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

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!

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

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.

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.

I cant find the wheel thing on my calculator.

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

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.

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

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.

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”

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

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 :)

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

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

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!

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!

Ten seconds in, and I realized that I'm on the wrong video. I'll see myself out.

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

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

@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.

Your wheel numbers explanation was brilliant

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.

I come across thousands of videos on RU-vid that really are mundane to me, and then one that is completely genius and this is one of the great ones. It's worth sifting through the others!

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.

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

I still see numbers with my eyes.

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 🙂

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!

Great great great video, I watched it from begining to end and enjoyed every single minute of it. Thank you so much for your hard work, although I learned about all these things before but putting them all together, the wheel form, and putting the mathematical theorem behind it... I mean simply WOW! Mathematics is the beauty of life😍

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.

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

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.

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. :-)

Maths was never fun like this Thank you for wonderful videos

Fascinating and perfectly explained with visual effects, thank you!

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

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

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

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 RU-vid to learn it. Thank you for this awesome video!

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.

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

This completely changed the way I see numbers

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

Thank you😊, I will watch it in full detail over the weekend

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.

well, this has officially blown my mind.

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.

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

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

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.

This is one of the best videos EVER made on RU-vid. Possibly life changing.

This is magnific! Keep it up, man!!!

You enlightened me Thanks Please make a video on engineering physics

I wish I had the same level of thinking as of yours, major prep. Excellent excellent excellent work. 😘😘😘😊👍👌🙏 From INDIA, LOVE. YOU...

Awesome video 👌, definitely helped me see numbers in a new way. Very grateful to you bro 🙏.

Wow! This is really AMAZING!

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

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.

Loved this episode; thank you!

Awesome! This helps a lot, now I can begin to understand those p^(some_value) maths often used in cryptography. Always wondered how do they actually calculate the obscenely large results... apparently there are some interesting behaviours with modulo arithmetics that 'reduces' the numbers into manageable ones.

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 ....

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

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.

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.

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.)

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

I always liked the way you explain things! I can easily follow, better yet I wanna follow!

When you already know number theory but still get mind blown

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!

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.

My topic for my IB Mathematics Internal Assessment was cryptography and I had to learn all about Modular Arithmetic by myself. Very fascinating to be honest.

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

"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.

Eye opening. Falling in love with math again. Thanks

That cryptography example at the end blew my mind. Diffie Hellman Protocol?--I might forget that, but now I won't forget that seemingly impossible problems can be solved by clever mathematics. And your visual teaching method is as impressive as always. I still don't think I could remember these concepts clearly just from watching the video, but the video would significantly speed up my learning if I was doing example work alongside it. Had a Calc 2 prof do something similar and now I wish that more of my subjects/classes were taught using this method.

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

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

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.

I appreciate the visual explanations in this video.

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?

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)

This is the most awesome math video I've seen! This is so inspiring. Now I'm going to start learning number theory :)

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.

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

44 seconds in and I'm confused already.

ultimate explanation. i am glad that this video appears in my search.

Dude you blow my mind! Thank you! You explain mathematics in a way my teachers couldn't....perhaps I wasn't listening 🤘

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

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

wow i just realize we could do modular arithmetic this easy! oh bye the way thanks for your hard work, i really appreciate this, please keep making more educational videos 👍👍👍

This is a fantastic video that I am much too hungover to appreciate fully. Kudos.

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 ;-)

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

What an amazing video! Thank you so much!

On every minute i learned something new. beautifully explained

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

Math videos relieves me of insomnia.

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.

Really cool! I haven't understand 100 % (I wish English would be my mother language...) but the wheel is impressive! I actually really like "to play" around with numbers... It also helps me to remember long numbers pretty quickly! I can pick up a 20 digit number in less than 2 min and won't forget it for some period of time!

Tbf this could be helpful to how education could be taught