This completely changed the way I see numbers | Modular Arithmetic Visually Explained

[@zachstar]

2:50 should be "For any composite number x one of its prime factors must be less than OR EQUAL TO its own square root." (the 'or equal to' part only would apply to primes squared but still needed to be included). I was so focused on my specific example and wasn't thinking lol. Thanks to those who caught it and hope you guys enjoy the video!

[@ratamacue0320]

Also "its" (as you have here), not "it's" (as in the video).

[@ratamacue0320]

5:18 should that be 9 modulo 7 = 2?

[@chessandmathguy]

@@ratamacue0320 the way it's phrased in the video is perfectly fine

[@zachstar]

@ratamacue0 both will work in regards to the 9 = 2 (mod7) or 9 (mod7) = 2

[@ratamacue0320]

@@zachstar I guess you're using it as a descriptor, not an operator.

[@Ratzfourtyfour]

This completely changed the way I don't understand numbers.

[@dogwithwigwamz.7320]

I agree. Why make understanding easy when you can make it hard ?

[@dogwithwigwamz.7320]

YT : "Euclid`s Algorithm." As of today ( October the 25th, 2019 ) click on the first video in the list - by "Learn Math Tutorials;" Its not my work. If it had been it would be called "Learn Maths Tutorials." To my mind it offers a far simpler explanation of the introductions to Modular Arithmetic.

[@charlesquarra5050]

this post has 365 likes, which equals the numbers of trips earth does around itself for every trip it does around the sun, while the moon does 1/28 trips around itself for every trip it does around the earth, since 28 * 13 = 280 + 28 * 3 = 280 + 30 * 3 - 2 * 3 = 280 + 90 - 6 = 364. Which means that the moon takes 13 trips around the earth plus one trip of the earth around itself for the earth to make a trip around the sun Hope that clears things up

[@lungflogger9]

agree, this makes NO sense. he may be factually correct but so what.....?

[@greatkingkay7954]

@Hassan Akhtar yea I saw perfect circles and rays coming from the center.

[@canadiannuclearman]

I was a machine designer for a few years number theory is geat for gear train design. Thanks for the video. I designed a concentric speed reducer once. The ratio was 6.0025 to 1. My boss said why not 6 to 1? I said because the square root of 6 is an irrational number. He asked why and i said because the number of teeth in the 1st gear is 20 the second is 49 thats on the same shaft as the 3rd gear that has 20 that drives the 4th gear with 49 teeth. Fun and interesting. Prime numbers with gears are cool too. If you have 2 gears with number of teeth 12 and 60 This means every tooth in the gear with 12 will match every 5th tooth and only that tooth per revelotion and not engage any others this increases ware on the teeth. But in the above 49 is divisable by 9 and 20 divisable by 2 and 5. There is no common prime between 20 and 49. Because 20=2×2×5 & 49=7×7. This means that each tooth of one gear will eventualy mesh with every tooth of the second gear. Therefore spreading ware over all the grear teeth.

[@zachstar]

I definitely learned something from this. Never thought about number theory being applicable to something like that but makes perfect sense. Thanks for sharing!

[@maxwellali]

Wow you fooking genius

[@johnyepthomi892]

Wow.. mind blown.

[@edstirling]

this is the most interesting thing i've heard in a while.

[@ntwede]

How does gear tooth wear depend on which tooth of the other gear it meshes with? I'd say it really only depends on how many times it contacts the other gear which depends only on the number of teeth in the gear (For a given number of rotations)

[@captainsnake8515]

Tip: if you’re a high schooler interested in competition math, modular arithmetic is one of *the* most important topics to study, since normal classes don’t tend to teach it much, but math competitions love modular arithmetic questions because they make for really interesting problems.

[@l1mbo69]

anyone interested would already know this

[@danpalu2308]

@@l1mbo69 what an arrogant comment. Surely someone who just got introduced to the concept of competition math mighy not yet now that. Heck, some may not even have known of competitive math until they read Captainsnake's post. And maybe Captainsnake's post thus inspired someone to take competitive math up, or maybe just inspired them to spend a little more time on learning math in the first place. So Captainsnake's post added value. But your post added only vitriol.

[@l1mbo69]

@@danpalu2308 first of all in countries with a screening round of sorts (US, China and India all come in this category) modular math really only becomes important after the first stage. So the comment holds true only for those that have already qualified or are confident will qualify the first stage. Just tautologically these people wouldn't be complete beginners. And even otherwise, i would expect someone interested in competition math to atleast go through the list of common topics, yk. If they haven't yet they will in the future irrespective of this comment. Otherwise they aren't really serious to begin with I don't really see how this could be taken as inspirational by anyone, but anyway my comment wasn't really supposed to be vitriolic but rather just casual, so sorry if it came off as so

[@l1mbo69]

@@danpalu2308 oh and i just remembered, this is not even true for all countries. Iirc, UK devotes a decent amount of time in covering modular arithmetic. Sounds pretty US centric to me

[@peamutbubber]

@@l1mbo69 not everyone goes to a good school or has access to good resources

[@dbaker280]

Holy shit. In 20 minutes you covered almost 70% of the topics on the syllabus of my number theory class.

[@alexv5581]

You must go to a shitty school.

[@CamMackay96]

Do you mean lecture? There's no way a whole semester course is covered by this video, this was about one lectures worth of material from my undergrad Number Theory course....

[@CamMackay96]

@No Name can't say I've studied fractals so idk what you want me to tell you, I'm perfectly willing to admit I have specialty areas, Number Theory being one of them. Since you brought it up, why don't you share with the class?

[@CamMackay96]

@No Name what does any of that have to do with fractals and primes...?

[@That_One_Guy...]

@@alexv5581 This is taught in College

[@wojocolebuilds]

The 12 spoke wheel reminds me of music theory and the circle of fifths, a model that visually represents harmony and dissonance between different tones of sound(music notes). The circle of fifths, comprised of the 12 notes of the chromatic scale, visualizes intervals that would fully revolve a musician around the chromatic scale. These intervals, despite whatever root note you start off with, are constant in all musical harmony and dissonance.

[@uaswitch]

it isn't incidental - if you work with addition on the entire spokes, then adding 1 repeatedly will cycle through every spoke exactly once with no repeats. The same thing will happen with adding 5 repeatedly, adding 7 repeatedly, or adding 11 repeatedly. If you view label the spokes as A, A#, ... through G#, then the adding 7 repeatedly is the circle of fifths, and the reason cycling through the circle of fifths involves every note exactly once is precisely due to the fact that 7 and 12 have no common factors. I use this example in my modern algebra class when we discuss cyclic groups.

[@shanaadams4456]

I put off learning modular arithmetic for so long because it looked dauntingly difficult. I can't believe it's this easy! Thanks for making stats much easier for me :)

[@DownWithBureaucracy]

This video game me flashbacks to math class. Started out understanding everything, feeling good about life, and then suddenly I'm lost. "So naturally we can see that..." no. No I cannot see

[@halasimov1362]

Reminds me of the harmony of 2 notes. Even when the 2 notes are moved too different octaves they still multiply and create a similar freq that would seem to fall in the same spoke if you will.

[@nell__byte]

That's what I was reminded of as well, though I was specifically reminded of the circle of fifths.

[@jamessloven2204]

Hi

[@insertname252]

“With that background you should now be okay with this theorem” Me:

[@funkahontas]

My calc II teacher in a nutshell lmao

[@seanhatton4013]

🤣🤣🤣

[@tinnnyz]

😂😂😂

[@lekhapratap1652]

My little bro asked me to explain to him tonight. Me: “I went to fucking music school. I don’t know number theory, kid.” I’m trying to understand but “excuse me. That’s like that asylum demon boss at the start of dark souls.”

[@waynethomas1726]

@@funkahontas I learned enough Algebra to be good with Trig but the more advanced Algebra, remembering the Quadratic equation...sucked! Of course it turned out that nobody wanted me to use my calculator to do anything. the computer could do it far more accurately by drawing the geometry. And then the classic "you don't use it you loose it" came into play. Only a few years out of the design field and I can't do trig either. This made my brain hurt.

[@bunberrier]

I cant find the wheel thing on my calculator.

[@XxBobTheGlitcherxX]

its the %

[@noobita4983]

Mod operator isn't available in most of the calculators

[@whatelseison8970]

@@noobita4983 It's possible to make the mod function from functions all scientific calculators do have. Namely arctan(tan(x)). They are both rising sawtooth functions. If you have a computer, and internet you have access to desmos. Google it. Use it. You won't regret it. It sounds like a fun puzzle. I wish I could be more help but my god... I'm just up way too late at this point. Good luck.

[@whatelseison8970]

@@noobita4983 x mod a = (a/pi)*arccot(cot(pi*x/a)). It can also be done the way I mentioned above but it's more work. I'm not going to type it out. See my work here: www.desmos.com/calculator/1btmjt4fdi

[@AlainNaigeon]

Thus you're the kind of guy who could NOT have written your calculator software.

[@siobhanbartz2688]

This video was super amazing. I now know that I am interested in number theory. You explain things in a way that all age ranges could understand. Honesty, I love your videos! Keep up the outstanding work!

[@margin-fades]

I’m new to mathematics. It couldn’t figure out why it is true that a number divisible by 9 also has the sum of its parts divisible by 9. This led me to modular arithmetic and your video. I haven’t absorbed it all in the first watch, but what I have absorbed has made numbers so much more interesting! Great teaching!

[@TheKradok]

2:08 was really confusing for me. I understood it as a! is not divisible by any number i where {i ∈ Z | i > a}. Completely missed that it's actually {i ∈ P | i > a}. After listening to it a bunch of times 2:36 made me realize what was going on.

[@basspuff514]

This is so fascinating. I love when seemingly really hard problems have clever solutions like this.

[@SM-qk7jv]

MajorPrep still making next-level videos. Keep up the great work.

[@wallonice]

@@fonfon575 His level is determined by a wheel with 23 sections, starting with 1. He got number 24

[@1.4142]

Who's majorprep?

[@MichaelPohoreski]

@3:56 An easy way to tell if an integer > 10 is divisible by 7 is to subtract twice the last digit from the remaining digits and check if the result is divisible by 7. Repeat if the result is > 10. For 119, rewrite as 11 - 2*9 = 11 - 18 = -7 which is divisible by 7.

[@Uthael_Kileanea]

THANK YOU!

[@michaelfruge421]

A professor once made us write out our work on graph paper. One character per cell. If the character drifted out of the cell, the grade was a zero. He specified every single minute detail. It was quite controlling. However. He didn’t specify what number system. I wrote the entire problem, and solution in Roman numerals because he didn’t specify Arabic numerals. He returned my paper with: “Touché 100”

[@Jacob-ye7gu]

He should have just given you a "C" to mess with you

[@m3xikanolokoruiz]

"and then the whole room gave me a standing ovation, and the hottest girl in class asked me to marry her"

[@CZghost]

Definitely a mad lad :D

[@jazz4dayz543]

@@m3xikanolokoruiz Bruh, I've had professors as he describes myself. It's not such an unbelievable story..

[@1oo1540]

m3xikanolokoruiz you tell all the best stories, I hear you’re super fun at parties

[@Nomenius1]

This would be incredible if I could remember it all the time

[@missionpupa]

Math is not about rote learning. Its about understanding. If you understand it, you never have to remember anything.

[@Tobable]

Feralz primes.

[@shanaadams4456]

Right? lol

[@MMABeijing]

first make sure you actually go over the content slowly and let your brains digest the finer details

[@shanaadams4456]

@Hassan Akhtar We all could've continued on happily if you kept your negativity to yourself. Why did you feel the urge to write that idea down and share it with the world? Just to belittle someone you don't know.

[@bobminion3438]

Hi @Zachary while watching your video i converted whole calendar into single 7 spoke wheel arrangement. Now i can easily predict dates, on which day it falls.( _Although there exist an algorithm but this visualization helped me_ ) Thanks for such intuitive videos 🙂

[@user-zu1ix3yq2w]

Incredible

[@varimas]

Thank you for showing me this application.

[@SeeMeOnTheTube]

How?

[@rimfethi198]

WoW you managed to make me understand in few minutes what my math teachers were never capable of ! Crazy how modules are actually so simple to understand Maths are always ten times easier when they come with the right image to explain them

[@SeeThat92]

Your wheel numbers explanation was brilliant

[@masontdoyle]

An engineer has become a number theorist. What a beautiful timeline we live in! In all seriousness though, when I was learning modular arithmetic for my Number Theory class there was no video of this quality on KZbin to learn it. Thank you for this awesome video!

[@Thrna_1]

Such an epic way to plug your sponsor btw, actually showing how what you teach on your channel can be a useful method to use on Brilliant's test, and that these are the sorts of topics covered by Brilliant.

[@macroxela]

I never quite understood Fermat's Little Theorem but with your visualization it all makes sense now. Thanks for explaining it in such an elegant way!

[@小张同学-v6i]

holy holy holy .... they say a genius creates good math, but need another genius who can explain it well!

[@louiseevans5752]

& need another genius to understand it !!!

[@Verinenkorppi]

Just another day of creating math at the math factory

[@HighestRank]

Xiao Zhang kzbin.info/www/bejne/jmfan56pp6qVntk

[@sircyborg]

This is actually insane. I don't have a high education in math, but my hobbies makes me use it on many occasions. Many times, I don't know any formulae, so I'll have to make my own (inefficient, but with accurate results) formula. Those videos remind me of those hours figuring out how to math. The person who came up with this particular trick must really have put in some work. Impressive!

[@rogerlow9107]

Maths was never fun like this Thank you for wonderful videos

[@vojtechstrnad1]

Well you obviously don't watch 3Blue1Brown.

[@cyanprint001]

@@vojtechstrnad1 All hail Grant.

[@gflow8357]

It has always been fun. You just have to look in yourself if your instructor doesn't know what is going on.

[@KipIngram]

Ok, this taught me some things. Deepest thanks, Zach! I consider myself "good with numbers," but there was some fresh material for me here. And super well explained, too. :-)

[@blacksky7091]

3:35 it should be less or equal right? example 4 = 2 * 2 sqrt(4) =2

[@zachstar]

Yep! My bad for forgetting perfect squares lol

[@blacksky7091]

@@zachstar squares of prime numbers

[@mohamedboulaich5450]

But 2 is less or equal to 2

[@mohamedboulaich5450]

@No Name i don't understand u I said even for perfect squares the proposition is right

[@ldonnell4437]

I'm taking a semester of Number Theory this year and this video has been a life saver!!

[@Ollivie13]

Tbh I still have difficulty taking all this in, but in I'm amazed that you point out things like this.

[@CamMackay96]

This is quite basic level undergraduate maths my dude! Get reading, your mind will be blown repeatedly!

[@funahead5426]

simplicity in your explanation is the key factor that attracts each and every people that watches your video for the first time also subscribe to your channel.

[@BKNeifert]

I always forget he's an engineer. He's very gifted, to have such humanity and also such a grasp of mathematics, too. That's a rare combination of skills.

[@alestane2]

3:50 You actually check all 4 primes, not just two. Just because you verify 2 and 5 using a simple divisibility rule doesn't mean this is not a check. And there is another such rule for 3, so you don't need to actually do the division for that one either (the sum of the digits of a number divisible by 3 is also divisible by 3). The same rule eliminates 3 at 4:10, and you can also use the rule for 11 (the alternated sum of the digits of a number divisible by 11 is also divisible by 11) to eliminate it.

[@OG-ds4iy]

Uhm...I lost u at “hello”, but still made me feel smart 😂😂

[@TheDreamingDriftersStories]

I looked at this at 7 min and thought this is enough info for a week and then I realized the video is 20 min. MY BRAIN EXPLODED

[@HA7DN]

We've learnt this in high school, but using this chart more of us would understand this.

[@matyaspoko]

This is probably the most comprehensive explanation of Fermat's little theorem I've ever seen!

[@thatnativeking1333]

I still see numbers with my eyes.

[@anilkumarsharma1205]

when number become experience then we sensing the numbers not merely see them like speed of light and intensity of laser light or power of hydrocarbons or hydroelectricity dam or hydraulic pressure machine power and many degree of time and randomness and temperature etc

[@prim16]

@@anilkumarsharma1205 beautiful job detecting danny's sarcasm, i cried

[@anilkumarsharma1205]

number are for information we see the numbers with eyes, hands, touch, sound or any other means so we know mann ki aankhon sey dekh saktey hain

[@whatelseison8970]

I see colors too sometimes

[@neonblack211]

Great comment

[@alanr4447a]

A minor thing I did with modular math: Given the integers 1 to 9, there are 84 combinations of three numbers picked from those 9. Some years ago I received a computer program where an array of 84 "items" represented all 84 of those combinations. The program would go through a succession of picking one "item", and putting it through the series of "tests", with A, B and C standing for the three numbers represented by the "item". It frequently came up in these "tests" to ask variously if a certain number of the 1 to 9, each called x for its occasion, was anywhere among the three numbers (A, B or C) represented, which the program would do by asking for each, "is A equal to x, or B equal to x, or C equal to x?" I used modular arithmetic to reduce those three questions to a single test question. When the "item" was initially chosen for examination, I would formulate for each of A, B and C (called "y") the value 2*y+15, and multiply the three values for A, B and C together, and call that N. In 2*y+15, 1 to 9 produces 17, 19, 21, 23, 25 , 27, 29, 31, and 33. The nice feature of these 9 consecutive odd numbers is that all 9 contain factors unique to them. Five of them are prime outright. 21 is the only one with a factor of 7, 25 has the only factor of 5, and 33 has the only factor of 11. And while 21 and 33 provide two factors of 3 between them, only 27 has a THIRD factor of 3. Thus, multiplying various of these numbers together cannot "inadvertently" produce one of the other numbers as a factor of it. With this N, then, to test whether any x was among the three numbers, I would just ask the one question, "does N MOD (2*x+15) equal 0?" If yes, then x was among the three numbers represented.

[@brodysdaddy]

3:45 you can do a faster check on if a number is divisible by 3. Add up the digits that make up the number....119 would be 1+1+9=11...11 is not divisible by 3 so 119 isn't. 219 is or 120 ....

[@Albert-fe8jx]

Enjoyed watching. I appreciated the effort put into designing and animating the visualizations.

[@guillev5420]

When you already know number theory but still get mind blown

[@homelessrobot]

modular arithmetic is so fascinating, especially in how it relates to machine arithmetic and memory addressing in computers. I guess it kind of makes sense thinking about it as 'wheel math' when you think about how generally useful things gears are in mechanical computation engines. Everything is always wrapping around, and complete circuits of one gear generally leads to some much smaller amount of movement in a larger gear, the evenness, odness, and primeness of the amount of teeth in a particular gear lead to broad reaching consequences in the behavior of the rest of the system.

[@lewismassie]

So it's called the digital root. I've been looking for that term for about 10 years

[@gauravcheema]

And the number of additions required to get to digital root is called the additive persistence of that number.

[@lordx4641]

@@gauravcheema well isn't this called vedic mathematics in ancient India? I mean I read this in vedic maths books

[@ChrisTian-uw9tq]

Woah 10 years is some time! I am clocking up 7 years :) Vedic Square the start point ;) Got any outputs online somewhere to inspire? What do you feel it is leading you to? Do you have some education behind you or is it self learned?

[@milanstevic8424]

@@lordx4641 also numerology -- but yep, digital root

[@lordx4641]

@@milanstevic8424 yes sir I read vedic maths nd Vedanta so that gives me a very deep rooted understanding of mathematics and it's structure or let's say patterns

[@a-levelmathstutorials9175]

you have earned yourself a subscription my friend, great way to visualise modular arithmetic

[@georgepaul6240]

This completely changed the way I see numbers

[@dessguy7199]

Did video title change after this comment?

[@obibellowme]

Keemu nope

[@mohammadfahrurrozy8082]

Loool

[@TasteMyStinkholeAndLikeIt]

Unless you are a math major, you'll forget all of this in 10 minutes

[@lordx4641]

Nothing special this is vedic maths

[@L0j1k]

Haha man... Imagine cooking up some teriyaki burgers, smoking a little dope, and then discovering a new video of mathbro talking about NUMBER THEORY ARE YOU KIDDING ME. Literally the best Sunday I've had in months.

[@zachstar]

amazing comment

[@L0j1k]

@@zachstar Amazing channel, so.

[@roylavecchia1436]

Dude, stay off the dope.

[@michaelpugh2617]

@@roylavecchia1436 let the dude live

[@grantyentis5507]

Now I remember why I hated math in junior high and high school. They always go too damn fast without giving a firm foundation to what's going on. It's like it starts making sense then they pull this twist that breaks the rule of what I thought I just learned. I have a headache now.....I'm going to eat cereal.

[@Roosyer]

Cryptography is sooo interesting. Thank you for talking about it!

[@timchallenge]

Number theory is probably my second favorite field of mathematics. It's absolutely fascinating!

[@mayabartolabac]

so what's your no. 1 favorite?

[@timchallenge]

@@mayabartolabac Chaos theory. Hands down, it's just fascinating to me.

[@mayabartolabac]

@@timchallenge i find chaos theory pretty fascinating as well idk my favorite field of math; i don't think i know enough math to pick one

[@Xabraxus]

I never liked using the modulo in programming because it seemed like something that could become difficult mathematically, this video helps immensely with abating that fear.

[@vapourmile]

This is so much better than all those "maths wow" newsfeeds. Sure, you can learn all kinds of messed up or surprising facts about maths, but you can watch those videos all day and never really understand the subject. This is actually useful maths you can learn and apply.

[@cjcote3490]

well, this has officially blown my mind.

[@Lumegrin]

14:40 this sponsorship was actually kind of funny "Another property _if-we-go-to-Brilliant's-site"_ Lmao

[@dogmeat7486]

I'd give every last cent i have to be able to remember this when i need it.

[@peggyfranzen6159]

🎇

[@jackdowning6804]

I’m a 5th grader I used to hate dividing by 3 or 9 now with new found knowledge I don’t hate that anymore

[@ryanrrree1744]

My brain could focus but at the same time couldnt I love these kind of videos tho Please make more

[@TheSandkastenverbot]

Cool, you have Schroedinger's brain! I'm envious ;-)

[@ryanrrree1744]

TheSandkastenverbot hahahaha

[@area51xi]

This is one of the best videos EVER made on KZbin. Possibly life changing.

[@rishi1679]

You enlightened me Thanks Please make a video on engineering physics

[@alexv5581]

Or you could do your own research. Its a useless degree option, are you trying to be an engineer or a scientist? Both different disciplines and mindsets. Are you an abstract thinker who likes to think why something happens? Or are you a practical thinker who likes to understand how something works? If you can be honest with yourself than one if these professions is for you. The world needs smart and good scientists and engineers, not some person who holds a degree and claims to be an engineer or scientist.

[@tiagocomputacao]

Finally! This video made me understand why modular arithmetic is useful.

[@spandansaha5663]

After watching this video *My brain has left the chat*

[@jbscott8914]

Wow

[@raymondfrye5017]

@Zach Starr: The "wheel math" you are demonstrating is ancient Sumerian,Babylonian and Hebrew mathematics. Originally, the number system was base 6 and was called a perfect number as the cycle of time could be subdivided into portions. It was much later,that the symbol of zero evolved and expanded the number system to ten and kept 1-6 for geometric figures and religious purposes.

[@EpicMathTime]

One of my favorite problems is a "word problem" that sounds simple, but is based in modular arithmetic. Asking people that haven't studied modular arithmetic to solve it is kind of interesting, because they try for a while and then suddenly they have the epiphany that there is a huge gap in their knowledge of arithmetic. Here is the problem: A group of 50 pirates finds a chest of gold coins. After the coins are distributed evenly among the pirates, they find that there are 6 gold coins remaining. After a heated argument about how the remaining 6 coins should be distributed, two of the pirates are killed. The coins can now be split evenly among the crew. What's the smallest number of coins that could have been in the chest?

[@Ennar]

Thanks for that problem. Boils down to a linear Diophantine equation.

[@abhavyakeerti8599]

So what's the answer, 336??

[@Ennar]

@@abhavyakeerti8599, no, you cannot divide 330 = 336 - 6 evenly among the 50 pirates. Let x be the number of gold coins. Then the problem tells you that 50 divides x - 6, and 48 divides x. So, x = 48n and 48n - 6 = 50m. To get relatively prime coefficients, divide by 2 to get 24n - 25m = 3. Since 1 = 25 - 24, then 3 = 3*25 - 3*24, so (-3,-3) is a particular solution of 24n - 25m = 3. Thus, all the solutions are given by (-3 + 25k, -3 + 24k), for some integer k. Since you are looking for the smallest positive n, you want k = 1, so n = 22 and x = 22*48 = 1056. If you need more convincing, look at the first 22 multiples of 48: {48,96,144,192,240,288,336,384,432,480,528,576,624,672,720,768,816,864,912,960,1008,1056}. Now, subtract 6: {42,90,138,186,234,282,330,378,426,474,522,570,618,666,714,762,810,858,906,954,1002,1050}. Only the last one is divisible by 50, so that one corresponds to the smallest possible number of coins.

[@abhavyakeerti8599]

@@Ennar thank you, I am an idiot, I did it the same way but forgot to check if 330 was divisible by 50. Also, isn't there a more accurate method than this hit and trial sort of method, I mean how long would you go on checking which is divisible, coz sometimes the no. may be too big Thanks again

[@Ennar]

@@abhavyakeerti8599, you are welcome. Actually, there was no trial and error, we know all the solutions of every linear Diophantine equation, it's precisely what I wrote. Check the wiki page en.wikipedia.org/wiki/B%C3%A9zout%27s_identity for more details.

[@stephenbrown40]

Nice.This Visualisation of Numbers has me hooked, need more. I have a visual memory, I can use this, still need to see the whole thing with the result. There are patterns that seem to fit visually. A few random number sequences interspaced with sin and - , could be an accelerating particle being tracked or plotted in a co-ordinate system or the projection point in a architectural design on an hill side. I need more.

[@hetgenie]

1:06 you don't need a calculator to see if a number is divisible by three. The digits 1+1+9 don't add up to a number that is divisible by three. So the number itself equally isn't. EDIT: 13:17 Nevermind ;-)

[@UTU49]

I've used this test for divisibility by 3 as long as I can remember. I might have learned it when I was about 10 or 12 or 14. Not sure. One of my best friends did not know the test, and he had taken 2nd year university courses in stats, calculus, and linear algebra. I can't imagine going through the amount of math I have in my life without knowing the test for divisibility by 3.

[@justincase5272]

The encryption you described is RPK, PKI, etc. Using a publically known algorithm, a message encrypted by one's public key can only be decrypted by one's private key. The "wheel" is essentially a toroid of finite longitudinal segments but infinite circumferencial segments , and this approach is considered "elliptical." But toroids are easy to crack. Chaotics are not, as they result in countless solutions from the public key but the private key can still decrypt it.

[@sadaghem2151]

I didn't understand anything but now I can say things nobody else is able to understand too

[@SauravKumar-xg4zr]

I have picked up random no. 587 to check whether it's prime or not. Using the above technique it took me less than a minute to conclude. Yes, it is Prime. Just Fanstanstice Knowledge, Thanks a lot man.

[@Ash-zm1vx]

Ah yes, I remember figuring out that every prime (except 2 or 3) is 1 more/less than multiple of 6, since 3 more/less is multiple of 3, 2 more/less is multiple of 2. Still I was surprised about how that could be applied!

[@lfestevao]

119 thing is you checked 4 numbers. As an adult, you are used to check the 2s and the 5s unconsciously. If you were used to the 3s as per the rule of algarism addition, this would also be automatic. And finally the 7s you actively check. Your vid is a good thing because it gives insight and a visualization for a concept many have abstractly. Which is easier and faster, just like the 2s and 5s check in the decimal base.

[@zsoltsz2323]

You can also immediately see if 119 is divisible by 3 if the sum of the digits is: 1+1+9=11, not divisible by 3. (On the other hand e.g. 252: 2+5+2=9, divisible by 3.)

[@pseudolemon8272]

he went over that

[@obibellowme]

Zsolt Sz watch the video before commenting lol

[@ErgoBytes]

Fantastic exploration of modular arithmetic, my textbook was not doing it justice but I could tell that this was an important topic i needed to grasp lol

[@nissangamba]

Math videos relieves me of insomnia.

[@edwardgraham2566]

I believe this is the Book by David M. Burton ::: Elementary Number Theory ... yes, a nice text!

[@KurNorock]

"We can answer yes with no intensive work required" Except for drawing the circle diagram specific to multiples of 7, and then figuring out and memorizing all the patterns.

[@Bee37351]

Modular arithmetic is such a deep and useful topic - it’s the basis of cryptography! Thanks for sharing 💕

[@jarrodanderson2124]

What is the Number Theory book you liked so much? Excellent video btw. I loved the wheel math graphics!

[@zachstar]

Thank you! And I actually used the book 'elementary number theory' by david burton (it says 'revised printing' on the front rather than an edition). I didn't use the exact one I showed a picture of but that just looked more visually pleasing for the video lol. I had found a free online version of the one I used and thought it was well written.

[@jsdp]

@@zachstar Cheers. Definitely going to check the book out now!

[@Aruthicon]

My statistics teacher lent Burton’s book to me a while ago, and I can say that it is amazing.

[@manganari8022]

at the beginning of this school year i discovered modular arithmetics in the notebook of an older student, the first thing i did with the power i was just given was make an unbreakable code to send messages to my friends, for those interested what i first did was converting text to morse so - and . then changing it to binary 0 and 1, and then attirbute any number wit a rest of a when divided by an integer x to , same for 1 with any number congruent to b mod x, and then the only thing you need is the key looking like this [x,a,b] to crypt or decrypt the message, it was stupid, it was not useful at all, it was not optimized either, But oh my god it was fun

[@tone618]

Lmao you discovered the basics of rsa encryption

[@manganari8022]

thx, thanks to you i'm going to spend my 6 hours left to sleep in order to learn about that and make a better encryption method, then code it in some way, make UI for both encryption and decryption, then, then hmm, what can i actually do with that...

[@Supremebubble]

Digital Roots are just another reason I loved the game 999.

[@ThErrandBoy]

You should try the game 69

[@BenjaminLeiby-e8w]

I'm taking design of computer algorithms in college, this is super helpful

[@domc3743]

119 isn't prime, consider writing 119 as 144 - 25 that is a difference of two squares then we have the factorisation (12+5)(12-5) = (17)(7)

[@dataandcolours]

Great observation! It's called Fermat factorization and it's one of the the reason why it's important that the two primes p and q that makes up the composite n in RSA-encryption shouldn't be too close to each other.

[@ross825]

It’s amazing you have 1.1M subs when I have to resub monthly care of the KZbin API

[@Naitasm]

14:02 "A slightly more official term for working with 9's though, is the digital root." *Nonary series flashbacks*

[@awpbaldyMC]

sudoku flashbacks

[@rocketgames9873]

Digital Root *Deltarune ch. 2 before-spamton teacups flashbacks*

[@manioqqqq]

@@rocketgames9873 Don't remind m- Oh. That's my alt.

[@larianton1008]

This was exactly what I was looking for. Some love for number theory, in a video form. I enjoyed this alot :)

[@ugexcelsus4812]

Tbf this could be helpful to how education could be taught

[@MMABeijing]

tbf as "to be fair"? strange use in that context

[@RealLifeKyurem]

Actually, for the example with 119, you can reduce the number of tries to just 1. Just check if it’s divisible by 3. Checking if a number is divisible by 2, 3, and 5 are easy enough. 7 and beyond are quite harder.

[@OwlexMyth]

You mentioned several times that given X, the relative numbers ended up in "the same section", when the referenced numbers (highlighted) were clearly not in the same sections.

[@joanagomes1898]

I think he means that if you add a number from section x and and a number from section y you always get a number from section z.

[@leecaraway706]

Ben doing this for 60 years, best I have ever seen!

[@BangMaster96]

Could you do a video on Tensors, like Rank 3, and Rank 4 tensors if possible, i am very confused on understanding their notations and visualizing them

[@tomkerruish2982]

Gravitation by Misner, Thorne, and Wheeler, suggests that tensors be thought of as linear machines with slots for vectors and 1-forms. Don't worry about trying to visualize a high-dimensional array of numbers.

[@seriouscat2231]

The tensors are not there because the theory of gravity would require them, but to hide problems in the theory. The expectation is that once you have invested so heavily in the math, you have no desire to turn around and be critical of it.

[@chrisreilly1290]

The wheel thing is pretty cool

[@bigpickles]

44 seconds in and I'm confused already.

[@cronobactersakazakii5133]

You mean 2×2×11 ?

[@bloodypommelstudios7144]

Decided to make a prime number generator while listening to this, haven't done this since I was a kid but using the square root as the maximum number and only using the 4 spokes of the mod 12 wheel made it exponentially faster.

[@johnnysparkleface3096]

I was keeping up with you nearly all of of the way through the first second.

[@itsmidtrib1569]

Johnny Sparkleface 😂 best comment

[@111abdurrahman]

This is the best video on number theory I've ever seen!

[@Katharinka007]

Exactly my Algebra 2 exam... I'm still having nightmares. :D

[@itsmidtrib1569]

Arrow princess I’m 6 years out of high school and still have nightmares about algebra 2

[@TowerofGuitars]

KZbin math videos are the visual classical music I’ve always been looking for.

[@naimulhaq9626]

It is mind boggling that Pythagoras 2600 years ago, declared 'numbers are everything'. Physical world have mathematical structure. Even 'cause' and 'effect' are related by a number. Empiricists wanted to know how cause and effect are related, but could not find one. Then they decided that they must be independent of each other, until the development of complex analysis, when it became clear that change in y due to change in x of a complex number z=x+iy when mapped in to the w-plane, are related by the complex number i.[ page 217 of Visual Complex Analysis, by Tristan Needham].

[@poojakori7113]

Thankyou so much....before this video I was like .....what the heck is that....how do I even solve anything like this..but now with this I can take a deep breath .....