Solving Seven - Numberphile

Рет қаралды 146,496

4 күн бұрын

Featuring Dr James Grime. See brilliant.org/numberphile for Brilliant and 20% off their premium service & 30-day trial (episode sponsor). More links & stuff in full description below ↓↓↓
More about 7's divisibility: • Why 7 is Weird - Numbe...
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James Grime on Objectivity: • Squaring The Circle (f...
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Пікірлер: 611

@numberphile 3 күн бұрын

See brilliant.org/numberphile for Brilliant and 20% off their premium service & 30-day trial (episode sponsor)

@FrankHarwald Күн бұрын

(this method really works because it's equivalent to computing the number mod 1001 which is 7*11*13)

@dunsparce4prez560 3 күн бұрын

We need a Numberphile video on why James doesn’t age

@raasttsaar9397 3 күн бұрын

He stopped counting

@imveryangryitsnotbutter 3 күн бұрын

He just counts his age in units of √-1

@Thrlta 3 күн бұрын

He's a vampire.

@eatpant1412 3 күн бұрын

Somewhere there's a painting of him slowly aging

@Mechanikatt 3 күн бұрын

Both "James" and "Grime" are 5 letters, so he'll forever be in his prime.

@unvergebeneid 3 күн бұрын

I have a heuristic way of figuring out if a number is divisible by 7. The answer is always "no" and you're only wrong 14% of the time!

@penand_paper6661 3 күн бұрын

Physicist detected.

@vampire_catgirl 3 күн бұрын

I like those odds!

@TheSheep1 3 күн бұрын

14.285714%

@wmradar 3 күн бұрын

"It's 80% accurate!" --yes I know it's 86, but let me make my references!--

@daemoneko 3 күн бұрын

14% of the time, it works every time

@yuhboi6816 3 күн бұрын

Instructions unclear - drew symbol, now my brother's soul is trapped in a suit of armour

@robbechristiaens6384 3 күн бұрын

FMA

@wmradar 3 күн бұрын

Yeah, mistakes like that cost you an arm and a leg. Thank you I'm here all week.

@Typical.Anomaly 3 күн бұрын

Necronomicon!!!

@peterbonucci9661 3 күн бұрын

@@Typical.AnomalyMake a sigil out of the path you take.

@ViolentSillyGirl 3 күн бұрын

Is your brother by chance William Afton?

@MatthewGlenEvans 2 күн бұрын

"I thought I invented a thing, but I didn't invent a thing" such a bittersweet feeling

@XaleManix 2 күн бұрын

Kind of like a magic square that isn't.

@electromika 11 сағат бұрын

@@XaleManixI wish there was a catchy word for that.

@CrashSable 3 күн бұрын

"So how do you see if a number (in base ten) is divisible by seven just by looking at it?" "Well, you divide it by seven and see if there's a remainder..."

@tBagley43 3 күн бұрын

you can also subtract multiples of 7 from your original number wherever you find them, and also change all 7s to 0s, 8s to 1s, and 9s to 2s if it helps. so for the number brady chose, I know that 3,714,289 will only be divisible by 7 if 3,000,002 is. so just start at 3, follow the arrows 6 times (taking you back to 3), and add 2 to get 5.

@Streunekater 3 күн бұрын

Nice optimization! Thank you.

@WWEMikano 2 күн бұрын

Brilliant!

@sabinrawr 2 күн бұрын

Okay that was actually pretty cool...

@sabinrawr 2 күн бұрын

3,000,002 is only divisible by 7 if 200,002 is, only if 60,002 is, only if 4,002 is, only if 502 is, only if 12 is, which it isn't. This reducing stuff is really fun. I feel like I'm running the division graph, just a bit inside-out.

@wmpowell8 3 күн бұрын

I actually derived this method myself while at a math competition. I knew I would be answering a question on modular arithmetic and could anticipate it being modulo 7. So before time started, I made a multiplication-by-3-modulo-7 table to aid me. It wouldn't be too far of a stretch to call this method "the universal divisibility test" since it can be generalized to any base.

@adamplace1414 2 күн бұрын

Alright, the red arrows being the way to multiply by 10 and keep the remainders got me. That "ohhh!" moment keeps me coming back to this channel as much as anything. Well done!

@Markus_8191 3 күн бұрын

"I have it on my phone." - brilliant!

@julietardos5044 2 күн бұрын

My teacher said I'd never have one of these things in my pocket.

@AaronFBianchiJupiter 3 күн бұрын

The fact that Brady has been using the notes from C to B instead of D to C sounds like he used the B Locrian scale and it feels like a sneeze that never comes

@jenkinsjason 3 күн бұрын

Tells us you have perfect pitch, without saying you have perfect pitch. 😊

@WhiteRussianDolls 2 күн бұрын

That would be an incomplete Dorian scale.. :( It should be C to C.

@James2210 2 күн бұрын

Especially when it goes down an octave instead of finishing the scale

@Tinybabyfishy 2 күн бұрын

I actually thought this was a really nice touch though! If the scale went to the root/octave again the resolution would make it feel like the process is complete. The lack of resolution/keeping the tension makes us feel the fact that the process continues, even without a visual cue. I get what you mean though, the lack of resolution definitely can be jarring.

@RobinDSaunders 2 күн бұрын

They could have resolved to a C [*] by starting with one before beginning to count, then climbing one note at a time as usual - but that might've been pretty confusing, by implying there was one more number than there actually was. Either way, kudos to whoever thought of adding scales to the illustrations. Both periodic patterns of seven objects, it's a neat idea :) [*] or the tonic of whichever major scale - I definitely don't have perfect pitch!

@DanielTompkinsGuitar 2 күн бұрын

All the pinging sounds for the numbers around the circle animations make a C major scale, and since there are only seven numbers, we never get the 8th number bringing us back to C. So each illustration left us hanging-musically. I kept mentally singing the C note after each illustration to complete the octave 😂

@monstronamaguederaz 3 күн бұрын

Legend says this method was discovered by a man called Stephen Strange.

@riuphane 3 күн бұрын

I knew it looked familiar!!!!

@Hamuel 3 күн бұрын

I was thinking that this would be a fun way to design a window

@kjh23gk Күн бұрын

Not James Rhodes? 🤣

@dionlobo4013 3 күн бұрын

I'm in awe of how someone discovered this

@avantesma1 3 күн бұрын

This is an application of modular arithmetic, which is a hugely important part of Maths as a whole. 🙂

@Kyle-nm1kh 2 күн бұрын

When you're a mathematician, things just happen

@Z3nt4 2 күн бұрын

It's not as if someone came up with the diagram all on their own by figuring out if you connect the numbers some magical way and do the thing you can go and tell whether a number is divisible by 7. This is just a flowchart of a very simple modular arithmetic algorithm conveniently shaped like a clock, which in turn just happens to be a modulo 12 graph. Or, Gauss did it. Whichever one you prefer.

@jaspermay5813 3 күн бұрын

3714289. Closest two digit number from the beginning divisible by 7 is 35. Remainder is 214289. 21 is divisible, so ignore it. Ignore the 42 as well, and you're left with 89, which is 5 more than 84. Seems easier than this ingenious circle trick.

@CorrectHorseBatteryStaple472 3 күн бұрын

This is just long division!

@phiefer3 3 күн бұрын

@@CorrectHorseBatteryStaple472 So is the circle, technically.

@PhilBagels 3 күн бұрын

Yes. I thought it was a bit funny that he would choose a number with so many multiples of 7 in it. You could take them out and replace them with 0's. Take out the 7, the 14, and the 28, and you're left with 3000009, which will have the same remainder.

@jaspermay5813 2 күн бұрын

@@CorrectHorseBatteryStaple472 Basically yes, except you don't keep track of the result.

@Kyle-nm1kh 2 күн бұрын

It's called short division

@quinn7894 3 күн бұрын

James Grime is back!

@javen9693 3 күн бұрын

They call him Dr. Grime cuz his math skills are nasty 🔥

@Z3nt4 2 күн бұрын

Dropping division bars.

@woody442 3 күн бұрын

The mesmerizing graphs are the best part of this. The math is nice but the visuals are the cake on top.

@RobinDSaunders 2 күн бұрын

As well as working for any divisor, this method also works for any base. But when the divisor is prime, there are close relationships with finite fields, and in particular discrete logarithms which have applications in cryptography. Kudos to whoever thought of using a musical scale for the illustrations, presumably because it's another periodic pattern of seven objects. It's a neat touch!

@deliciousrose 3 күн бұрын

I never clicked so fast! ETA: I find the pattern is fascinating and wondering what it would like for bigger numbers. Found a blog writing about it showing beautiful patterns, thank you for introducing this ❤

@nicomal 3 күн бұрын

Link, please

@LesCish 3 күн бұрын

By pattern do you mean the symmetric (n-1)-sided star, incomplete at its base? And yes, link, please.

@zach.0 3 күн бұрын

I tried drawing this and accidentally summoned a demon

@mal2ksc 13 сағат бұрын

You're supposed to stand OUTSIDE the circle and do the summoning INSIDE the circle. Live and learn!

@kedarsharma487 3 күн бұрын

It's the same as keeping track of only the remainder while dividing. This can be extended to any number

@Orange_Affinity 3 күн бұрын

Yeah, it's modular arithmetic at its core.

@V4ndalite 2 күн бұрын

Was gonna say, isn't this just long division? If you want to skip any kind of mental math other than adding one at a time, you could append a second set of numbers around the circle that represent that digit of the quotient... From the top going clockwise you'd have "+1", 1, 2, 4, 5, 7, 8. The "+1" is there for when you start counting around the circle clockwise. If roll-over the clock from 6 to 0, add 1 to the most recent digit of your quotient. All other properties discussed here would be the same... you'd still land on the remainder when you were done, but now you've not only figured out divisibility, you've also found the quotient.

@DaTux91 2 күн бұрын

"I thought I'd invented something but turns out it already existed." - Nearly every 19th century mathematician after checking Euler's collected works.

@smylesg 3 күн бұрын

2:07 It's two away from a multiple of seven (timestamp concurs).

@danielpretorius2430 3 күн бұрын

What do you mean

@kaisle8412 3 күн бұрын

@@danielpretorius2430 James says the number is "5 away from a multiple of 7," but really the closest multiple is only 2 away (2 more, or 5 less, are both multiples of 7).

@Pedro999Paulo 3 күн бұрын

This 7-divisibility method appear in a maths olympics when I was 12, nostalgic to see it again.

@OneTrueBadShoe 2 күн бұрын

OG Numberphile! James Grime is wonderful.

@shinigamisteve5607 3 күн бұрын

Beautiful! I also like that you only have to show half of the clock, as the rest of the clock is symmetric. That is because for mod n, 10k is congruent to n - 10(n - k)

@uniqueName496 3 күн бұрын

Thanks!. I was wondering if it is always symmetric, now you've confirmed it

@jackmojo 3 күн бұрын

Snape: "Turn to page 394." Harry: "But that's not evenly divisible by 7..."

@ATMLVE 3 күн бұрын

Can't hear the number 394 any other way

@marklonergan3898 Күн бұрын

"you can do it in your head - multiply by 10, add the next digit, multiply by 10, add the next digit" I mean, that essentially is how you do long division already. Find the remainder, append a further digit onto the end, that is just multiplying by 10 and adding it.

@lukevideckis2260 3 күн бұрын

division graph for 10 makes sense: edge i->0 for all i in [0,9], so basically multiplying by 10 get's a remainder of 0, the last digit has to be 0

@Erichwanh 3 күн бұрын

Banana math person! Banana math. Banana.

@PunzL 3 күн бұрын

Singing banana even 🍌🎵

@Zejgar 3 күн бұрын

S 2(ing) Ba 2(na)

@SeriousAlexej 3 күн бұрын

@Erichwanh 3 күн бұрын

@@Zejgar 2Ban2Nana

@agrajyadav2951 3 күн бұрын

Sus

@annesorensen4004 3 күн бұрын

😂 “you can do that in your head”… I dont think i . !!!SQUIRREL!!!! Can

@TaladrisKpop Күн бұрын

Since 10 has a reminder of 3 in the division by 7, it is simpler to multiply by 3 instead of 10 at each step (when doing the algorithm without the division graph)

@ricardolordelo3881 3 күн бұрын

I can’t “work” right now, but it seems promising, with lots of potential… Will definitely come back to this video.

@wtfpwnz0red 2 күн бұрын

I would rather carry remainders and divide by 7 in my head. This looks like so much extra work

@fhnn 3 күн бұрын

The graphic for this is great!

@dibenp 3 күн бұрын

0:28 I thought Brady was gonna start rattling off the digits of pi. 😂

@JeremyForTheWin 3 күн бұрын

Now i really want an animation of sequential division graphs morphing into each other. I'd be interested to see what someone with a geometry focus has to say about the graph layouts and how they relate

@JeremyForTheWin 3 күн бұрын

@@JorgetePanete no actually it's my feral lizard brain that is interested

@ngwoo 2 күн бұрын

I've played around and made a bunch of these, every single one is a pleasing design. I'd love to see that too.

@Bodyknock 3 күн бұрын

When working with modular arithmetic tables I like to shift the base numbers to be as symmetric around 0 as possible because it often makes the underlying symmetry stand out and can simplify the math sometimes too. So for instance, working modulo 7, I’d use the number -3, -2, -1, 0, 1, 2, and 3 instead of the numbers 0 through 6. This works because -1 is congruent to 6, -2 to 5, and -3 to 4 modulo 7, so the diagram you make is identical but with 6, 5, and 4 swapped out for -1, -2, and -3 respectively. But now you can immediately see why the diagrams in the video are symmetrical around 0, it’s because adding 1 is going to be the mirror process of subtracting 1, for example. Using negative numbers as your base also works well for example when you’re drawing up multiplication tables or power table for the same reason. Multiplying by -2 is the same as multiplying by 2 and flipping the sign, for example, so the columns for 2 and -2 will always be symmetrical in a modular multiplication table.

@stco2426 2 күн бұрын

Numberfile never disappoints.

@SaveSoilSaveSoil 3 күн бұрын

This is a beautiful way to look at things. Thank you.

@MindstabThrull 3 күн бұрын

Suddenly Matt Parker shows up in a Numberphile video where we don't even see him! Is that a Parker Video now?

@billcook4768 3 күн бұрын

If we bring in Matt, we have to worry about numbers that are “Parker divisible” by 7.

@MindstabThrull 3 күн бұрын

@@billcook4768 Would those be numbers that, when squared, are just shy of being divisible by 7? Parker and his squares...

@matthewcodd2939 2 күн бұрын

@@billcook4768reals between 7n +/- 1/2 are all Parker divisible by 7

@jasoncox7244 2 күн бұрын

The graphs are so pleasingly symmetrical too

@Matthew-bu7fg 3 күн бұрын

I love not only that James explains how this works but also why this works This is going to be so useful for me. So many lessons I do with upper primary students where I teach them how to find primes where I tell them to ask themselves whether a number is in the 2, 3, 5 or 7 times table by way of check. For those students that are not too savvy with their mental arithmetic, this will make a huge difference!

@nutherefurlong 2 күн бұрын

Really love this. Great illustration of of the mechanisms behind it

@zh84 3 күн бұрын

Splendid video. Thoroughly clear and entirely new to me.

@blackholesun4942 3 күн бұрын

Very cool diagram. Thanks for sharing 😊 Pi day was 3 months ago! Numberphile must have a large catalogue of pending videos

@backwashjoe7864 2 күн бұрын

Plot twist: the value of Pi is going to be used in an upcoming Tau Day video this week!

@cinderwolf32 3 күн бұрын

This is beautiful, I love it!

@walidability 3 күн бұрын

This is really wonderful

@amethystklintberg7436 22 сағат бұрын

Oh my goodness, I discovered this circle picture too! (As a way to investigate the divisors of Mersenne composites.) I’m actually doing my master’s in number theory on it right now! 🎉 Finally, I have a corroborating video to point to, that shows this diagram is helpful to mathematicians! 🙏

@NkThrasher1 Күн бұрын

Now we need a computerphile video talking about if algorithms like these are efficient for determining divisibility or if their value is primarily being human friendly.

@boRegah 3 күн бұрын

The thumbnail alone tells me, this will be a great episode Edit: It was

@ambrosia777 2 күн бұрын

Absolutely putting this into my arithmetic arsenal

@rage4dorder 3 күн бұрын

Incredible video!

@julianr2736 3 күн бұрын

Wow! I didnt expect this, this was a very nice discovery!

@LunaMail 3 күн бұрын

I can't wait to take this method to parties!!!

@samburgess7924 2 күн бұрын

I like that the 'multiply by ten' is simply a part of our base 10 system.

@tldreview 3 күн бұрын

His first sentence is filled with more energy than my entire existence

@itaipee 3 күн бұрын

Amazing simple visuals

@Roccondil 2 күн бұрын

I just love how symmetrical the sigils end up being!

@ajedrezandres11 Күн бұрын

this video is just too awesome

@matthewsaulsbury3011 2 күн бұрын

Wow, this is really neat! It's neat that the arrows come out symmetrical, too!

@martybando1668 Күн бұрын

An example of the brilliance that brought me to Numberphile years ago! I’ll attempt to replicate this algorithmically in python.

@marc-andredesrosiers523 Күн бұрын

Great job!

@maxprilmuller9461 2 күн бұрын

Really nice trick! Thanks. Easy to do, really surprising!

@csababekesi-marton2393 3 күн бұрын

At 0:42 I've solved the task quickly: I've set together the number of 3,714,289 as 3,500,000 + 210,000 + 4200 + 84 + 5. So, the number is not divisible by 7 and the remainder is 5.

@B.M.0. 3 күн бұрын

NICE! you can divide the place value by any number and continue down.... wow

@csababekesi-marton2393 2 күн бұрын

@@B.M.0. Thanks. Oh, come on, my 4th grade "method" simply beats the method presented in the video that's all. But I have to admit that the enneagram-like graphics are admirable.

@hamouz1999 3 күн бұрын

James is back ❤❤❤❤

@zsteinkamp 2 күн бұрын

Beautiful!

@FoodFestTelevision Күн бұрын

Another fascinating video 👍

@MridulGupta94 2 күн бұрын

What's amazing is that you can do it in any base, not necessarily 10.

@GGoAwayy 2 күн бұрын

Knowing how this works makes it easy to create a clockface with shortcuts for any number base.

@wbgookin 3 күн бұрын

I like the symmetry of these!

@DqwertyC 3 күн бұрын

The point around 7:45 about multiplying by 10 is really cool! Back in school, I tried coming up with a rule for 7s myself, and found that if you treat the number like it's base 3 (i.e. add the one's digit, then 3 times the ten's digit, then 9 time the hundred's digit and so on), then the resulting number would have the same remainder as the original when dividing by 7

@ChongFrisbee 3 күн бұрын

That was a great one! I just wanted to point out a further way of looking at that weel.. instead of seeing which digits multiplying by 10 goes, you can use the same resoning to see where the different powers of 10 lead you. 10->3, 100->2, etc. That way you can arrive at reducing the number to another with the same remainder by this weighted sum. As a bonus to make calculations easier, you can use "negative remainder" (for instance 10,000 = 1429*7 -3).. and also 999,999 is divisible by 7 so every 6 digits can be tought of separately then just add the remainders

@Thoran666 2 күн бұрын

A beautiful way to solve a division by 7.

@heaslyben Күн бұрын

Ah! So this was one of the macroprocessor elements involved in the PI computation! I love that.

@MasterJack2 2 күн бұрын

"We will show you how to know it's divisible by seven" "Step 1: conjure Satan"

@JRMAlabama 3 күн бұрын

About 45 years ago, as a 19- or 20-year old programmer, I had to divide a 14-digit account number by 7 to generate a check digit, which was the remainder. (I was programming in Microsoft BASIC on an 8-bit processor, the 8085.) Using the sequence that your arrows follow, 1-3-2-6-4-5 to multiply the acct # digits from right to left, one generates a number which is congruent modulo 7 to the original. The largest possible calculated number (each digit being a 9) was 414, which easily fit into a 16-bit integer and the division was then simple. Why does this work? Because the weights are the remainder mod 7 of each power of ten: 1 mod 7 = 1; 10 mod 7 = 3; 100 mod 7 = 2, etc. Same as the old "casting out 9s", for which every power of 10 modulo 9 is 1.

@YoavZilka 3 күн бұрын

We need a shirt with the division graphs for a bunch of “annoying” numbers!

@Utesfan100 2 күн бұрын

13x7x11=1001, so we can use a method similar to the 11 divisibility rule. 3-714+289=392-714=-322, which can be reduced mod 7 (or 13) in many ways.

@angelodc1652 2 күн бұрын

3-714+289=292-714=-422

@ChrisWalshZX 3 күн бұрын

You could make a graph for other bases (hexadecimal etc) if you build your graph by multiplying by 16 (10h) .

@raphaelgonzales3481 14 сағат бұрын

I love it ! You can also generalize it to other bases ! Instead of multiplying by ten, you can multiply by something else. Lets define C(n, k) the diagram for the divisibily by n in base k. It's an oriented graph. Open Questions : how many cycles in the graph ??? In base 10, there's 2 for Seven, 6 for Eleven, 3 for Thirteen... It seems totally random !

@5hape5hift3r 3 күн бұрын

I was looking into this just yesturday.

@r.a.o.laster5984 2 күн бұрын

So, I'm a big fan of base twelve. I made a base twelve number wheel and made the shortcuts like you did. It circles back on 0, 4, and 8, and 6 goes straight to 0. It's friggin beautiful, and I think it would be good material if you made another base twelve video.

@thelanavishnuorchestra 3 күн бұрын

This is intriguing. Fun video.

@topherthe11th23 Күн бұрын

0:36 - What I'd do with 3,714,289 if I had to do this in my head: Subtract 700,000 because it's an obvious multiple of seven (7 times 100,000). That leaves 3,014,289. Subtract obvious multiple of seven 14,000 (7 times 2, times 1,000) leaving 3,000,289. Subtract obvious seven-multiple 280 (7 times 4, times 10), leaving 3,000,009. Subtract 2,800,000 (7 times 4, times 100,000), leaving 200,009. Subtract 140,000 (7 times 2, times 10,000), leaving 60,009. Subtract 56,000 (7 times 8, times 1,000), leaving 4,009. Subtract 3,500 (7 times 5, times 100), leaving 509. Subtract 490 (7 times 7, times 10), leaving 19. Subtract 14 (7 times 2), leaving 5. The remainder for 3,714,289 is therefore also 5, because everything we removed was a multiple of 7, and so the sum of everything we removed is also a multiple of 7. If 3,714,289 has a remainder mod 7 of 5, then 3,714,289 minus 5 (i.e. 3,714,284) is a multiple of 7, and my calculator verifies that 3,714,284 is EXACTLY 7 times 530,612.

@topherthe11th23 Күн бұрын

In a way, that's really just the same as doing long-division EXCEPT you never write down digits of the quotient, keeping track ONLY of remainders.

@ibrahiymmuhammad4773 3 күн бұрын

the brilliance is that it shows the mind can be operated manually

@Hawksmoor42 3 күн бұрын

Initially I thought "OK, we're counting mod7..." As James described how the process works I realized "Oh, this is just long division with a graphical assist!"

@kanewilliams1653 2 күн бұрын

beautiful diagram!

@cleanerben9636 3 күн бұрын

this is so cool and clever

@davidioanhedges 3 күн бұрын

That this works for any number and 9, 5 and 2 are just special cases is enlightening

@RobinHouston 3 күн бұрын

The graph that James drew is essentially a finite automaton, and there is a lovely theorem (due to Stephen Kleene) which says that a set of strings that can be recognised by a finite automaton can also be recognised by a regular expression. So… that means there must be a regular expression that matches all the multiples of 7, and only multiples of 7. I defy you to write such a regular expression by hand! I calculated one from the graph, which I am sure you would enjoy very much; but sadly KZfaq will not allow me to post it. I am sorry. (I did post a video of the complete comment that KZfaq won’t let you see on the social network formerly known as Twitter, in reply to Numberphile’s post of this video.)

@pooiyx 3 күн бұрын

Please do another video on these division graphs covering how many types there are, what symmetries they have, etc.

@storyspren 3 күн бұрын

I love how it looks like some sort of magic circle you'd find used in a wizard's spellbook :D

@numberphile 3 күн бұрын

Don’t tell anyone.

@nightknight498 3 күн бұрын

The different patterns you get as you draw graphs for different numbers are really interesting to look at. Now if you'd excuse me, I'm going to sit down over there and draw a bunch of circles for no particular reason.