[Discrete Mathematics] Modular Arithmetic

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We introduce modular arithmetic, the function that outputs remainders and separates them into equivalence classes.
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Hello, welcome to TheTrevTutor. I'm here to help you learn your college courses in an easy, efficient manner. If you like what you see, feel free to subscribe and follow me for updates. If you have any questions, leave them below. I try to answer as many questions as possible. If something isn't quite clear or needs more explanation, I can easily make additional videos to satisfy your need for knowledge and understanding.

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1 янв 2015

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Комментарии : 98

@river9a 5 лет назад

"If you learn anything in Discrete Math, make sure its this." *throws out terms and formulas without going over first*. Now majorly confused.

@syremusic_ 4 года назад

This is not a bad video. It just requires a little bit of prior knowledge. Very useful for reviewing. Thank you

@terriblast8076 7 лет назад

not so clear

@iangriffith7327 6 лет назад

You say equal, but you mean are equivalent to right? They don't equal each other but have equivalent congruencies.

@Hi_im_metric 8 лет назад

hey Trev, we did congruent mods in our course is there a quick video on that? or is it a natural extension of what you've said in this video? We do a few proofs using congruent mods. Love the videos :)

@KDLieutenant 4 года назад

Sometimes even the textbook makes more sense

@castlefrank1724 6 лет назад

this is the first time your lecture is as bad as my professor's hahaha

@kamui-tg8wf 6 месяцев назад

false

@panickp Год назад

I think the last two minutes are the most valuable, at least for me, that's when I actually got it. Thanks for the video!!

@patrickmayer9218 Год назад

That moment when you realize you've been incorrectly assuming what the equivalence symbol means related to mod. arithmetic for the entire semester. XD Thanks so much for the video! Great stuff!

@hoony30 3 года назад

after reading the book named Discrete mathematics and its application from Rosen, this video makes sense, even though there are some mistakes that trev has made haha Thank you

@pedrofyury1420 2 года назад

You are litteraly saving my semester

@briefcasemanx 8 лет назад

very confused. isn't 18 mod 3 equal to 0?

@adrianmarkelov3470 7 лет назад

It's congruent though to all other numbers that also mod 3 = 0. All of these numbers are congruent and form a "congruence class". Also note that the union of all of the congruence classes partitions the original set in this case it partitions the integers

@bossinator_gaming7382 6 лет назад

He was mistaken. 18 is CONGRUENT to 12(mod3) meaning 3|(18-12), which is denoted by a triple-dash sign rather than an equals sign.

@evancornish656 6 лет назад

Yeah this explanation is VERY imprecise. Don't take this video very seriously. The video from MIT open courseware is better.

@xenialxenagogue5978 3 года назад

Ya it also is 0, it's a class of 0

@That_One_Guy... 4 года назад

Generalization : a mod b = a - b.floor(a/b); works even for a < b

@kaylahjones2230 2 года назад

This video was so helpful, thank you so much

@nicolaevasiliu9430 Год назад

Thank you so much! Clearly explained! Great!

@MSneberger 6 лет назад

at 4:09 you say "one divided by four is going to have a remainder of 1." Given the discussion after, and that (0 * 4) + 1 = 1, I am guessing you meant to phrase that differently since 1/4 does not have a remainder of 1. {EDIT} Oops! I see no that 4 goes into 1 zero times but that leaves a remainder of 1!

@samlowe2926 5 лет назад

Ok so when you say a- b and you assign those letters by defining it as 3/(18-12) you make 18 a. But when you converted it to modulus form you put 12 where a should go?

@samlowe2926 5 лет назад

Never mind, my impatient brain couldnt wait the 2 seconds to 1:16 to see you correct it lol

@Leo-ig6yl Год назад

I am confuse on 4:32, for -3/4, isn't the remainder is -3? Why it is under set [1] ?

@doctorscalling9479 3 года назад

Very well explained

@khadijahflowers5566 6 лет назад

Hey, If anyone has an answer, please let me know, but the question is this: When solving a linear congruence, how do you determine how many solutions it has? Someone told me that if we have something of the form: ax =(pretend this is the equivalence sign) b (mod m) then the amount of solutioms was determined by the gcd of a and m. Is that right?

@xenialxenagogue5978 3 года назад

I don't really know but with an example I found, it seems it is wrong, again not sure though, take a=5, b=3 and m as 2, like u see, a and m hav gcd=1 Now, m|ax-b for the solution, means 2| 5x-3, now this is true for all odd values of x (obviously, since odd-odd is even) Now, if it is like gcd≠1, then let a= 10, b=7 and m=4 So, 4| 10x-7, now this is false for all integer values of x, since 10x-7 will always give a number ending with 3 or with 7 (for x as -ve) So, with 2 very basic examples I think your statement is wrong, maybe it was something different but similar

@giomik9400 Год назад

I know I'm late, but... Proposition: Let ax ≡ b (mod n) be a linear congruence. ax ≡ b (mod n) has solution ⟺ GCD(a, n) | b Proof: The congruence ax ≡ b (mod n) has solution ⟺ ∃ x ∈ Z such that ax ≡ b (mod n) ⟺ ∃ x ∈ Z such that n | ax - b ⟺ ∃ x ∈ Z such that nh = ax - b ⟺ ∃ x, h ∈ Z such that ax + n(-h) = b ⟺ the equation ax + by = b has solution ⟺ MCD(a, n) | b □ Notes: - Every solution of a linear congruence is also called "particular solution". - If a particular solution exists, then an infinite amount of solutions exists.

@luisriveragonzalez9052 2 года назад

1:08 should be 18 is CONGRUENT to 12(mod3)

@breakdancemedia3229 5 лет назад

Thank you very much...

@thenvsnowtv1811 6 лет назад

I'v been following all your cources and i love your way of teaching in all of your videos. But i can sadly say that this video was'nt as good as you usually are. I know you can do much better man :)

@ram-tv5701 6 лет назад

Exactly!

@pianoforte17xx48 3 года назад

I never found his videos helpful anyway. He never goes to the concept. He just explains how to use the rules.

@dbella804 6 лет назад

Thank you

@YacineBenkirane 7 лет назад

Isn't q = r in most, if not all, cases?

@aryanjoshi3342 Год назад

Not at all, he just happened to pick exaples where q=r

@einnorw 6 лет назад

ok i thought i was the only one who found this vid super confusing and not helpful until i read the comments lol . jfc.

@hajrahissaniti 4 года назад

thank u loved the way you explained 9 mod 4. i wish if you had shown anything for negative numbers too. say -97 mod 11. also if we have to find a huge number like in thousands or hundreds, writing all numbers will take forever. is there a quicker way?

@ephoenix7 5 лет назад

How does 1/4 have remainder 1? 4:10

@ayanaxhye Год назад

4 divides 1 zero times with a remainder of 1. essentially 4 cannot go into 1. hope that makes sense

@jonathantosiocello 4 года назад

My professor's explanation of this is still way worse than this, if it makes you feel better, TrevTutor

@armans47 4 года назад

the video wasnt as clear as all your other ones :(

@olyadmulisa6825 7 лет назад

more clarity please, you can do better

@thetimephysician8589 4 месяца назад

this is not an introductory video on modular arithmetic lmao, but I se it's video 78 in the playlist, so I'm not sure if you need to watch all 78 videos before this. but you prob should have mentioned that at the beginning

@ecneics7660 3 года назад

I can't realize where u r writing

@Tzubaki 6 лет назад

Now this is all very easy examples. Let me hit you with the question from my exam: (5+55+101)*(576*555 + 100000000002) mod 5 is equal to? Possible answers 0, 1, 2, 3, 4. I cannot use a calculator, only use pen and paper. What to do?

@n0handles 5 лет назад

The answer is 2 You can apply the (mod 5) to each of the terms. So: ( 5 (mod 5) + 55 (mod 5) + 101 (mod 5) ) * ( 576*555 (mod 5) + 100000000002 (mod 5) ) and then equate each of the terms' values ( 0 + 0 + 1) * ( 0 + 2 ) which equals 2 576*555 is one term, so the modulus is applied to both as a whole. Multiplying the last two digits, will give the last digit in the product, in this case it is zero (6*5 is 30). When there's mod 5, you just have to look at the last digit, and if it ends in a multiple of 5 (5 or 0).

@leafslizer2376 4 года назад

@@n0handles thx I tried solving it without your answer but I got tricked by the '576*555' your explanation were a blessing

@xenialxenagogue5978 3 года назад

This question is also as easy as the examples😂, see 567*555 gives a number with last digit = 5, now if 1000....2 is added, then last digit comes out to be 7, now then u multiplied a number 55+5+101, so last digit is 1, now two numbers multiplied, one with last digit 7, one with last digit 1, so last digit comes out 7 again, we divide by 5, then remainder will come 7-5=2 :) So answer is 2

@Thebourrin777 Год назад

Sorry but this was very confusing. I had to find another explaination on Google to finally get it. Maybe you should have started with the modulo operator ? Then you could have drawn the parallel between a (mod b) and a mod b. From your example, 9 (mod 4) ≡ [1], I would also have 9 mod 4 = 1

@isam3l3 4 года назад

thanks, loved the explanation

@pointer7pl Год назад

i do not catch it

@outofbody4788 6 лет назад

At 1:00 it should be 18 = 12 mod 3

@178890905 5 лет назад

Ikr?

@xenialxenagogue5978 3 года назад

It can be what he wrote as well, cuz even if there's that -ve sign, it'll still be divisible by 3, cuz 3(-2)=-6

@Grantmadeachannel 8 лет назад

I think at 4:50 you want [1] {-5, -3, 3, 5} the one shouldn't be there?

@sarabeatrizschwarz215 8 лет назад

It is supposed to be there. 1 = 4*0 + 1, so 1 is the remainder.

@jp-gy3vh 3 года назад

So I’m more confused now than I was before...

@carlossantillana8058 8 лет назад

Thank you so much, you cleared it up for me!

@tabrissss 9 лет назад

What's the name of the book you mentioned?

@Trevtutor 9 лет назад

Jonatham Vicens Either "The Book of Proof" which you can find on google, or a text written by a dude named Grimaldi which you can probably find illegally as well. Not sure where I mentioned it, but those are the books I'm using for this series.

@Trevtutor 8 лет назад

+shadowofforms I know. I said "or" :)

@zhhhh01 3 года назад

very confusing how you introduce modular in the beginning

@mohammadminhaziftekhar4074 4 года назад

not so clear..

@philperry6564 Год назад

12 is certainly not 18 mod 3, nor the other way round.

@arnavgodhamgaonkar585 Год назад

he meant congurent simple writng mistake

@juicyrush10 6 лет назад

I have no idea what your saying

@TheRoddie16 4 года назад

12 = 18 mod 3? Am I missing something?????

@xenialxenagogue5978 3 года назад

Yes, ig u r missing-ve sign

@buiducvuongk16dn15 2 года назад

Instead of equal it should be a congruent sign

@ayanaxhye Год назад

3 divides (18-12) is true because 6/3 = Real number

@CeCegurrrl258 4 года назад

Our lord and savior during these times of corona virus making discrete math classes online when you have a bad teacher

@xenialxenagogue5978 3 года назад

Unlike most, I loved this one, kinda helped me a lot to understand how we calculate remainders in big cases, I'll just make the ap, and find the number I need eehehe

@shamirabbas7706 3 года назад

sorry.....but I can't understood.

@kevinsibiya 3 года назад

Went from 0 to 100

@shainatodi9123 Год назад

now

@Anwaar.m.a 7 лет назад

Am I the only one here who hates mods.. -_-

@bananabananae 7 лет назад

Anwar Al-Amrad Yes, you are the only one :-)

@ai378 6 лет назад

No, you aren't :3

@yousifakram7981 5 месяцев назад

what

@lazor6526 Год назад

this video confused me more. i have to watch another video

@hohtz 7 лет назад

Useless and not clear.

@ethannetumala2073 4 года назад

tangina wala ako naintindihan

@firozmehedi9225 3 месяца назад

After 3:47 cant understand anything. Man why did you asume we already know this? Seriously this lecture was bad for me. Had to learn basics from somewhere else then came here again to learn the rest.

@marsabun Год назад

sorry

@noitnettaattention 4 года назад

Actual MATH is most important for us, keep your proofs to your "heavenly" demigods residing in untouchable Olymp called Professors...

@marioksoresalhillick299 3 года назад

Huh? This is math...??? I'm confused...

@amalfi615 2 года назад

not explained very well at all and full of errors

@ganeshgump5034 7 месяцев назад

Not one of your better videos you kinda rushed through the topic w/o much explanation

@trava4156 4 года назад

ok.... and this is important WHY?!

@luisriveragonzalez9052 2 года назад

in CS, binary/Hex

@zstar8397 9 месяцев назад

Hey hope you are doing alright just I wanna say that GOD loved the world so much he sent his only begotten son Jesus to die a brutal death for us so that we can have eternal life and we can all accept this amazing gift this by simply trusting in Jesus, confessing that GOD raised him from the dead, turning away from your sins and forming a relationship with GOD. Nice vid btw