The Unusual Mathematics of Modular Division
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- Опубліковано 23 кві 2023
- In the land of "mod 60" (which works like minutes on a clock) "1 divided by 7" is 43, while "1 divided by 6" is impossible! Let me show you the strange patterns of "modular division", a deep mathematical concept which we can visualize through clocks and calendars....
I had hoped to have this episode out a couple days ago, but my computer broke over the weekend. I got a new one, but had to re-edit this episode. Now things should be back on track with another new episode this weekend. And make sure you're also subscribed to my @Domotro channel which has lots of shorts, livestreams, and bonus videos in between the main episodes here.
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In case people search any of these words, some topics mentioned in this video are: clock math, modular arithmetic (particularly mod 60 and mod 7), modular addition, modular subtraction, modular multiplication, modular division, coprime (also known as relatively prime) numbers, which operations have the property of being "closed" in which realms of numbers, and more!
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DISCLAIMER: Do not copy any uses of fire, sharp items, or other dangerous tools or activities you may see in this series. These videos are for educational (and entertainment) purposes.
I had hoped to have this episode out over the weekend, but my computer broke. I got a new one but had to re-edit this episode. Now things are back on track and I'll have another new episode this weekend. And make sure you're also subscribed to my @Domotro channel which has lots of bonus content in between these episodes.
Shouldn’t have put that water bath in front of the desk there 😉
I cannot possibly imagine you breaking something.
I can not express how cool it is to have found this channel. This particular video covers exactly what I think about all day and this is so cool!!
Great video as always, grade -2 is already shaping up to be awesome
This is real imaginary stuff! 🥳
(pun 99.999…% intended)
I loved the gag with the white boards. First time you pulled the one up just to have it fall and pull out a new one. Second time you tried writing on one and it fell so you just started writing on the other, lmao
I thought i've checked out every cool educational channel on youtube.. Then I found this one.. You guys have earned a fan..
Elementary school math: We're going to learn about division! Yes, not all numbers divide evenly into each other, but you don't need to worry about that! We'll just use remainders!
Middle school math: It's time to learn about decimals and how to do *real* division, not that remainder cop-out nonsense.
College-level math: Actually, remainders are the defining aspect of modular mathematics, which is super important for many mathematical fields.
I may have forgotten nearly all the trig I learned in high school, but life truly does work in cycles.
Even when doing "proper" division in middle school, you may have studied long division at some point which relies on - you guessed it - remainders. (Specifically, long division is: "what's the whole number part of this division, carry the remainder onto the next line, hop down one decimal place to essentially divide by a number that's ten times smaller, repeat.")
Lovely. (7)(43) - (60)(5) = 1 and (2)(4) - (7)(1) = 1. For me, moves around the clock give the clearest intuition of the Euclidean algorithm.
Instead of moves round the clock for a clear intuition, I used repeated sequences of 0 1 ... 58 59, laborious though it is. Count the asterisks below for a demonstration that 1 ÷ 7 = 43 in mod 60, or more precisely (1 mod 60) ÷ 7 ≡ 43 mod 60
0 * 1 2 3 4 5 6 7 * 8 9 10 11 12 13 14 * 15 16 17 18 19 20 21 * 22 23 24 25 26 27 28 * 29 30 31 32 33 34 35 * 36 37 38 39 40 41 42 * 43 44 45 46 47 48 49 * 50 51 52 53 54 55 56 * 57 58 59 0 1 2 3 * 4 5 6 7 8 9 10 *11 12 13 14 15 16 17 * 18 19 20 21 22 23 24 * 25 26 27 28 29 30 31 * 32 33 34 35 36 37 38 * 39 40 41 42 43 44 45 * 46 47 48 49 50 51 52 * 53 54 55 56 57 58 59 * 0 1 2 3 4 5 6 * 7 8 9 10 11 12 13 * 14 15 16 17 18 19 20 * 21 22 23 24 25 26 27 * 28 29 30 31 32 33 34 * 35 36 37 38 39 40 41 * 42 43 44 45 46 47 48 * 49 50 51 52 53 54 55 * 56 57 58 59 0 1 2 * 3 4 5 6 7 8 9 * 10 11 12 13 14 15 16 * 17 18 19 20 21 22 23 *24 25 26 27 28 29 30 * 31 32 33 34 35 36 37 * 38 39 40 41 42 43 44 * 45 46 47 48 49 50 51 * 52 53 54 55 56 57 58 * 59 0 1 2 3 4 5 * 6 7 8 9 10 11 12 * 13 14 15 16 17 18 19 * 20 21 22 23 24 25 26 * 27 28 29 30 31 32 33 * 34 35 36 37 38 39 40 * 41 42 43 44 45 46 47 * 48 49 50 51 52 53 54 * 55 56 57 58 59 0 1
Division is seen as repeated subtraction till 0 is left. Division in mod n is performed on a sequence 0 1 2 ... n-1 repeated as many times as needed. In the above example the subtractions start at the far end, which is 1, the number to be divided, and move leftwards and upwards in groups of 7 till 0 is left.
Wow! Great production quality! You really upped your filming and editing game. Love it! 🤗
Thank you for this very fascinating and intriguing intro to mod-math (not to be confused with the infamous math-mod 😄!).
I like the candor and positivity of the instruction here
There is an application of this principle in Vernier scales. Within the same distance, one scale will have N markings and another will have either N+1 or N-1 markings, which guarantees that the two are coprime so the intersection of the two covers the whole range.
Love the implication that you're just like a crazy homeless math man living in a tent somewhere
Nice, I was just wondering about this!
That kicked so much ass.
and then all that was left was the Combo Cl
If you remove the "w" from the wall the COMBO ASS is on, it is "all".
Slender salamanders are super stellar
spilling the birdseed was a perfect metaphor to learning math. as the seed is falling there's total chaos, but they always settle themselves down to the lowest common denominator;)
right on man!
I put a hint/simple solution to the 3 squares with 2 angles adding up to the 1st angle proof under the name KGG in puzzle discussion. I hope that was the correct place to put that.
If you mean on the Combo Class Discord server, yeah that’s the right spot to post solutions/hints/whatever related to the weekly puzzles :)
I mean. funniest moment in youtube history? i'm dying at 9:12
Bro you're weird and I love your channel.
As a cryptography nut, it's good to see modular arithmetic.
Keep it up homie, thanks for sharing the knowledge.
I thought it was neat that I thought of cryptography when I was trying to understand how to perform the calculation, because I noticed the asymmetry in difficulty, but didn't know that it was used for encryption until I thought about it and looked it up.
calling the congruent sign "modular equals" makes it a lot easier to understand. Thanks for that! :)
You should make a video about the intersection of modular arithmetic and complex (maybe even hyper complex) numbers.
I think you could have some cool takes on this.
I dimly remember once I was testing some software written by a colleague and it worked fine, but then suddenly some huge numbers popped out. That's why, it was using modular arithmetic internally.
10:00 If we extend to rational numbers, we get six values for c. at values like 1/6+10m
I feel like I'm learning advanced mathematical concepts at my late grandmother's house, which is a bit comforting if not strange.
This makes me wonder, since 3² and 4² in mod 7 both work out to 2, can we say that sqrt(2) = 3 and 4 in mod 7?
under this system, the square roots of 3, 5, and 6 would be undefined, which makes me wonder which mods have the largest proportions of defined square roots as well
3² and 4² are 2, so √2 = 3 and 4. 2² and 5² are 4, so √4 = 5 and 2. 1² and 6² are 1, so √1 = 1 and 6. 0² and 7² are 0, so √0 = 0 and 7. Yes, this seems to give no possibility of definition to the square roots of 3, 5, and 6. Is that what you mean?
I've also been looking into cubes. In mod 10 for example, all digits 0 through 9 cubed work out to be distinct, namely 0, 1, 8, 7, 4, 5, 6, 3, 2, 9, so each has its cube root.
I understood this quite quickly before he got into the explanation, however I still couldn't figure out how to perform the calculation itself easily. I'm guessing it's not easy. In fact wouldn't this be effective for cryptography because of it's asymmetric difficulty? yeah I just looked it up, and it seems like it is (at least at a glance).
I'm not sure either how he gets to 1 ÷ 7 = 43 in mod 60. He just says at 8.55 "It 𝙩𝙪𝙧𝙣𝙨 𝙤𝙪𝙩 that if I go 7 minutes 43 times [round the clock] it will be congruent to 1" as he writes out the expression (7)(43).
He's quite correct, since 7 times 43 is 301, which also equals 5 times 60, remainder 1. But how he got that 43 in the first place I don't know. However I know how I did it, and it wasn't by smart calculation with multiplicative inverses,. I laboriously proceeded on the assumption that division of a number by n is "secretly" subtraction of n repeatedly from the number till 0 is left, and the answer is the number of subtractions. 12 ÷ 4 = 3 means if you subtract groups of 4 numbers from the sequence starting with the 12, 0 * 1 2 3 4 * 5 6 7 8 * 9 10 11 12, as marked by the asterisks, you get 3 asterisks, so 3 subtractions. Modular division is the same except you do this operation on the modular sequence repeated enough times - like going round the clock enough times - until 0 is left. In this case we have a repeated sequence of 60 digits, 0 1 ... 58 59. In the example below we start at the end, 1, and subtract groups of 7 working leftwards and upwards till only 0 is left. We have to do that 43 times, so 43 asterisks.
0 * 1 2 3 4 5 6 7 * 8 9 10 11 12 13 14 * 15 16 17 18 19 20 21 * 22 23 24 25 26 27 28 * 29 30 31 32 33 34 35 * 36 37 38 39 40 41 42 * 43 44 45 46 47 48 49 * 50 51 52 53 54 55 56 * 57 58 59 0 1 2 3 * 4 5 6 7 8 9 10 *11 12 13 14 15 16 17 * 18 19 20 21 22 23 24 * 25 26 27 28 29 30 31 * 32 33 34 35 36 37 38 * 39 40 41 42 43 44 45 * 46 47 48 49 50 51 52 * 53 54 55 56 57 58 59 * 0 1 2 3 4 5 6 * 7 8 9 10 11 12 13 * 14 15 16 17 18 19 20 * 21 22 23 24 25 26 27 * 28 29 30 31 32 33 34 * 35 36 37 38 39 40 41 * 42 43 44 45 46 47 48 * 49 50 51 52 53 54 55 * 56 57 58 59 0 1 2 * 3 4 5 6 7 8 9 * 10 11 12 13 14 15 16 * 17 18 19 20 21 22 23 *24 25 26 27 28 29 30 * 31 32 33 34 35 36 37 * 38 39 40 41 42 43 44 * 45 46 47 48 49 50 51 * 52 53 54 55 56 57 58 * 59 0 1 2 3 4 5 * 6 7 8 9 10 11 12 * 13 14 15 16 17 18 19 * 20 21 22 23 24 25 26 * 27 28 29 30 31 32 33 * 34 35 36 37 38 39 40 * 41 42 43 44 45 46 47 * 48 49 50 51 52 53 54 * 55 56 57 58 59 0 1
Can’t wait to show my friends how 13=1!
but 1! = 1 🤔
@@harriehausenman8623 1! = 13 in mod 12
@@hkayakh 😁
13 = 1 mod 12
Haven't watched yet but saved for later. Just want to say believe modular division how the Rainman - "guessing" the weekday any birthday fell on - calculation derived. Another method but much slower, an algorithm based on the perpetual calendar. Looking forward to viewing & t/y the YT.
Why are u so smart and love the video keep going
Is it merely a coincidence that when you talked about consistent systems that you were holding a RING, which fields like Z/pZ are? Are you going to do bezout's lemma next, as I imagine that's what you were hinting at at the end for finding multiplicative modular inverses?
Nicely done. Would have been interesting to see this type of concept being used on a base we are all used to using. (Mod 10)..
In mod 10, we could divide by 1, 3, 7, or 9. An example would be 1 divided by 3 being congruent to 7 (mod 10). Maybe I’ll show more examples in some future video about last digits of numbers, since mod 10 is like the last digits numbers have in base ten
In mod 100, 1/7 is also 43 :D
@@ComboClass And vice versa, swapping 3 and 7.
I think you'll really enjoy this Mathologer video: ua-cam.com/video/6ZrO90AI0c8/v-deo.html
base 10 sux 😆
Hello, i love your work and wanted to say it is thoses videos that make me love math ,so thank you!
Do you know any similar channels like yours? I really love this kind of content !
Thanks! Well, I have a second channel @Domotro that's like a more casual version of this one and has lots more bonus content. As far as other math creators, I don't know of any who have a very similar style/presentation to me, but I can recommend a few who are just great math channels in their own way. A few of my personal favorites are: Mathologer, 3Blue1Brown, and Stand-Up Maths
@@ComboClassnot to mention NumberPhile! Otherwise, you named everyone who immediately came to my mind 😅
Waaaaiit 6:45 isnt a division always kinda a multiplication in disguise? Lol
You could think of the normal integers as being mod infinity where in this version of infinity the only Coprime to infinity is 0
But couldn't you just think of the normal, that is unmodulated, integers as being mod 0?
@@chrisg3030 I suppose so,still infinity is the only coprime kind of
11:15 is not 100% accurate. Because 4/2 is congruent to 5 (or 2) mod 6. And b=2 and d=6 have the factor 2 in common.
Proof that it can be congruent to 5:
2*5=10 is congruent to 4 (mod 6)
I believe the rule is actually a≈bc(mod d) only has solutions if gcd(b,d)=gcd(a,b) or something like that.
But what if we write (1)*[7^(-1)] as (1)*{7^[59(mod60)]} AKA 7^59(mod60)? Does it give the same result then?
Hi. It was interesting, but I didn't get why 1/7 has singular defined solution (as in the video it's 43)..what about 103? Or any 43+60*n where n is integer?
Isn't it because 103 isn't a number in mod 60? 103 = 43.
@@marasmusine ok, I got it now. So the answer must be integer between 0 and 59.
Yeah I tried to clarify that on one of the title cards that flashed on stream, but to explain clearer: the equation could be solved by “any number congruent to 43 mod 60” but 43 is the only possibility between 0-59. Whenever the modular division is defined, that will be the case, where an entire “congruence class” fits the answer but has just 1 representative number in the mod’s range
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That's why 1 + 1 ≡ 0 🤖
Why thumbnail mod 300
oh it's mod 60
Isn't 1-3=58
oh it's nod mod
1/0 now
I don't think 1/2 exists rn
Nice video but i think it could be doen without mentioning week days and minutes