expected you to answer in unary, you'd just repeat back what I said, "111111".
The OEIS sequences all have tags that describe them, one of them being "base", which means the sequence is dependent on the base used. Surely it wouldn't be difficult to filter these sequences out, should someone try.
is it just on the video or does it seem that on the bottom of the right side of the curve the numbers form some horizontal lines? Is there a reason for this? Hmmm, maybe just coincidence... ;)
This is what is known as the difference of two squares. It is useful when factoring two-term polynomials. a^2 - b^2 = (a+b)(a-b) In your example, (x^2) - (1^2). (a+b)(a-b) = a^2 + ab - ab - b^2 = a^2 + 0ab - b^2 = a^2 - b^2 That is why it works, as you were wondering.
There are 3 numbers at the far right of the graph, between 8000 and 10,000, that only appear in sequences once each? I'm really curious to know what numbers those are! Also, at the far far left end of the graph, there is a huge gap (due to the log scale of the y-axis) between the first few numbers and the rest of the set... what are the numbers on either side of that divide? Actually, is this data series published anywhere for public download?
Nah. the smallest number of an otherwise uninteresting set just isn't as interesting as the least interesting prime number fits the sequence n^2 - 1. Which is, of course, 3. There are those who say none of the numbers are interesting, of course, but then those people aren't mathematicians making number sequences, now are they?
IIRC there's a process of transforming a number so: If it's even, divide it by two. If it's odd, multiply by three and add (?) one. If it eventually ends up at 1, it's interesting. It was 30 or so years ago that I heard this, so the details may be a little off, but all numbers tried, ended up as 1. But some blow up to be very big before it happens. In any case, who's to say what's interesting? The successes of both Pokemon Go and Frozen indicate that tastes vary widely.
I love how the paper is set in the font Computer Modern, which is the default font of LaTeX, the typeset language/program most scientists seem to write their papers in. :D
Someone ought to create the sequence of numbers that don't appear in the Online Encyclopedia of Integer Sequences. Or, actually, better not. This might implode the Internet.
great explanation, i wonder if people start focusing on the numbers in the gap, they will be more likely to discover newer connections more easily and quicker
I don't find that very surprising, the sequences of numbers are neither distributed evenly, nor are they independent. They are representations of groups of properties people came up with, often leading to similar properties and series, thus establishing a dependence. What this really means is that for those numbers that are in the lower band, we have simply not yet found classes of "interesting" sequences yet.
+Dennis Lubert I agree - that's the cultural bias. The easiest way to 'fix' the distribution is to add a second set of sequences which are the inverse of the others - add those into the distribution and it would flatline at 2x the current number of sequences. Which demonstrates the artificial nature of analysis of a very limited sample set, although the bias is valid.
@VladHPsuper Gaming It's a play on the "triggered" meme, which is centered around mocking "SJWs", leftists/intersectionalists, LGBA people, and trans people as irrational and discrediting their complaints about issues which affect them, such as misgendering, mockery of poor people or minorities, stereotyping and mockery of LGBT people, and so-forth.
The numbers remain the same regardless of the base you chose to represent them with; we use base 10 out of commodity, but these sequences of integers are the images (or 'results') of certain functions, ie, the fibonacci sequence is F:N->R (that is read as, the function grabs natural numbers and 'outputs' real numbers); F(n)=F(n-2)+F(n-1), for n in the Natural numbers, given that F(0)=0, F(1)=1. The images of these sequences are independent of the m-base used to represent them.
+Tejas Naik I would think so! I was about to comment on it, glad someone else, though a year earlier, saw it. Interesting how natural Zipf is, isn't it?
This is a pretty well known algebraic principal, the difference of two squares. Basically, x^2-y^2 = (x+y)*(x-y). If you replace y with 1, you get your formula. If you apply the algebraic rule of thumb, first outer inner last (foil) to multiply (x+y)*(x-y) you get: x^2-x*y+x*y-y^2. The -x*y and +x*y cancel out and you get x^2-y^2.
When I was about 16, I decided to become a great mathematician by submitting all these worthless sequences I came up with myself to the OEIS. After a few of them got published, I started submitting variations on them--using them as functions within each other, changing variables, etc. Most of those were published until I got a message from Neil Sloane himself telling me to quit it, with an exclamation mark. Five years later, that's still probably the coolest thing that's ever happened to me.
To give you an idea of how bad these sequences were, one of my (rejected) submissions was: a(n) = n - 47
"counting, but it starts at -46"
brilliant
Well... 32 used to work for °F temperatures, so why not? -- Today it's fallen out of use, except for one former British colony ;)
@@achtsekundenfurz7876 One self-liberated colony of an irrelevant rump state, you mean.
And fahrenheit is a more human-relative temperature than c*lcius.
was about to like this until I realized it was at 0 likes
I was trying to join oeis when I was like 12, but they never accepted me since I couldn't prove my identity. I used to regularly lookup sequences I came up with to see if OEIS found them yet and occasionally I'd find new sequences but could never submit them
Dr. Grime you are the most cheerful, funny and emphatics guy I have ever seen.
It is impossible to see one youtube video of yours without becoming in a good mood and with spirits up. GOD BLESS YOU !
fft2020 Cliff too! The Klein bottle guy!
😬😬😬😬😬😬😬😬😬😬😬😬 Nonono Nonono Nonono Nonono Nonono Nonono Nonono Nonono Nonono Nonono Nonono Nonono
I want to like this ... but it already has 333 likes. Wouldn't want to make this number less popular now
Wait till u see Parker
Let's make a sequence of all the numbers that don't appear in OEIS, and add that sequence to OEIS ^_^
Jeoshua Collins Fair points, but you missed the main one: adding the sequence to OEIS would be an instant paradox.
aditsu If whatever code you used for the equation actually pulled from the OEIS directly, and failed to ignore itself, it would be an infinite loop... tho it might take a while to realize that's what happened, as even without that problem, it's a hard problem to solve... I don't mean like calculus, either... rather one of those insoluble problems that mathematicians call "Hard" in a rather cheeky "It's actually possible, but could never be solved given all the time in the universe" manner.
It's too bad your comment reply isn't where it should be. I was going to post it to /r/iamverysmart
Where should it be? I don't know if you meant me or him, but have at it, as far as I am concerned.
I got a notification saying there'd been a reply to my year-old comment but it seems that it only exists in the email notification. I wouldn't have mentioned it if it was there.
I'm glad they got Neal Slone on Numberphile after this was made... he's my favorite one to watch
So... mathematicians using math to figure out what math mathematicians most often use in math... how meta can we get here? Any one have an equation to find that?
+jtfroh f(derp) = (derp^2herp x π) + (erhamgerd x derp^e)
+celestus87 g64^g64
+celestus87 are these variables or constants?
Not completely obvious.... the base 10 guys are clearly cultural.... Mathematicians study numbers without an obvious cultural bias so perhaps this gap is less meta mathematical that one would assume at first sight.
But who knows.....
g65... I win...
the graph seems to fit Zipf's law as well.
yep...per bak's self-organizing criticality !!
A/x for some constant A would perfectly fit zipf's law. this is A/x^1.33, so it's pretty evident
dammit i came here to say that
thank you for watching - we appreciate it!
bruh
"When the Internet came along, that was a nice opportunity to put it online."
- Love these vids :D
I wish I had Dr. Grime for maths analysis at university. Love this guy.
He's always smiling! tt's nice to see someone excited about their work for a change, especially with everyone being so grumpy these days.
Bruce K Watch the Klein bottle video lol. That guy is the happiest guy I've ever seen.
leon bushnell and the other people were
His enthusiasm is quite infectious. Unfortunately, I have no idea what he's talking about.
Barba Ro about numbers and math and stuff
Barba Ro
( ಠ_ಠ) aahh, ok, i see...
( ಠ_ಠ) I still dont get it, guess ill watch another one.
Barba Ro I think that's why I like this channel.
someone made a graph about 'popular numbers' used in database where were tons of number sequences. the thing was that some numbers are more mainstream than others
someone made a graph about 'popular numbers' used in database where were tons of number sequences. the thing was that some numbers are more mainstream than others
This is always my favorite guy to watch on this channel. Like others have said, his enthusiasm for the material is quite infectious and refreshing ;)
My professor played this video for class today. The video stopped to buffer many times, and at each point, we all burst into laughter at the hilarious expressions that you give.. Keep it up man!
Yesss! I found a channel for my love of math and numbers!! Best day ever!
hey! did you end up going the math path?
What I love about these channels is that they provide proper sources. Often peer reviewed journal articles or papers. A lot better than the usual circumstances with videos
1729 is the 1729th number. Thank me later.
Cob Canon
MindBlown.gif
Cob Canon *natural numbers
+Ulisses Rodrigues
Its the 0th number
Actually, it depends on which number you start at, and the rate of growth.
1729 is the 1729th natural number, the 1729th whole number, and it would be near impossible to count its place on the integer sequence or rational number sequence.
"Or is it?
Hey Vsauce, Michael here."
When you call a number boring and that number ends up being remembered by everyone for how interesting it is.
Feels bad man.
A number cannot be boring, for if it is boring, then having the property of being boring makes it interesting.
@The coo - king Hey it is not Thomas Hardy, Thomas Hardy was a writer, this man is Godfrey Harold Hardy, a mathematician.
When you realize the guy whos Sloane gap is named after, Neil Sloane, has been on numberphile. Everything is connected
Ramanujan was the most wonderfully creative maths genius. It's so sad that he didn't live longer.
And then, years later, the man himself appears on Numberphile
I hate it when all the popular numbers band together,,,,and beat up on the nerd numbers.
lik dis if u kry evry timme
+WowplayerMe But the nerd numbers are the popular numbers. . .
,,,,
DUN DUN DUNNNN!!!!!!!!!!!! (cue dramatic film music! >:) XD)
Revenge of the nerd numbers.
I just wanted to say thank you to Brady and everyone else that puts together the videos from Numberphile, Sixty Symbols, Periodic Videos, Compterphile, etc... These videos are very entertaining, and always informative! Thanks!
Am I the only one to notice that this guy is the happiest man on earth?
I think the guy who painted happy trees on PBS has him beat.
@@uruiamnot That man isn't on earth he's in the happy little clouds he painted for his friends Mr Rogers and Robin Williams.
That man's name is Bob Ross ☁️
Don't forget cliff stoll and tom Crawford
*the happiest man on TH-cam
I feel like the purpose of that last bit was just to show his completely endearing fascination with maths culture. It captured my heart.
very funny watching this after niel sloane joined numberphile
Being an electronic engineer of trade and an electronic engineer of heart, math is a part of most solutions in my world.
Your presentations of complicated math to us amateurs is just pure entertainment.
Thank You
/JD
Like if you're against number discrimination.
+Cvijetko Livadic I was waiting for this
34,562 FTW!!!
Ordinary numbers matter!!!
Prime numbers are overrated. Drain the swamp.
there are no ordinary numbers
I want to hug you so much for being as enthusiastic about math as I am!
So the popular band is the one with less numbers flocking to it? Wow, I must've been really popular back in high-school!
With 1600 comments I wish there were a search feature to see if someone else mentioned this, but:
This reminds me of Goldbach's Comet.
this guy is so enthusiastic to numbers , that i wanna give his childhood a hug :/
Multiplication*
this was dr. grime's way of calling out the basic numbers
So, what is the least interesting number?
boumbh That is kidn of a paradox, the least interesting number would be very interesting just because it is the least interesting xD
boumbh 27
Alejandro Tejeda I hope you are joking, it is three to the power of three! Like in "may the power of three be with you"!
boumbh Didn't get the simpson reference?
Alejandro Tejeda No :-( ... Girls just want to have sums...
These videos are always so intriguing, and yet so calming. I should see how often I fall asleep after one of these videos over other videos...thank you for intriguing me in math when school ruined it for me, more or less
It is a great read!
I watch all of these videos, and I almost never know what they are talking about, but I always feel like I've learned something.
Did they compile a list of the ordinary numbers? It would be quite interesting to see that. A OESIS entry could even be given.
Dr Grime called it the Encyclopedia of Integers (rather than Integer Sequences) right after I was pondering the potential and growth rate of a wiki with a page for each integer and now I feel like I have to devote the rest of my life to this.
Most of the time I have no idea what they`re talking about, non whatsoever!
Yet I`m fascinated!
These are the kind of topics are the reason I subscribed to this channel.
Find a number thats a sum of three prime numbers, and its also a quotient of 2 cubed numbers.
59+2+3 = (8^3)/(2^3)
A geometric version (can't post it, you'll just have to imagine):
Take a grid that is x by x units. It has an area of x^2.
Remove the top row. You now have a rectangular grid that is (x-1) by x.
Turn the loose row 90 degrees and stick it on either side as a new column. You now have a rectangular grid that is (x-1) by (x+1) with 1 additional square sticking up. Its area would be (x-1)(x+1) + 1
So since no squares were ever added or removed:
x^2 = (x-1)(x+1) + 1
or
x^2 -1 = (x-1)(x+1)
Here's my conjecture: Numbers with larger amounts of simply definable characteristics will appear more due to sequences being based on defined characteristics.
Furthermore, as the complexity of the characteristics increases, the frequency of the more common numbers will change only slightly, as they have already been established in the earlier stages of complexity.
The ordered bands are a side effect of the basic properties of numbers and rules.
i never understand anything you say but i still watch your videos, i think it's just your enthusiasm about everything lol
Is it possible to write a sequence of numbers that has no valid mathematical rule?
For example, 3, 73, 132, 1972, 4, 41,... might look like random numbers but is there a way to prove that there is no mathematical link? (If indeed there is none.)
CowLunch I know... let's ask the NSA? They'll tell us which number sequence
is random and truly beyond their control. ; )
CowLunch Polynomial interpolation is a powerful thing.
CowLunch Any mathmatical rule, no matter how it might look like, must build from an finite numer of charakters and therefore all of the rules combined are cuntable infinite.
But your sequences are all infinite which makes the set of all of them uncountable infinite. So there musst be an uncountable set of sequences which do not follow any mathmatical rule :)
CowLunch Isn't there a paradox that says the first number that isn't interesting becomes interesting because of that fact?
It is not possible. Writing a list of numbers is in itself a 'rule'.
I love finding a sequence that isn’t in OEIS
that one dimple is driving me crazy
+jayla duhaney WHY!!!!!!
Numberphil make me always smile. You have the passion that i missed in the mathematical study
A mathematician of all people shouldn't speak so lightly of the blatant discrimination and subjugation of numbers. It's okay to be ordinary
This is the best thumbnail on numberphile ever.
And with that, math has proven that mathematicians play favorites. XD
Correct. That's something I noticed rather recently as well. It relates to the difference of two squares. (x^2) - (a^2) = (x+a)(x-a). In the case that you pointed out, a = 1. However, this also works when a is equal to any real number, including negatives and decimals. (x^2) - 4 = (x-2)(x+2) ; (x^2) - 9 = (x-3)(x+3) ; (x^2) - 16 = (x-4)(x+4) ; etc.
If you know a lot of perfect squares, it can save time when multiplying. Instead of trying to figure out what 39 x 41 is, square 40, then subtract 1.
I hate you guys!
I watched this video before bed. I dreamed about it till I woke up at 3 am and had to think about it for a solid hour.
Thanks for tickling my brain ... fsss ;-)
The second I saw TFIOS I literally screamed.
Proof by contradiction that all positive integers are interesting: assume that not all positive integers are interesting. Then, the set of positive integers that are not interesting is non-empty. By the Well Ordering Principle, that set has a least element. So, there is a smallest, non-interesting number. ISN'T THAT INTERESTING! ~_^
Assume not all positive integers are interesting: OK
Then, the set of positive integers that are not interesting is non-empty: But who cares?
Contradiction averted.
Lol, I just finished writing this exact thing. Then I scroll down to find your comment. Yours is clearer though.
Solving the paradox:
‘Interesting’ (or ‘boring’) are undefined terms. It is like you were saying: Not all people are beautiful. Take the set of ‘not beautiful’ people. There must be one which is the most beautiful of them.
Silly. Beauty and interest are 1. subjective ; 2. Continuous rather than categorical ; 3. non measurable.
What is the smallest ‘big’ number?
@@MarcusCactus It's not a paradox, it's a math joke. About your continuous thing though, in the case ok this joke, interest is not continuous: a number is either interesting or it isn't.
Also, your argument about beautiful people doesn't work the same way as the interesting thing. Being the most beautiful non-beautiful people just makes you that. The most beautiful non-beautiful person. Whereas being the smallest non-interesting number MAKES that number interesting. You don't have that contradiction with your beauty example : being the most beautiful non-beautiful person doesn't make you beautiful. That's why the joke works.
Philippe Carphin : something is not ‘either interesting or not ‘. It is more interesting than this and less interesting than that. Continuous fuzzy subjective ordering.
Quand il y a paradoxe, il faut chercher l’erreur dans l’énoncé ou les prémisses implicites. Ici, l’erreur est de poser que ‘intéressant’ est une catégorie binaire bien définie. Le paradoxe est une preuve par l’absurde, non pas que tous les nombres sont intéressants, mais qu’on ne peut les trier en deux classes ainsi définies.
Nice job on finding it. It's actually a really useful formula that is used a lot. Normally it is taught in algebra classes, and only sometimes is it actually explained.
It would be interesting to study Sloan's Gap's evolution in time. My wild guess is that it was wider in past and narrows in time. The gap represents the amount of subjectivity in our perception of abstract. The lower band is white noise and upper band is human capacity of perception.
These videos are great!
Looks zipfy
+christopher mccaul It does to an extent. Just that annoying gap that makes it a little odd. It's like high school heirarchy for numbers with that top band.
+Mr Rishi The Cookie is it zipfy?
Muhammad Abdullah *shrugs* ¯\_(ツ)_/¯
christopher mccaul for it to be zipfy, in the equation of that curve, the power of the 'n' in the denominator should be 1 or closer to 1 (like 1.002 or something like that)
Indeed, it's quiet close to Zipf's law, except Zipf's law is lambda * n^(-1), whereis this is closer to lambda * n^(-1.33)
Excellent question. You can expand the multiplication (x-1)(x+1) to x^2 - x + x - 1. The two x's cancel out, leaving just x^2-1, so they're equivalent.
Put another way, imagine you have 25 (x^2) stones laid out in a 5 x 5 grid. Throw the top right stone away (the minus 1). Now take the four (x-1) stones at the top, and place them on the right. Now you have a grid of four (x-1) rows high and six (x+1) columns wide, 4*6=24=25-1. Same logic works for any size grid.
What I'd like to know is, what types of numbers usually appear in the midst of the gap?
+NoriMori Yeah, I'm curious about that as well.
To me it looks like there's a small band just above the bottom band but quite a bit below the top band of numbers inside the gap.
And they look interesting to me. =)
***** Look at the graph, there are a few numbers there.
Even though they're few and far between they seem interesting to us.
We'd like to know more about them, *because* they're so few.
And yes, we did understand the point of the video...
+Luredreier But if we started showing interest in them, those might get elevated to top band, leaving behind a gap. :)
skr47ch 無限 True, but only *after* they've been mentioned x number of times in that encyclopedia they mentioned.
Them being in the middle of the gap would only cause them each to be mentioned once more each in the lists I think.
Luredreier that makes sense.
Six years later and now you can watch videos with Neil Sloane on Numberphile.
Actually, you can show that all whole numbers are interesting. Proof:
- Assume that one or more of the whole numbers are uninteresting.
- Then, there must be a smallest uninteresting whole number.
- Surely, the smallest uninteresting number is interesting by virtue of it being the smallest uninteresting number, which leads to a contradiction.
- Therefore all whole numbers must be interesting.
You contradicted yourself. Your first assumption assumes at least one whole number is uninteresting, while your conclusion concludes that all whole numbers are interesting. uninteresting ≠ interesting
Sam Spencer The contradiction is the point. It's called a reductio ad absurdum.
What he was showing in the video is not what you have supposedly proven. The video suggests that while all whole numbers are interesting, some are more interesting than others. Let's say we have f(n) = x, where n is some number, and x is a numerical indicator of how interesting the number is. Now suppose that a number is considered "popular" if x is greater than some value y. Now you state that if f(n) = x, then the smallest n where x < y should be more interesting than it really is, thereby making the statement of x + z > y. This may sometimes be true, but it is not guaranteed.
Right now you have 1729000 subscribers. Exactly. Wow.
fun fact: in Russian language the prime numbers are called "simple" or "ordinary" numbers lol
I mean they are the simplest numbers in a way (# of factors being only two, while the others have more)
calling them simple makes a nice reflection to simple groups, it's a pretty pleasong name actually
In Soviet Russia, numbers prime you.
@@Etevoldo 1 has only 1 factor
Prime numbers are called “simple numbers” in many languages, for example Russian
Me: “What’s the equation for the gap? What is it? Tell me!”
Grimes: “Culture.”
Me: “Oh.”
What is the sequence of unsequenced numbers?
(I'm just trying to Russell up trouble, here!)
I want to give this fine fellow a hand shake for all he's taught me via these videos.
This might sound like a weird question, but is there a way to express numbers without using a base? Because from this video it would appear that whatever base you use, certain numbers will immediately take on a significance purely because you are expressing the numbers in that particular base.
In other words, can any significance that the base has be filtered out when looking at the "interestingness" of numbers?
From what I understand from other Numberphile videos, mathematicians usually only consider a discovery serious if it works in ALL bases. Otherwise it seems like a recreational math trick, something just for fun. A lot of the integer sequences on the OEIS *are* just for fun, so no surprise that they include visually interesting base-10 stuff. Sorry you had to wait 100 years for a reply, lol. (In binary.)
It depends on how pedantic you get about what a "base" means. There's unary, which in some sense is base 1, but it doesn't really behave like other bases as there's no concept of "place value". Unary is just having a mark for every bit of value. So, 1=1, 2=11, 3=111, 4=1111. When you add them, rather than having to do some computation, you just concatenate them. So, 11 + 11 = 1111 (2+2=4 in unary). But of course multiplication can be a bit slow if you want to work purely in unary. You can sort of just make a multiplication table. if you want to get the result of 111 * 1111 (3*4), just make a rectangular array of 1's that is 111 wide and 1111 tall and concatenate all those 1's together and you have your answer.
And if you're working purely in unary, counting becomes a kind of pointless exercise as if I asked you how many 1's are in: "111111" and expected you to answer in unary, you'd just repeat back what I said, "111111".
The OEIS sequences all have tags that describe them, one of them being "base", which means the sequence is dependent on the base used. Surely it wouldn't be difficult to filter these sequences out, should someone try.
It"s always nice to throw around theories and have them proven or disproven so you can expand your knowledge base. Thank you.
I'm curious to know what kind of gap we'd see if we get rid of all the sequences that pay heed to the number's base.
Me too, I'm not really interested in sequences that only apply to base 10. I usually skip over those videos on Numberphile.
+
love your videos, well done Brady for all the channels and James is one of my favourites. Inspiring stuff guys.
is it just on the video or does it seem that on the bottom of the right side of the curve the numbers form some horizontal lines? Is there a reason for this? Hmmm, maybe just coincidence... ;)
This is what is known as the difference of two squares. It is useful when factoring two-term polynomials.
a^2 - b^2 = (a+b)(a-b)
In your example, (x^2) - (1^2).
(a+b)(a-b) = a^2 + ab - ab - b^2
= a^2 + 0ab - b^2
= a^2 - b^2
That is why it works, as you were wondering.
There are 3 numbers at the far right of the graph, between 8000 and 10,000, that only appear in sequences once each? I'm really curious to know what numbers those are!
Also, at the far far left end of the graph, there is a huge gap (due to the log scale of the y-axis) between the first few numbers and the rest of the set... what are the numbers on either side of that divide?
Actually, is this data series published anywhere for public download?
Joseph Fernando Those were my thoughts exactly when I saw this.
Cannot believe he recommended that book. I just finished it a week ago. It seems like it's popping up everywhere.
All numbers are interesting! Consider the opposite: Some numbers are uninteresting. Of that set, what number is smallest? That is now interesting. QED
Nah. the smallest number of an otherwise uninteresting set just isn't as interesting as the least interesting prime number fits the sequence n^2 - 1. Which is, of course, 3.
There are those who say none of the numbers are interesting, of course, but then those people aren't mathematicians making number sequences, now are they?
IIRC there's a process of transforming a number so: If it's even, divide it by two. If it's odd, multiply by three and add (?) one. If it eventually ends up at 1, it's interesting.
It was 30 or so years ago that I heard this, so the details may be a little off, but all numbers tried, ended up as 1. But some blow up to be very big before it happens.
In any case, who's to say what's interesting? The successes of both Pokemon Go and Frozen indicate that tastes vary widely.
Numberphile jz posted a video about the Collatz Conjecture!
you have to love that guy on how much joy he has in mathematics
oh you like the top band, how cute... I prefer the boring numbers, you've probably never heard of them.
I liked 1729 before it joined the top band. Back when it meant something. :)
There is no such a thing as a boring number.
I love how the paper is set in the font Computer Modern, which is the default font of LaTeX, the typeset language/program most scientists seem to write their papers in. :D
Someone ought to create the sequence of numbers that don't appear in the Online Encyclopedia of Integer Sequences.
Or, actually, better not. This might implode the Internet.
jakykong I can do it right here:
Every integer appears in the encyclopedia
great explanation, i wonder if people start focusing on the numbers in the gap, they will be more likely to discover newer connections more easily and quicker
I just got an ad for people who are bad at math... xD
2:55 He either means positive or nonnegative integers. There's an infinite # ways to right an integer as a sum of integers.
All whole numbers are interesting. Consider the set of uninteresting whole numbers. It must have a smallest member. That is interesting.
.
I like how the curve also appears Zipfian (and most likely it is a Zipfian distribution) :)
Ziph's law.
Thanks !Happy new year!
I don't find that very surprising, the sequences of numbers are neither distributed evenly, nor are they independent. They are representations of groups of properties people came up with, often leading to similar properties and series, thus establishing a dependence. What this really means is that for those numbers that are in the lower band, we have simply not yet found classes of "interesting" sequences yet.
That still doesn't really explain the gap though, just that there should be a range of numbers.
+Dennis Lubert I agree - that's the cultural bias. The easiest way to 'fix' the distribution is to add a second set of sequences which are the inverse of the others - add those into the distribution and it would flatline at 2x the current number of sequences. Which demonstrates the artificial nature of analysis of a very limited sample set, although the bias is valid.
Thanks for the book recomandation. Just finished it and was one of the best books ive ever read.
Sloan is integerist.
As a 256, I feel offended.
@VladHPsuper Gaming
It's a play on the "triggered" meme, which is centered around mocking "SJWs", leftists/intersectionalists, LGBA people, and trans people as irrational and discrediting their complaints about issues which affect them, such as misgendering, mockery of poor people or minorities, stereotyping and mockery of LGBT people, and so-forth.
The numbers remain the same regardless of the base you chose to represent them with; we use base 10 out of commodity, but these sequences of integers are the images (or 'results') of certain functions, ie, the fibonacci sequence is F:N->R (that is read as, the function grabs natural numbers and 'outputs' real numbers); F(n)=F(n-2)+F(n-1), for n in the Natural numbers, given that F(0)=0, F(1)=1.
The images of these sequences are independent of the m-base used to represent them.
13*13*13 = 2179 which uses te integers from 1729 :D
+relworP
*the digits
thanks
You meant 2197.
Is it possible that the curve has something to do with Zipf's law ?
+Tejas Naik I would think so! I was about to comment on it, glad someone else, though a year earlier, saw it. Interesting how natural Zipf is, isn't it?
so whats the best number overall
+0yeme0 -1
( ͡° ͜ʖ ͡°)
69
0
64
I find it interesting that apparently no numbers from 0 to 10,000 appear zero times, though 3 of them appear only once.
frikkin pleb numbers git gud scrubs
The formal definition of a Carmichael number says for 1 < x < n-1, so it specifically excludes x=1.
Dr. Grimes pupils are always so dilated he must always be tripping on acid.
Math, not even once.
Pupils dilate when someone is intellectually or otherwise aroused
This is a pretty well known algebraic principal, the difference of two squares. Basically, x^2-y^2 = (x+y)*(x-y). If you replace y with 1, you get your formula. If you apply the algebraic rule of thumb, first outer inner last (foil) to multiply (x+y)*(x-y) you get: x^2-x*y+x*y-y^2. The -x*y and +x*y cancel out and you get x^2-y^2.