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

►My Setup:

[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!

[Ratzfourtyfour]

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

[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.

[captainsnake]

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.

[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

[dbaker]

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

[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.

[bunberrier]

I cant find the wheel thing on my calculator.

[halasimov]

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.

[shanaadams]

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 :)

[insertname]

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

Me:

[Nomenius]

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

[michaelfruge]

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”

[basspuff]

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

[SM-qkjv]

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

[bobminion]

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 🙂

[siobhanbartz]

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!

[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 YouTube to learn it. Thank you for this awesome video!

[SeeThat]

Your wheel numbers explanation was brilliant

[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.