Cyclic Numbers - Numberphile
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- Опубликовано: 26 окт 2013
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142857 is the most "famous" of the intriguing cyclic numbers.
Featuring Dr Tony Padilla from the University of Nottingham - / drtonypadilla
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Memories of finding this weird number while playing with calculator. Thanks for this-
Same here. Also; it’s kind of cool, how the blocks of 2 digits are multiples of 7 (and also 14, and they always double): 14 = 14 * 1 = 7 * 2; 28 = 14 * 2 = 7 * 4; 56 = 14 * 4 = 7 * 8; but, because the next one would be 112, which is a 3-digit-number; so, the leading 1 carries over to the block: ”56”, which then becomes ”57”; and the next one, after that, would be 224; so, once again, the lead 2 carries over, and the (1)12 becomes ”(1)14”. The next one, after that, would be 448; so, once again, the lead 4 carries over, and the (2)24 becomes ”(2)28”. That’s, why it repeats, as well. 😎
His genuine excitement over this is honestly beautiful.
One thing to note is that it's always a cyclic permutation (as you might expect from the name) but it's a subset of permutations (which is the word he used in the video). Ex: 142857 -> 125748 is a permutation, but not a cyclic permutation. But general types of permutations never occur here, it's always cyclic; yet another amazing fact.
7:43 also 2 cyclists numbers outside the window
I was just about to comment that, but then you already had. XD
How convenient
more coming
give it to me daddy
Still waiting
daddy plz
how many numbers are divisible by 7 in 1-10? 1
how many numbers are divisible by 7 in 1-100?14
how many numbers are divisible by 7 in 1-1000?142
how many numbers are divisible by 7 in 1-1000?1428
... you get the picture :)
kingofdogs49 The number of times any number (y) divides another (x) is, brace yourself, x/y without the remainder. I think you just blew every 1st grader's mind.
I uh... think you were trying to point out that at every second step, the number is a multiple of 7? But the thing is, your conjecture fails at the very next iteration.
frtard I thought it was neat...
+kingofdogs49 Ahh I see. as you approach infinitey, you can say that "all" numbers are divisible by 7.
+kingofdogs49
1/2 = 0.5000000...
how many numbers are divisible by 2 in 1-10? 5
how many numbers are divisible by 2 in 1-100? 50
how many numbers are divisible by 2 in 1-1000? 500
how many numbers are divisible by 2 in 1-1000? 5000
... you get the picture :)
1/3 = 0.33333333...
how many numbers are divisible by 3 in 1-10? 3
how many numbers are divisible by 3 in 1-100? 33
how many numbers are divisible by 3 in 1-1000? 333
how many numbers are divisible by 3 in 1-1000? 3333
... you get the picture :)
Literally works for any number. Even decimals.
I noticed this when I was 12 in physics class. I used to impress friends by using it to give the exact recursive value of any integer divided by 7 in my head.
That's a pretty cool use for it!
This is like ASMR satisfying for the brain. Absolutely incredible.
Ahhh numbers. Like a spring breeze through my mind.
I am fascinated to see people so enthusiastic about maths... I wish i had that kind of enthusiasm during high school :)
Brady, may I make a suggestion?
You should do a video (or series of videos) on Slide Rules. Not only are they a cool old method of doing calculations, they demonstrate LOTS of mathematical concepts.
Logarithm properties, orders of magnitude, significant figures, etc etc
These things all but disappeared back in the late 70's when electronic calculators became affordable. They're still readily available on ebay (even the huge demonstration models), but their use is becoming a lost art.
Here from the future: he's done quite a few and they're great!
When I found it accidentally on my calculator , I thought I was gonna get a fields medal
good to hear!
I haven't noticed all the other things about the cyclic numbers, but I noticed the perplexity of 1/7 long ago.
I first saw that 14 was half of 28, so I kept going with that.
I saw that 28 was almost half of 57, but realized that if I used 28.5714... and doubled it, I'd get, exactly, 57.1428...
Found that all multiples of a permutation was just another permutation with the decimal moved.
I'm glad this video was made though; now I know why.
Cyclists passing the window at 7:40 as they talk about cyclic numbers. Must be some kind of meta-cycle.
Or Epi-cycles ;-)
Ha! I remember reading about this number (142857) in the Man Who Counted by Malba Tahan when I was like 13 yo. I've been fascinated by it ever since and when I started watching numberphile I hoped for them to explain facts about it. I still sit idle sometimes with a calculator and play with it. Thanks for the video Brady!!
man, I have EXACTLY the same story!
I first read about it when I was 7
⁰
I remember reading that pi is 22/7 in grade 5 or 6, then I found out that pi is irrational and that it means that the number doesn't end OR repeat. Then I sat down and decided it to calculate 22/7 by hand and found 3.142857142857... .
I learnt later that pi is ACTUALLY 3.1415926... .
That's when I found out about 142857.
Also I feel lied to about pi NOT being 22/7. :/
@@adarshmohapatra5058 that's actually something I didn't know, I guess it's a simple and fair approximation, thanks for sharing that 😁
thanks for watching
I used to remember the decimal for 1/7 = 0.142857 as "un-for-tu-nate fifty-seven". I suppose that applies here just as well.
I used to sing that number to the six-note repeating riff in Chicago's "Color my World".
I'm a simple man.
I see 142857, I click.
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Đýļåň àňđ Xğűţhĕ
Surely 9 is the smallest cyclic number
Yes
9×2=18 1+8=9
9×3=27 2+7=9
9×4=36 3+6=9
9×5=45 4+5=9
9×6=54 5+4=9
9×7=63 6+3=9
9×8=72 7+2=9
9×9=81 8+1=9
9×10=90 9+0=9
Technically 3 would be
@@anirudhsilai5790 no 3×2=6 which is not 3.
(10^1-1)/1, so it fits the pattern.
Heyy that's my favorite number, 142857!
What about -1/12
@@DanB-sh3wt Stfu
infty, blob.
Same
mine too, I even used it as a password on club Penguin, I think.
I can tell you that 'cause it's ded ;-;
1/97 also displays a cyclic number. And it's also the powers of 3, until they encroach on each other and mess up.
Muzik Bike - Geometry Dash and stuff let me guess: you searched up cyclic or were on one of cyclic' videos and this was in the related list
No actually, just boredom in maths class and noticing what the recurring digits of ⅐ were.
⅛
This is such an awesome video! Cyclic numbers are so cool, thanks for teaching us about them!
This channel makes me so excited to take a number theory class
Same as with the law of large numbers. Over infinity samples of some test, you can still derive some percentage of those infinity numbers that are technically successful. Infinity is a limit, not a number.
13 works a little different because its period is of 6 digits in stead of 12.
The cyclic must be (10^6-1)/13, which is 076923.
The multiplication works with 1,3,4,9,10,12 to give its permutation as a result.
The reason the cyclic numbers have length (p-1) is because the digits correspond to all non-zero remainders of mod p, and by definition there are (p-1) of them.
Also, n/p and (p-n)/p have digits shifted halfway from each other.
Of all your channels Brady, the people of Numberphile are by far the biggest enthousiasts :D
Is the percentage of primes which are cyclic the same for all bases or is there some calculable relationship?
+Matt Williams Even square bases have no cyclic primes. Odd square bases only have the cyclic prime 2. For bases that are not perfect powers, the 37.395% might hold.
That "one more quirkiness" , 857^2 - 142^ 2 = 714285, got me hooked and inspired me to go on performing the same operation. 714^2 - 285^2 = yeah 428751. However 751^2 - 428^2 = 380817, and in fact I didn't get another permutation until I reached 560196 (the 30th such number I derived) on which 560^2 - 196^2 = 275184.
(The rules are the square of the smaller 3 digit number is subtracted from that of the larger, and where the entire number consists of only 5 rather than 6 digits, such as 41769, it splits 2 on the left and 3 on the right, or as if it begins with a 0, so we then do 769^2 - 041^2)
Carrying on after that I got to 115479 at number 34. 115^2 - 479^2 = 216216, so that's where I had to stop. Phew.
Explanations welcome.
Matthew Schellenberg What about 587412? (Btw: I made a slip up in my comment above. 714^2 - 285^2 = 428571, not 428751. No wonder I went off on that wild goose chase. Then 571^2 - 428^2 = 142857 again)
Thank you for doing a video on my favorite number!!!
I love this guy! He reminds me of Wallace and his, obvious, love for what he's doing makes me happy.
I've long been amazed by this number, and we thank you for delving into helping us understand more about it.
This is the first of your videos ive watched and im sorta a number/math fanatic and that was amazing it interested me so much and it was amazing how that works and how you explain it. Thanks!
8:53
describes math pretty damn nicely
They all forget that 142857 squared = 20408122449. Which is composed by 20408 and 122449. BUT 20408 + 122449 = 142857. Illuminati confirmed ?
142857 squared is the same as 142857 times 142857, so it's explained
We all thought that “Illuminati was confirmed” at the demonstration of the start of the video, then he explained it right after, so I don’t think it’s any different ; P
@Zachary Hunter numbers do not have to be rearranged when multiplying by a number that can be written as 7n+1 where n is an integer
Notice 1×142857=142857
I kept thinking throughout the video: "Is there a formula to create these numbers?" and then there it was. Its really cool to see the reasoning behind it :D
You're right about those infinities being of equal cardinality.
If you could randomly select one of all primes, then there's ~37% chance that it will be a cyclic number.
I'm simply saying that it's possible to find a ratio of numbers in a sequence when there is the right kind of pattern, even if that pattern is infinite.
I think this is actually an advertisement for Sharpies and brown paper… xD
LOL
Those number sequences showing great properties are always so good-looking when represented on some sort of 'graph'
A very similar system, "Vortex Math" uses the same set up of the 9 digits around a circle, but the core idea in that system is to regular math but then always reduce everything back to the digital roots.... There are TONS of beautiful patterns and neat little tricks that can be found with it.
If you start at 1 and just keep doubling it goes: 1-2-4-8-7-5-~ (1 - 2 - 4 - 8 - 1+6 - 3+2 - 6+4, 1+2+8 = 1, and the pattern repeats forever. There's those same numbers again!
If you take the digit digital root of all the multiplication tables you end up with sets that mirror each other (as they also mirror each other across the a vertical line of symmetry, where the 9 sits... take your 2s and 7s tables: 2-4-6-8-1-3-5-7-9 is the reverse of 7-5-3-1-8-6-4-2-9. The 1-8, 3-6, 4-5, tables are all mirrors of each other. It's pretty neat. There's tons more to it!
This was the episode I was waiting for since Numberphile started. Loooove 142857..
Amazing, Brady!
I read about this like 10 years ago, this started my love for math. It's awesome.
I've been waiting for them to make a video about this for the longest time! Finally, a video about the wonders f the number 7
Hi numberphile, thanks for answering this question
Absolutely incredible
FINALLY a video on 142857 ! I love this number!
I did some messing around with this number, discovered the number was generated from two digit doubling of 7, with a third digit etc. overlapping.... 7 14 28 56 (+1)12 (+2)24 (+4)48 (+8)96 etc. becomes 7 14 28 57 14 28 56 (+1 carried over from the next doubling to make 57 again).... If you keep doubling 7 but only displaying 2 digits, with the hundreds and thousands and ten thousands, etc. overlapping the previous numbers it reads 714285 repeating to infinity (presumably).
I have absolutely no understanding of this....just love to listen to someone talk about something they are passionate about
Thanks Sho.
I always liked the decimal expansion of 1/7 , because of all the multiples of 7 that appear in it: 14, 28, 56, 112 , 224 , 448 , 896 , 1792. Basically, each time, the digits just carry over, so you get your: 0.142857142857... i.e. 56+1 = 57 , 12+2 = 14, 24+4 = 28, 48+8+1 = 57 (from the 96+17+1 is the extra 1 carry over), etc. It was a neat property.
142857 had always been my favorite number. So it's cool they made this video
your logic is impeccable, i can't believe i didn't see it.
And next to it being cyclic it appears that the decimals of 1 to 6 divided by 7 represent some tweaked table of multiples of 7-with-a-little-extra: 0,7(142857...); 0,14(285714...); 0, 28(571428...); 0,42(857142...); 0,56 (+0,01142...=0,57142857); 0,85(714285...). That's why I love 7. It comes handy when you're trying to do some estimates and calculate just a little faster than others around you.
Fascinating stuff.
That is truly amazing
Anytime ;)
Thanks for producing!
Seen it like 3 times. In fact, that's the reason I chose to get a couple slide rules for myself (they can be had for as little as $15).
I'd still like to see a series of videos about slide rules from numberphile, possibly with Dr. Grimes himself.
Love this stuff
love this vid! also rip lou reed
Soooo, amazing!
OMG. I've been using this number for years, and didn't know 10 percent of these properties!! Well done lads.
Numberphile Hey hi, I think you messed up the arrows at 3:23, it's a cyclic number but your arrows are showing a permutation not a cycle (though, of course, there is a cycle, its just your arrows are not showing that cycle)
Gurmeet Singh I noticed that too.
It's like they've missed the whole point; that the digits are in sequence.
Still, 5/7 marks for trying
jaaa... it´s an inocent mistake, but since that´s the whole idea of the video, I believe it is worth to fix it...
this is amazing. with this you can show, witch number can divided by 7 and witch dont
Good question. We can say that a number is cyclic if each cyclic permutation of the number is an integer multiple of the number.
Since we know that 142857 is a permutation of 428571, and it is not an integer multiple of 428571, it is not cyclic. You may notice then, that another corollarial property is that the first digit in the number has to be the smallest that occurs in the number.
However, since 428571 * n = 142857 * 3 * n, 428571 will exhibit some similar properties sometimes.
We need more of Tony Padilla! And the other classics, such as Grime, Matt Parker, etc.
I found most of these properties a day I was bored in a history class. I have to admit I had a thing for dividing numbers by 7 at the time. There was really something special.
Thanks!
omg.. this is too mindblow!!!
I found out about this via Neo: The World Ends With You. There is one character who commonly speaks with mathematical terminology.
I love how I always find this when dividing by 7
There is another little quirk about the number. You can see that the first two digits are 14 (7*2), the second pair is 28 (7*4). But the next number really 56 (7*8) however it is written 57 because the next number is 112 (7*16) and the 1 overlaps into the 6 in 56. the next number again is 224, which again, overlaps the 2 into the last number of the previous pair. It continues like this.
thank you very much sir
Yay! old fashioned brown paper vid! keep them coming :)
The way we notate numbers can be done in different bases. The standard is base 10 (or decimal), because we count 10 numbers before we increase the value of the next digit. If we were to count in a different base, say base 8, we would count 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21; which translated to decimal are the numbers 0-17.
Different bases are used for different things. For example: computers work because of their ability to process and store information in binary.
Wow. Well, done.
It has been decided that zero is even. This is due to a number of reasons- mainly that if it wasn't even, there would be lots of problems with many theorems and that zero actually has all the properties of an even number. The definition of an even number is that it can be divided by 2 and leave no remainder. Divide 0 by 2 and you still get zero, but no remainder. Thus, 0 is even,
Dr. Padilla broke my brain. Somehow he made it all make sense.
I'll have to sleep on this.
I just now realized how that worked... it is 1/7th less than exactly 1000000, meaning multiplication will always give similar permutations. I discovered that a while ago, but never put 2 and 2 together.
This video should only have 142857 views!
I'm just too old!! I immediately recognized the opening number as 1/7. Fascinating anyway!
I notice the dash through 7 in hand-writing, where a tilted 1 might look like a 7. Sometimes there is a line under 1 (like it is standing on a platform). It is just another way to make it perfectly clear to others what you are writing.
Awesome !
Memorizing this number became easy when I noticed that pressing 857 on a calculator is the same pattern as pressing 142, except upside down. And the two patterns slot neatly into a nice little block that makes my Tetris OCD happy.
Great video, I just wonder why some primes give cyclic numbers while others don't. What is it about these primes that their decimal repetitive sequence increases directly with its magnitude? I would have thought that this would work for all primes.
With the exception of the primes that divide the base, 10, namely 2 and 5, all primes have reciprocals that repeat infinitely; the ones they show are those p whose repeating lengths, L(p) = p-1.
L(p) is always a divisor of p-1; when it is < p-1, the multiples of the repeating part, R, will not always be cyclic permutations of R. Instead, there will be k different numbers of length L(p) whose cyclic permutations will occur in the multiples of R, where p-1 = k·L(p).
Simplest example: p=3. L(3)=1; p-1=2; k=2. The repeating parts are the single-digits 3 and 6.
More interesting example: p=13. L(13)=6; p-1=12; k=2. The repeating parts are R=076923 and R'=153846. Each of those appears in one of its six cyclic permutations, in the multiples of R from 1 to 12.
As for why the length of the repeating part of 1/p can't be any longer than p-1, think of calculating 1/p by long division. At each step there's a remainder. And it can't ever be 0, because that can happen only when p divides some finite power of 10, which can only happen when p is 2 or 5, which we've excluded.
So of the p possible remainders, only p-1 of them, at most, can occur. And so the division can't go more than that many steps before some remainder repeats. And when it does, the quotient, 1/p, must repeat.
Now once you get to that first remainder that = 1, after L(p) digits, that means that p divides (10^L(p) - 1).
But There's a famous theorem that says that p divides (10^(p-1) - 1).
And because of that, the remainder = 1 after p-1 digits; and the only way that can happen is for L(p) to divide (p-1).
Fred
That's amazing.
WOOT! you listened! You guys rock. :)
Absolutely mind blowing!!! :-D
it is very interesting. thank you
Yay, they finally did my request for my favorite number! :3
aaarrrrgg!!! why is your favourite 3!!!
You did not mention the prime number 137. 1/137 produces a cyclic number of only 8 numbers. 00729927. In fact when you add any of the numbers in a cyclic set, it always equals nine.
My time of birth is 14:28 (and I like to believe 57 seconds :P ). Waited for this video since I started watching Numberphile. Thanks, Brady. I love Your channel.
Mind blown!
You sir deserve a cookie
Wow, awesome indeed!
Let r be the repeating section of 1/p for any prime p (so for p = 7, r = 142857). You can cycle the digits of r to the left by n by calculating r * (10^n mod p). For example, to cycle 142857 to the left by 1, calculate:
142857 * (10^1 mod 7)
= 142857 * 3
= 428571.
7 produces a cyclic number because all the natural numbers less than 7 can be expressed as 10^n mod 7. Starting at n = 0 and increasing, 10^n mod 7 starts as "1, 3, 2, 6, 4, 5" and repeats.
This makes me curious about cyclic numbers in other bases! The list of primes for which the reciprocal gives the sort of repeating decimal expansion (is it still called that if it's not in base 10?) needed for a cyclic number should vary depending on the base, right? I'd imagine intuitively that there are some bases for which there are none, but are there any with a finite number of primes that do this instead of an infinite but particular list? I'm curious about what the relationship is between the base and which primes work.
noticed this while screwing around on my ti-83 during high school. always thought it was a cool sequence