He really looks like he does these videos with passion for math, he looks like he was born for math and he's so happy that he went for math in his life.
This reminds me of a little math trick my private math teacher used to show me decades ago: 12345679 x 54 = 66666666... 12345679 x 63 = 77777777... 12345679 x 72 = 88888888... ...and so forth. Works backwards as well. (45, 36, 27, you probably understand this pattern by now) You basically omit the 8 and follow this adding pattern. I don't know whether that's a popular trick or not, but I thought it was cool!
I remember finding that really cool when I was younger, but know you can kind of see why which makes it less interesting 12345679 *9a = 1111111111*a and you can easily see that makes the repeated "a"
+Molun Another way to look at it also gives me some peace. Consider the simpler example of 1/81 = 1/(9*9) . If you do it in two steps, first dividing 1 by 9, you get 0.1111111.... Not at all odd, when you look at the pattern emerging by normal division. You divide 10 by 9 in each step, and get the REMAINDER 1. Which is why you'll have 10 to divide in the next step. Then, to get to the final result, you divide the endless row of 1's by 9. Now your first step yielding something non-zero, you'll be dividing 11 by 9, getting remainder 2. Next step, 21/9 leaves remainder 3. See the pattern? Yup, you get ...4,5,6,7, in the remainders. So why no digit 8 in the sequence. Because after the remainder 8, you'll be dividing 81 by 9, so you suddenly no longer have a remainder, and so the sequence starts over.
+Molun By the way, following my previous comment, I get a suspicion of what's going on. Could it be that te reason that I get the neat sequence, is that 9 is the smoking gun here? It's special in the way that it's the highest single-digit number in the decimal system. Boy... I have to go now. I must make some calculation... say, in base 12, and using 11 (which will be the highest single-digit number there)...
1÷(101×101)=0.00009802960494069208 It's funny because the number that comes after 9 decreases by 2 while the number infront of nine increase by 2. Weird
Not only that. Even two digit numbers decending from 98 and ascending from 02 (9802 9604 9406 9208 9010 8812 8614) all the way to 0298. Then starting at 0099 doing the reverse with odd numbers 9901 9703 back to 0000 and then repeating every 400 digits.
Considering that 1/9 itself is an infinite decimal expansion of 1's, it makes sense that dividing by a number consisting of a string of 9's would end up creating another infinite decimal.
If you call '0000000' repeating decimal (understand if you don't) then technically every rational number will end in a repeating decimal, especially since if you check the numbers in a different base it might not be 000000
Nicely done! Another fun thing to try, that's related, is x/(1-x-x²) = F₀x⁰ + F₁x + F₂x² + F₃x³ + ... where Fᵢ is the i'th Fibonacci number. If x is taken as a positive integer power of 0.1, say, 1/10ⁿ, you get Fibonacci numbers until they 'bump into' one another by having more than n digits. Also, what you've shown, is a way to get a counting sequence; the sequence of positive integers, with that 'jump' near the end. You can carry it another step, and get the sequence of triangular numbers by cubing, instead of squaring, the n-9's denominator. So 1/999³ = 1/997002999 = .000 000 001 003 006 010 015 021 ... And with a 4th power in the denominator, you get the pyramidal numbers, and so on...
Fascinating explanation! Years ago, when HP came out with the first scientific calculators, I found something similar while playing on my new HP35. It gives the powers of 2: 1/49, 1/499, 1/4999, etc.
Something else about the number 9 as well. If you add up all the digits in a number and that number can be divided by 9 then the bigger number can also be divided by 9. Example: 7011 >> 7 + 0 + 1 + 1 = 9 Proves you can divide 7011 by 9 = 779 Example.2) 52017102 5+2+0+1+7+1+0+2 = 18 then 1 + 8 = 9 Proof 52017102 / 9 = 5779678 The bigger number can be as long as you like. :-)
One of the best in the series, well done! There is a variant I learned at school: You type 12345679 in a calculator, then 'Give me a digit 1..9', say you select 5. 9 times 5 gives you 45. So you look clever and then multiply 12345679 with 45, and get all 5's....
The paper is brown because it is recycled paper, and paper is made of a renewable resource. Aren't you being a bit picky here? I mean I understand that we use more paper than might be renewable in terms of tree growth right now, but still it is recycled paper anyways.
I always hated math class and never felt like I learned anything no matter how hard I tried, but watching someone this enthusiastic about math and numbers actually make me want to learn more. I never had a teacher that made math engaging, here I feel like I am actually learning.
Every episode on this channel is like a rare kind of cigar or some quite classic whiskey, that you can just enjoy while you brood on the interesting facts of life, just having this in your little free time every day. Moreover, this is a rather healthy way of enjoying your moments
A small "proof" of the formula he stated: if you know that the geometric series is 1 + x + x^2 + x^3 +... = 1/(1 - x) for |x| < 1, than you can write it as a function f(x) = 1/(1-x). Take the derivative f'(x) = -1/(1-x)^2 * (-1) = 1/(1-x)^2. You can proof that the left side converges "uniformly" to the right hand side term (it is because it as a "power series" ) and when you have a uniformly convergence, you can take the derivative in every summand. so the derivative of the left side is 1 + 2x + 3x^2 + 4x^3 + ... There you go, that is the formula.
noticed this years ago when I accidentally hit the return twice dividing 1/9. Spent a while trying to suss it out and figured it was just another little mystery of the number 9. Nice to see that explained so well. But it is still a little mysterious :P
Another interesting thing happens with 1/997002. Every pack of three decimal digits, including the first one which would be the integer 001, is the double (and this multiplication by two is in fact the difference between 999 and 997, which is two, and is also the value of the last three digits which is 002; same applies to 998001, and so on) of the previous pack plus one. 1,0030070150310631272555120250511 (001 * 2) + 1 = 003 which is the first pack after the comma. (003 * 2) + 1 = 007 which is the second pack. (007 * 2) + 1 = 015 which is the third pack. (015 - 2) + 1 = 031 which is the fourth pack, and so on. With 996003, it results in the same, but multiplying by 3: (1 * 3) + 1 = 004 | (004 * 3) +1 = 013 | (013 * 3) + 1 = 40 | 040... and again. Very interesting stuff indeed, at least for me (I'm not a mathematician at all).
And also, the divisor is the one who makes the rules. The X digits packs are directly related to the divisor's lenght divided by 2. A 6 digits long divisor (998001, 997002 ...) generates 3 digits packs in the decimal side (6 / 2 = 3) -> first pack is 002. An 8 digits long divisor (99980001 ...) generates a 4 digits pack in the decimal side (8 / 2 = 4) -> first pack is 0002. It's still awesome :D
I didn't care too much about the recurring thing or the camera work, but in a few minutes of watching this I finally understood what infinite sets are all about, so I have to say great video
The brown paper is perfect. Functional and pedestrian. As to why a problem goes viral, we have to consider the nature of fascenation, of lacking some element of understanding and to some degree being outside complete comprehension. Once something (or concept, etc.) is thoroughly understood it becomes ordinary. You might be able to make a simple formula of this, Doctor. Would love to see your uncertainty equation.
I am happy for the internet to make this cool little fact so viral that Dr Grime explains it to me. The amazing fact is just awesome, but knowing the mechanism behind the fact is really cool. I love the pause before he said the word 'formula' like it was something you should not say in a company of ladies. And I could share his excitement when I made a division that ended with 0,31415926
Didn’t watch this video before now, and I’m happy I didn’t. I worked on this for a 24 hour project for math loving middle schoolers (along with decimal expansion proof, harmonic series and other convergent/divergent series). Decimal expansion proof was the most fascinating one and I got it after 6 hours!
Not sure if you feel that way because of the reasons connected to it being the last digit in our base ten (such as in this video). I mean their are other reasons why nine is an amazing number being three squared, it has many neat properties in it's own right.
123,456,789 ----------------- 9,999,999,999 will give us the decimal we're looking for. Now notice they the are both divisible by 9. For the denominator, this is easy to see. It comes out to 1,111,111,111 when divided by 9. The numerator is tricky at first glance, but try counting by 9's above 99. 108, 117, 126, 135, 144, 153, 162, 171, 180, 189. Notice how this sequence is a bit off from when you count by 9's below 99? Instead of 9, 18, 27 we get 8, 17, 26. So every time the ones places goes back to 0, there will be a shift. This pattern is key to why 123,456,789 is divisible by 9. If you divide it by 9 you will get 13,717,421. Interestingly this is almost a prime number, but it actually has two other factors, 3607 and 3803 which are both prime.
Yes, the easiest way to see the divisibility by 9 of the numerator, is by the technique of "digital roots;" that is, the sum of the digits, 1 through 9, is 45, and the sum of *its* digits is 9; so the original number is divisible by 9. Dividing by 9, gives what you've shown, and another digital-root check shows that it is now indivisible by 3. So then it's on to larger prime factors, as you say, which in this case, there are none that divide both terms.
And if you do 1/FFE001 a similar thing happens in base 16. Does this rule carry on for every base that 1/(b^n-1)^2 where b is the base and n is an arbitrary number gives you a number like this in that base? A little testing with WolframAlpha would seem to indicate yes.
Fun Fact: If you split the denominator of your fraction [your (((10^x)-1)^2)], you get both the missing decimal segment (8, 98, 998, etc.) and and the first "whole" number of the series (1, 01, 001, etc.).
These guys actualy put high definition WW2 footage into their video! geuss what? its actualy an interview with hitler himself! if you go to 4:45 you can hear him answer the question "do you like the jews?"
I just thought of a way to think of this: rep-digit numbers with 9s to the negative 2nd power include every number half as long as the answer to that expression except the first half of that square number. I realized it when James pointed at the 998 in 998,001. This sounded super complicated when I was typing it so 81 is missing the 8; 9,801 is missing the 98; etc.
Wow, I had noticed this sort of pattern just from messing around with my calculator way back in 7th grade or so. It’s cool how now, so many years later, I now finally know why it works the way it does.
I know this is an old video, but can you go through why 1/499 appears to be the powers of 2, in 3 digit increments (i.e. 0.002004008016032...)? I'd love to see why that is true; a little different from sequential digits. It appears to do some overlap after' 064'. Tried it out with additional and reduced number of 9s and all seem to be variations with different significant places like your video here.
Maybe you have already figured out the answer by now but here it is anyway:- First, Observe that 1/499 = 2/998 Next, 1/998 = 1 / (1000-2) = 1/1000 * [ 1 / (1 - 2/1000) ] Next, We can convert the term 1 / (1 - 2/1000) into a GP series as 1 + (2/1000) + (2/1000)^2 + (2/1000)^3 + ... = 1 + 0.002 + 0.000004 + 0.000000008 + 0.000000000016 + ... = 1.002004008016032064128256513026..... Finally, 1/998 = 1/1000 * 1.002004008016032064128256513026..... = 0.001002004008016032064128256513026..... Thus, 1/499 = 0.002004008016032064128256513026..... This idea carries over to similar patterns. For instance, 0.003009027081243731.... = 1/997 etc.
I'm gutted that none of my Maths classes were like this at school - showing you something INTERESTING and then explaining WHY it happens and showing you HOW to work it out for yourself is so, so satisfying compared to... "this is this formula. This is what you do with it. This will be on the test."
+Lisa B (Clo) Schools are terrible places - day prisons for kids. I only started learning when I left and studied at the local college. Absolutely hate schools
Just wondering: Is there any reason why every time i multiply 625 by itself, i get a number ending in 625? 625*625= 390625 625*625*625= 244140625 625*625*625*625= 152587890625 Is this some profound thing?
I'm not 100% sure of why it is, but it probably has something to do with the fact that 625 = 5^4 or 25^2, and every odd numbered multiple of 5 has 5 at the end of it.
because it has a pattern. Any power of 5 will end in 5. Any power of 25 will end in 25. Any power of 625 will end in 625. Any Power of 390625 will end in 390625. And the sequence of these numbers is 5, 25, 625, 390625 ...... and so on Alternatively, I should have written as 5^1, 5^2, 5^4, 5^8, 5^16 and so on.
It works in any base. B=base, B-1= your last symbol which is always the “carry on”, so B-2= your skipped number. So, 1 over (Bsq)-1 = repeating places using all symbols in your base except B-2. Try it!
What about 1/7? The decimals contain the first 3 even 2-digit multiples of 7 (14, 28, 42) and 1 more than the second 3 2-digit multiples of 7 (57, 71, 85). Is there a video for that yet?
MegaMinerd I see that different:(_ period) 1/7 is 0._142857 right now sort the numbers by their sizes->124578 2/7 will be the same like 1/7 but it starts with the next number (size): 0._285714 The same with 3/7 and....0._428571, 0,_571428, 0,_714285, 0,_857142
look up cyclic numbers. there is even a wikipedia article about the number 142857. it boilds down to a geometric series and can be done in principle with any fraction, but it looks rather nice with 1/7
Brendan Matthews I don't like your derogatory attitude. use your braina dn realize that some poeople not even know this, I didn't and I like math, some people are young or learning and these can lead to further study, something simple is not at all, too simple because it's something to learn for curious minds that ignore it, and great minds sometimes miss the simplest details because all their thinking is based in complex thinking, so simple is there as a eventually needed reference.
***** "Your proof is reliant on a mistake...... I dont know.. just saying it is a fairly unimpressive proof- I am no mathematician" :D 100/3 = 33.333333333.. is not a mistake, if so, make an opposite proof.:p the mistake to make is to assume the 9s in 0.999... end at one point,same for 0.333... because they do not. Hence 1 -0.99999999999= 0. If 0.99.. were smaller than 1 you would be able to put a number in between 1 and 0.999999 that is bigger than 0.99999... meaning the average of the two. but since this is not true, there is no number in between either.(1+.999999999)/2 = 1 Take an example that's smaller and works: 0.9 just 1 9. -> (1+0.9)/2 = 0.95 . 0.9
***** ye it's more logic after you get it :D. I also like how you get this string of natural numbers in digitals when dividing 1 by 81 :D! It's mysterious. :D Or the fact what you mention with the -1/12 thing :D. :D It's a funny thing!:D
The ancient approximation of PI of the Egyptian was 256/81.The result is approximately 3.16. The value 3.16 can be used to test the primality of a number. The method that I discovered using this value can clearly demonstrate the characteristics of prime numbers.
A mysterious and beautiful recurring decimal also results from 1/1089: 0.0009182736455463728191000... with its palindromic segment consisting of successive outer to inner pairs in the 1-9 natural number sequence summing to 10. James discusses the sequence 12345679 and asks why it's missing the 8. One consequence is that this sequence sums to 37 and is also exactly divisible by 37 (another cool number which is subject of a Numberphile vid). Finally a connection between 37 and 1089: reverse and add 1089 three times to get 40293, then divide by 37 and you get 1089.
What a great video! Everything explained was so interesting. The whole time I was wondering, why the 8? Really cool that you saved that one for the end :)
I feel like I could sit down and correlate this to electrical binary logic "memory errors" or maybe the right nomenclature would be something like "buffer over-run". Little bits of data carrying forward until finally it culminates in an error value.
Actually, the numeral 0 in England, is "nought," not "naught." Pronunciations of both of those are the same, and the meanings are close, but the latter one means, "nothing" in a non-numerical sense. As in, "After a few hours, the furnace contents were naught but ash."
i was gonna ask if this was a re-upload because i remember seeing it before, then i realized i was not at my subscriptions box, i was at the initial page with all the recommendations. oh well, it was worth watching again.
He seems to be the happiest person alive
+DASMARC You can count on that, or maybe he can......
He really looks like he does these videos with passion for math, he looks like he was born for math and he's so happy that he went for math in his life.
+Oryon
*meth
+DASMARC I really dig this guy. I'm just starting to watch these videos but his excitement gets me into it, and I usually hate math, haha.
look at this guy
nein, nein, nein, nein, ..., nein
+Bahadır Onur Güdürü im here
Lol
Plüschi he was saying that as i read i lol
Hans....
Was about to upvote, but the fact that the comment is 999 makes me want to let this pleasure to someone else
This reminds me of a little math trick my private math teacher used to show me decades ago:
12345679 x 54 = 66666666...
12345679 x 63 = 77777777...
12345679 x 72 = 88888888...
...and so forth. Works backwards as well. (45, 36, 27, you probably understand this pattern by now) You basically omit the 8 and follow this adding pattern. I don't know whether that's a popular trick or not, but I thought it was cool!
it is cool. At least I think it is.
I remember finding that really cool when I was younger, but know you can kind of see why which makes it less interesting
12345679 *9a
= 1111111111*a
and you can easily see that makes the repeated "a"
rgqwerty63 Still a neat little algorithm.
6666666606... *
Why is 8 excluded?
10 (the director): "Hey 9, keep that one for me"
9 (the manager): "Hey 8, keep that one for me"
8(the trainee): "okay"
brilliant
@@notlsa hello notlsa geometry dash
@@notlsa hello notlsa geometry dash
@@notlsa hello notlsa geometry dash
@@notlsa hello notlsa geometry dash
who else cries themselves to sleep
+Molun I know right
+Molun Another way to look at it also gives me some peace. Consider the simpler example of 1/81 = 1/(9*9) . If you do it in two steps, first dividing 1 by 9, you get 0.1111111.... Not at all odd, when you look at the pattern emerging by normal division. You divide 10 by 9 in each step, and get the REMAINDER 1. Which is why you'll have 10 to divide in the next step. Then, to get to the final result, you divide the endless row of 1's by 9. Now your first step yielding something non-zero, you'll be dividing 11 by 9, getting remainder 2. Next step, 21/9 leaves remainder 3. See the pattern? Yup, you get ...4,5,6,7, in the remainders. So why no digit 8 in the sequence. Because after the remainder 8, you'll be dividing 81 by 9, so you suddenly no longer have a remainder, and so the sequence starts over.
+Molun By the way, following my previous comment, I get a suspicion of what's going on. Could it be that te reason that I get the neat sequence, is that 9 is the smoking gun here? It's special in the way that it's the highest single-digit number in the decimal system. Boy... I have to go now. I must make some calculation... say, in base 12, and using 11 (which will be the highest single-digit number there)...
I really want to like the comment but I can't (look at the likes).
Satis-fraction plz plz
0:55 Lol skip a few: 007...996
IM LAUGHING SO HARD 😂😂😂😂😂 and this is like years ago 😂😂😂😂😂😂
"""""a few"""""
1÷(101×101)=0.00009802960494069208
It's funny because the number that comes after 9 decreases by 2 while the number infront of nine increase by 2. Weird
cool
lies
that name........
+Filip, hover above that name and read what the pop up says. hahaha this dude!!
Not only that. Even two digit numbers decending from 98 and ascending from 02 (9802 9604 9406 9208 9010 8812 8614) all the way to 0298. Then starting at 0099 doing the reverse with odd numbers 9901 9703 back to 0000 and then repeating every 400 digits.
Considering that 1/9 itself is an infinite decimal expansion of 1's, it makes sense that dividing by a number consisting of a string of 9's would end up creating another infinite decimal.
At least at the beginning.
@@yaboi7034 are you sure about taht
If you call '0000000' repeating decimal (understand if you don't) then technically every rational number will end in a repeating decimal, especially since if you check the numbers in a different base it might not be 000000
So your saying that 7 didn't ate 9
my life is a lie
+Marcin Gołda i see what you both did right there too
+Joseph Victor I see what you three did there.
+Abdumalik Mustapha i see what you four did there
+Lucian Crosby I see what you FIVE did there.
+NoahVN I see what you six did there.
1+1=2
Holy cow!
Edwin Camuy And 2-1=1
Alejandro Matos Anguis Dude!
Obaida Daraghmeh loominarty
RedHairdo And one minus one is not at all :)
"Six, seven... Skip a few
Then we've got 996"
How
Why
Searched for this exact comment
@ The Fleepo | Hey, they just said that because doing the whole thing would be VERY BORING. So they intended that to be *fast forward until*.
Nicely done!
Another fun thing to try, that's related, is
x/(1-x-x²) = F₀x⁰ + F₁x + F₂x² + F₃x³ + ...
where Fᵢ is the i'th Fibonacci number. If x is taken as a positive integer power of 0.1, say, 1/10ⁿ, you get Fibonacci numbers until they 'bump into' one another by having more than n digits.
Also, what you've shown, is a way to get a counting sequence; the sequence of positive integers, with that 'jump' near the end.
You can carry it another step, and get the sequence of triangular numbers by cubing, instead of squaring, the n-9's denominator. So
1/999³ = 1/997002999 = .000 000 001 003 006 010 015 021 ...
And with a 4th power in the denominator, you get the pyramidal numbers, and so on...
Fascinating explanation! Years ago, when HP came out with the first scientific calculators, I found something similar while playing on my new HP35. It gives the powers of 2: 1/49, 1/499, 1/4999, etc.
Oh my Euler! Your comment is even more interesting than the thing in the video!
that awkward moment when he starts speaking german 4:45
It's English. 9 != nein
you know I was kidding, right? it's important to me that you know that
+Prerendered renderation No 9! = 362,880
+Aamir Quraishy != means not equal to LÖL
+0 you must be fun at parties
>My satisfaction when I actually understand why the 8 is missing
Tim Fischer I love that the answer is literally the trope of 'carry the #'
My satisfaction when I see Asterix
Pingu's lost cousin right there 0:28
4:46 And then he became German
Nein! Nein! Nein! Nein! Nein! Nein! Nein! Nein! Nein!
Doch! Doch! Doch! Doch! Doch! Doch! Doch! Doch! Doch! Doch!
When riddle says .999999999999 whatewer amount of nines 51% the speed of light
Something else about the number 9 as well.
If you add up all the digits in a number and that number can be divided by 9 then the bigger number can also be divided by 9.
Example: 7011
>> 7 + 0 + 1 + 1 = 9
Proves you can divide 7011 by 9 = 779
Example.2)
52017102
5+2+0+1+7+1+0+2 = 18 then 1 + 8 = 9
Proof
52017102 / 9 = 5779678
The bigger number can be as long as you like. :-)
I got 998,001 problems but a terminating decimal ain't one. Hit me!
you were adopted
But you do repeat yourself.
*_*Falcon PUNCH™*_* yeah that's right noobs I trademarked it hahahahahaha
One of the best in the series, well done! There is a variant I learned at school: You type 12345679 in a calculator, then 'Give me a digit 1..9', say you select 5. 9 times 5 gives you 45. So you look clever and then multiply 12345679 with 45, and get all 5's....
5:21 "That you will learn at some point"
*Learns in the video.*
lol
Lol
1/998001
Pretty easy formula ...
NOOT NOOT
since when did you turn into pingu?
+Xyko Yeah i always found the way he says 0 weird
They're British.
But he alternates between saying zero and naught at 6:52
knought, hes reptilian
I do that too, but it's well weird.
i know they love their brown paper, but seriously unless they're storing them like manuscripts after using them, why not dry erase board?
+jared carter I think they do store them and sell them online if you're a fan.
Chemicals are used in the production of every paper.
THEY'RE THE SACRED TEXTS
The paper is brown because it is recycled paper, and paper is made of a renewable resource. Aren't you being a bit picky here? I mean I understand that we use more paper than might be renewable in terms of tree growth right now, but still it is recycled paper anyways.
Also, dry erase boards are terrible.
I like that 111,111,111 x 111,111,111 = 12,345,678,987,654,321. It's a simple one but it's still neat.
+Hopesedge There was a thing called OTTO numbers but after a counter example was found such as 111111111111x111111111111 it was trashed
tommy chan Interesting, I had no idea. Thanks for the free knowledge!
+Hopesedge Thank you for appreciating it :-D
+Hopesedge 98,765-43,210= 55,555 and 43,210+56,789= 99,999
4:45 "NEIN NEIN NEIN NEIN NEIN NEIN!!!!"
I always hated math class and never felt like I learned anything no matter how hard I tried, but watching someone this enthusiastic about math and numbers actually make me want to learn more. I never had a teacher that made math engaging, here I feel like I am actually learning.
That is "nought" how you say zero! :P
That is "naught" how you spell naught! :P
Is it usually spelled "naught" in the UK? I'd only ever seen it spelled "nought" (which was not that often), and the dictionary has both.
They're both correct haha. I just saw the opportunity for yet another pun and took it
if you press the caption option on youtube, it spells it "nought"
"Naught" means to ruin, while "nought" is a circle.
why am i watching this on a friday night?
That would be a Sunday, time zones might make that a Sunday or a Monday, according to publishing date I am seeing. A Friday is not possible.
Because you're awesome! :)
coz you're smarter then those spending their night in a night club vomiting.
we're all nerds, so we don't have anything else to do
i'm watching this on a saturday night
he is the happiest nerd I have seen lol
Nerd is someone that values things that have no real life value. Like Video gamers. You can't call some1 nerd because he is a mathematician.
if I want to know that, i google it
John Milionis define life value
John Milionis Video games have no real life value? Actually, they do have one. It's called "fun".
Penguin236
yea, its like saying that movies, books and music have no value
🎼 Carry on my eighth spot sum. You'll be nine when we are done. 🎼
Ben Hudelson underrated
LOL
4:11 watch your fingers!
nein nein nein nein squared
"Let's get you more paper."
3:30
Get this man his brown paper, stat!
noot noot noot
Pepijn its naught not not
Hu huh . not not.
yee
Noombah
Noot
This guys enthusiasm made me binge watch Numberphile videos, even though I hated math in school
0.9999...
Repeating number is 9, so
0.999...= 9/9 = 1 right?
No, not at all. do math
Andrew Rollette Already did, show me what's wrong.
9/9 is 1.
0.9999999... and so on will never be one.
Ahh, alright. Thank you for clearing it up for me.
Apologies to the OP for saying things wrong.
Michail Bialkovicz there's nothing wrong. mathematically, 0.999... (recurring) is exactly equal to one.
I like how that number and repeated decimals have such an elegant explanation. Maths!
Никита Лубин math*
@@DepFromDiscord no only americunts say it like that. It's maths.
@@DepFromDiscord UK spelling
@@DepFromDiscord I know ”Math” is older, but someone has to put America back in its place.
Every episode on this channel is like a rare kind of cigar or some quite classic whiskey, that you can just enjoy while you brood on the interesting facts of life, just having this in your little free time every day. Moreover, this is a rather healthy way of enjoying your moments
A small "proof" of the formula he stated:
if you know that the geometric series is
1 + x + x^2 + x^3 +... = 1/(1 - x)
for |x| < 1, than you can write it as a function f(x) = 1/(1-x).
Take the derivative f'(x) = -1/(1-x)^2 * (-1) = 1/(1-x)^2.
You can proof that the left side converges "uniformly" to the right hand side term (it is because it as a "power series" )
and when you have a uniformly convergence, you can take the derivative in every summand. so the derivative of the left side is
1 + 2x + 3x^2 + 4x^3 + ...
There you go, that is the formula.
Thank you for explaining!
noticed this years ago when I accidentally hit the return twice dividing 1/9. Spent a while trying to suss it out and figured it was just another little mystery of the number 9. Nice to see that explained so well. But it is still a little mysterious :P
Another interesting thing happens with 1/997002. Every pack of three decimal digits, including the first one which would be the integer 001, is the double (and this multiplication by two is in fact the difference between 999 and 997, which is two, and is also the value of the last three digits which is 002; same applies to 998001, and so on) of the previous pack plus one.
1,0030070150310631272555120250511
(001 * 2) + 1 = 003 which is the first pack after the comma.
(003 * 2) + 1 = 007 which is the second pack.
(007 * 2) + 1 = 015 which is the third pack.
(015 - 2) + 1 = 031 which is the fourth pack, and so on.
With 996003, it results in the same, but multiplying by 3: (1 * 3) + 1 = 004 | (004 * 3) +1 = 013 | (013 * 3) + 1 = 40 | 040... and again.
Very interesting stuff indeed, at least for me (I'm not a mathematician at all).
And also, the divisor is the one who makes the rules. The X digits packs are directly related to the divisor's lenght divided by 2.
A 6 digits long divisor (998001, 997002 ...) generates 3 digits packs in the decimal side (6 / 2 = 3) -> first pack is 002.
An 8 digits long divisor (99980001 ...) generates a 4 digits pack in the decimal side (8 / 2 = 4) -> first pack is 0002.
It's still awesome :D
so satisfying to see a logical explanation to something initially presented as a quirky pattern in an arbitrary scenario!
I didn't care too much about the recurring thing or the camera work, but in a few minutes of watching this I finally understood what infinite sets are all about, so I have to say great video
Great explanation! The missing 8 was particularly cool.
The brown paper is perfect. Functional and pedestrian. As to why a problem goes viral, we have to consider the nature of fascenation, of lacking some element of understanding and to some degree being outside complete comprehension. Once something (or concept, etc.) is thoroughly understood it becomes ordinary. You might be able to make a simple formula of this, Doctor. Would love to see your uncertainty equation.
...do you mean "fascination"...
Yes.
James Jacocks I rather like your spelling though.
Duh.
Why...ur not cool
Dear Numberphile, get a whiteboard
Best of regards,
YT viewers
one viewer!
some people love the brown paper!
+Numberphile aw C'mon :D
+Tanzim Ahmed i love the brown paper
#BrownPaperTheBest
Brown paper 4 lyfe
I am happy for the internet to make this cool little fact so viral that Dr Grime explains it to me. The amazing fact is just awesome, but knowing the mechanism behind the fact is really cool.
I love the pause before he said the word 'formula' like it was something you should not say in a company of ladies.
And I could share his excitement when I made a division that ended with 0,31415926
Didn’t watch this video before now, and I’m happy I didn’t. I worked on this for a 24 hour project for math loving middle schoolers (along with decimal expansion proof, harmonic series and other convergent/divergent series). Decimal expansion proof was the most fascinating one and I got it after 6 hours!
9 is an amazing number!
and 6 too..
Nein
Giving us the real facts.
Not sure if you feel that way because of the reasons connected to it being the last digit in our base ten (such as in this video). I mean their are other reasons why nine is an amazing number being three squared, it has many neat properties in it's own right.
If you want to include the 8.
13,717,421
--------
1,111,111,111
Anthony Vanover how did u find that????
123,456,789
-----------------
9,999,999,999
will give us the decimal we're looking for. Now notice they the are both divisible by 9. For the denominator, this is easy to see. It comes out to 1,111,111,111 when divided by 9. The numerator is tricky at first glance, but try counting by 9's above 99. 108, 117, 126, 135, 144, 153, 162, 171, 180, 189. Notice how this sequence is a bit off from when you count by 9's below 99? Instead of 9, 18, 27 we get 8, 17, 26. So every time the ones places goes back to 0, there will be a shift. This pattern is key to why 123,456,789 is divisible by 9. If you divide it by 9 you will get 13,717,421. Interestingly this is almost a prime number, but it actually has two other factors, 3607 and 3803 which are both prime.
Yes, the easiest way to see the divisibility by 9 of the numerator, is by the technique of "digital roots;" that is, the sum of the digits, 1 through 9, is 45, and the sum of *its* digits is 9; so the original number is divisible by 9.
Dividing by 9, gives what you've shown, and another digital-root check shows that it is now indivisible by 3.
So then it's on to larger prime factors, as you say, which in this case, there are none that divide both terms.
ffggddss There's a game called "Nine Hours, Nine Persons, Nine Doors". For anyone that doesn't understand digital roots, try playing this game.
I immediately came to this video since I've wondered what 999x999 is since I was very little, and now have 998,001 memorized in my head.
I saw this number (1/998001) in Arkham Knight in the last Riddler room (Final Exam).
I always thought 7 ate 9 but I guess 10 ate 9
And if you do 1/FFE001 a similar thing happens in base 16.
Does this rule carry on for every base that 1/(b^n-1)^2 where b is the base and n is an arbitrary number gives you a number like this in that base?
A little testing with WolframAlpha would seem to indicate yes.
And 1/110001 in binary
Rule of thumb: When ever there is a propertly relatated to 9 in base 10, it mostly likely is also a thing in other bases as b-1.
Fun Fact: If you split the denominator of your fraction [your (((10^x)-1)^2)], you get both the missing decimal segment (8, 98, 998, etc.) and and the first "whole" number of the series (1, 01, 001, etc.).
"Nice number". That's a nice expression. Only someone into art of mathematics would say that.
I really like numberphile, I hope it sparks others interest in maths as much as it does mine.
These guys actualy put high definition WW2 footage into their video! geuss what? its actualy an interview with hitler himself! if you go to 4:45 you can hear him answer the question "do you like the jews?"
I just thought of a way to think of this: rep-digit numbers with 9s to the negative 2nd power include every number half as long as the answer to that expression except the first half of that square number. I realized it when James pointed at the 998 in 998,001. This sounded super complicated when I was typing it so 81 is missing the 8; 9,801 is missing the 98; etc.
anyone else got a Dirk Gently vibe from this guys and the way he explains things? :D
everything is connected
@@LaGuerre19 on a beach in the Bahamas.
You can be great with numbers... but I see much room for improvement in matter of paper usage :)
Wow, I had noticed this sort of pattern just from messing around with my calculator way back in 7th grade or so. It’s cool how now, so many years later, I now finally know why it works the way it does.
One of the most satisfactory things in the video id the fact that the audio is insanely nice even if he is in a classroom
I know this is an old video, but can you go through why 1/499 appears to be the powers of 2, in 3 digit increments (i.e. 0.002004008016032...)? I'd love to see why that is true; a little different from sequential digits. It appears to do some overlap after' 064'. Tried it out with additional and reduced number of 9s and all seem to be variations with different significant places like your video here.
Maybe you have already figured out the answer by now but here it is anyway:-
First, Observe that 1/499 = 2/998
Next, 1/998 = 1 / (1000-2) = 1/1000 * [ 1 / (1 - 2/1000) ]
Next, We can convert the term 1 / (1 - 2/1000) into a GP series as 1 + (2/1000) + (2/1000)^2 + (2/1000)^3 + ...
= 1 + 0.002 + 0.000004 + 0.000000008 + 0.000000000016 + ... = 1.002004008016032064128256513026.....
Finally, 1/998 = 1/1000 * 1.002004008016032064128256513026..... = 0.001002004008016032064128256513026.....
Thus, 1/499 = 0.002004008016032064128256513026.....
This idea carries over to similar patterns.
For instance, 0.003009027081243731.... = 1/997 etc.
nein nein nein nein nein 4:45
Downfall!
I didn't understand a single thing that was said but I did poop on myself during the video...
I'm gutted that none of my Maths classes were like this at school - showing you something INTERESTING and then explaining WHY it happens and showing you HOW to work it out for yourself is so, so satisfying compared to... "this is this formula. This is what you do with it. This will be on the test."
+Lisa B (Clo) Schools are terrible places - day prisons for kids. I only started learning when I left and studied at the local college. Absolutely hate schools
@@johnvonhorn2942 Yep 😒.
Just wondering: Is there any reason why every time i multiply 625 by itself, i get a number ending in 625?
625*625= 390625
625*625*625= 244140625
625*625*625*625= 152587890625
Is this some profound thing?
I'm not 100% sure of why it is, but it probably has something to do with the fact that 625 = 5^4 or 25^2, and every odd numbered multiple of 5 has 5 at the end of it.
Cory Mason Interesting!
It's about as profound as our solar system having exactly eight planets
because it has a pattern. Any power of 5 will end in 5. Any power of 25 will end in 25. Any power of 625 will end in 625. Any Power of 390625 will end in 390625.
And the sequence of these numbers is 5, 25, 625, 390625 ...... and so on
Alternatively, I should have written as 5^1, 5^2, 5^4, 5^8, 5^16 and so on.
Nitin Grewal dude, any odd powers of 390625 will end in 390625; however, any even powers would end in 890625
If I ever end up teaching math, I will start every day with something like this.
Hi miss Holt
It works in any base. B=base, B-1= your last symbol which is always the “carry on”, so B-2= your skipped number. So, 1 over (Bsq)-1 = repeating places using all symbols in your base except B-2. Try it!
4:44 "Nine over 11" i r8 9/999999999 m8
What about 1/7? The decimals contain the first 3 even 2-digit multiples of 7 (14, 28, 42) and 1 more than the second 3 2-digit multiples of 7 (57, 71, 85). Is there a video for that yet?
MegaMinerd I see that different:(_ period)
1/7 is 0._142857 right now sort the numbers by their sizes->124578
2/7 will be the same like 1/7 but it starts with the next number (size): 0._285714
The same with 3/7 and....0._428571, 0,_571428, 0,_714285, 0,_857142
look up cyclic numbers. there is even a wikipedia article about the number 142857. it boilds down to a geometric series and can be done in principle with any fraction, but it looks rather nice with 1/7
ah, yes An, my favourite number.
It also works in different bases of numbers too. 1/7^2 in base 8 comes out the same as 1/9^2 in base 10 for example
People are starting to use their brains and realizing how interesting, interesting science facts are.
Brendan Matthews
I don't like your derogatory attitude. use your braina dn realize that some poeople not even know this, I didn't and I like math, some people are young or learning and these can lead to further study, something simple is not at all, too simple because it's something to learn for curious minds that ignore it, and great minds sometimes miss the simplest details because all their thinking is based in complex thinking, so simple is there as a eventually needed reference.
*****
You're a bitter attention seeker loser.
You don't use your brain....
*****
You're a bitter attention seeker loser.
*****
You're a bitter attention seeker loser.
another proof for 0.999... = 1, if you want 0.99999 you would do 9 / 9, which is 1
+brandon carter or also this :D.
3 * 100/3 = 100
3* 33.33 = 100
-> 300 = 3 ( 3* 33.33)
300 = 3* 99.9999
100= 99.999
1= .999
:)
***** yea it is
*****
proof:
0.99999999(neverending)=x
0.0999999999(neverending)= 0.1x
->0.9x = 0.9
x= 1
1 = 0.9999999 (neverending..)
divide by 3 :
1/3 = .999999999999 /3
1/3 = 0.333333333333333
multiply by 100:
100/3 = 33.3333333......
*****
"Your proof is reliant on a mistake...... I dont know.. just saying it is a fairly unimpressive proof- I am no mathematician" :D
100/3 = 33.333333333.. is not a mistake, if so, make an opposite proof.:p
the mistake to make is to assume the 9s in 0.999... end at one point,same for 0.333... because they do not. Hence 1 -0.99999999999= 0.
If 0.99.. were smaller than 1 you would be able to put a number in between 1 and 0.999999 that is bigger than 0.99999... meaning the average of the two. but since this is not true, there is no number in between either.(1+.999999999)/2 = 1
Take an example that's smaller and works:
0.9 just 1 9. -> (1+0.9)/2 = 0.95 . 0.9
*****
ye it's more logic after you get it :D.
I also like how you get this string of natural numbers in digitals when dividing 1 by 81 :D! It's mysterious. :D
Or the fact what you mention with the -1/12 thing :D. :D It's a funny thing!:D
This kind of fascinating stuff is the kind of thing that makes me want to get a mathematics major
please do it for 1/7 and 7 multiples that's also an interesting mathematical series
Hey do you get the same effect if you convert it into hexadecimal?
watching this high
was a mistake
Why?
Isnt it mindblown?
an incomplete haiku
Not not 1 not not 2 not not 3...
He said "Naught"
the british say naught instead of zero
+Maniac of Doom
I'm british and I don't...
well some do at least
Mr. Rapenstein RUDE! I SAY ZERO AND I AM IN BRITAIN
The ancient approximation of PI of the Egyptian was 256/81.The result is approximately 3.16. The value 3.16 can be used to test the primality of a number. The method that I discovered using this value can clearly demonstrate the characteristics of prime numbers.
when u ask a german to lend u money 4:46
don t get me wrong. no problem with germans. only joking
This sounds like a kid's show at the start. NortNortNortNortNortNortNortNortNortNortNort.
*Nought/Naught
I wish he is my school teacher
1:30 shout out 👋
1-1=noot
Technically, 000 through 099 are not 3 digits they are 1 or usually 2 digits
All I hear
NIEN! NIEN! NIEN!!!!
Kenzo Qui its nein not nien😧
A mysterious and beautiful recurring decimal also results from 1/1089: 0.0009182736455463728191000... with its palindromic segment consisting of successive outer to inner pairs in the 1-9 natural number sequence summing to 10. James discusses the sequence 12345679 and asks why it's missing the 8. One consequence is that this sequence sums to 37 and is also exactly divisible by 37 (another cool number which is subject of a Numberphile vid). Finally a connection between 37 and 1089: reverse and add 1089 three times to get 40293, then divide by 37 and you get 1089.
Nice
6, 7 skip a few *continues at 996*
What a great video! Everything explained was so interesting. The whole time I was wondering, why the 8? Really cool that you saved that one for the end :)
Skip a few
skips 990 numbers
In other words, the reason the 8 is missing is because of decimal notation.
I feel like I could sit down and correlate this to electrical binary logic "memory errors" or maybe the right nomenclature would be something like "buffer over-run". Little bits of data carrying forward until finally it culminates in an error value.
why does he say not in place of zero
Maurizio Pavel British stuff
Maurizio Pavel It's "naught". It's like saying "nil" or "null" or "oh", just easier than saying "zero" over and over again
thanks
Actually, the numeral 0 in England, is "nought," not "naught."
Pronunciations of both of those are the same, and the meanings are close, but the latter one means, "nothing" in a non-numerical sense.
As in, "After a few hours, the furnace contents were naught but ash."
Get a whiteboard mate. Your paper is so annoying
How?
+Alex Martinez he writes on wallpaper
+eGilroy You didn't answer the question. Why do you find it annoying that he uses paper instead off a whiteboard?
it's called flipchart paper
This videos is just awesome!
Your love for numbers is unmatchable :)
i was gonna ask if this was a re-upload because i remember seeing it before, then i realized i was not at my subscriptions box, i was at the initial page with all the recommendations. oh well, it was worth watching again.