How does Modular Arithmetic work?

Question 6 from Tom Rocks Maths and I Love Mathematics - answering the questions sent in and voted for by YOU. This time we explore modular arithmetic through the familiar example of the 12-hour clock.
Full playlist: • How many ping-pong bal...
Q1: What is the probability I have the same PIN as someone else?
Q2: How long would it take to sink to the bottom of the ocean?
Q3: What is the gravitational field of a hollow Earth?
Q4: What is the best way to win at the board game Monopoly?
Q5: What are the most basic Mathematical Axioms?
Q6: How does Modular Arithmetic work?
Q7: What is the Gamma Function?
Q8: How many ping-pong balls would it take to lift the Titanic from the ocean floor?
Q9: What is the graph of x^x?
Q10: How can you show geometrically that Pi is between 3 and 4?
Produced by Dr Tom Crawford at the University of Oxford.
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[@TomRocksMaths]

There are now 10 videos in the series for you to enjoy - all available here: czcams.com/video/by8Mf6Lm5I8/video.html

[@marnierogers3931]

Just watched a 30min uni lecture explaining this twice ... still didn't get it. Watched the first 4 mins of this and understand completely. Cheers!

[@adrianalapuerta35]

I'm genuinely really impressed by how well you explained this. well done and thank you.

[@TomRocksMaths]

You're very welcome Adriana :)

[@doodelay]

I love how THOROUGHLY you explained the concept long before you gave the equation. I will definitely take this pedagogic lesson along with me now

[@TomRocksMaths]

Thanks!

[@ChacingBugs]

Holy crap, always assumed modular arithmetic was super confusing, but you explained it so well it actually seems like fun.

[@ziaulhug9666]

I was really longing for this kind of explanation.
Everyone were like into the numbers but you told us what the number is about
I love it!!

[@TomRocksMaths]

Thanks Zia - glad you enjoyed it!!

[@exoticcoder5365]

You teach so well, it's my first time to understand clearly what it is comparing to other videos ! Thank you sir !

[@TomRocksMaths]

You're very welcome :)

[@adhiyanthaprabhujeyashanka2091]

I have seen so many videos, but no one explained it as clear as you did, what I liked is the fact that you also included examples with negative numbers, and adding that a congruent to b (mod n) where b is negative is also true, but we don't usually put -ve values for b(the remainder when a is divided by n)

[@Nanis-World]

WHERE HAVE YOU BEEN ALL MY COLLEGE LIFE!!!??? This was the easiest explanation out of like 7 videos I wasted time watching and you explained it so effortlessly and it ACTUALLY makes sense :) Thank you!

[@TomRocksMaths]

Glad I could help :)

[@ibehsuccess]

The best explanation for modular arithmetic, love your class

[@twentylulu4372]

Though this is a few years late, thanks a ton for this video! I was struggling with the idea of modular arithmetic, but the clock example really opened my eyes to how truly simple this is!

[@ihrtashleylu]

i always knew that modular arithmetic had something to do with remainders, and im a visual learner so i loved the way you used the clock as an example. thanks this was so clear and showed me that i shouldn’t be scared of this concept cuz i always thought it was so hard ‼️

[@vincentmessina9947]

Fantastic explanation, by far the best out there. Well appreciated Tom!

[@TomRocksMaths]

Glad it helped Vincent!

[@sanmore101]

Ha. Trying to understand this for months....your clock example is just what I need. Thanks a lot. Keep up the good work

[@TomRocksMaths]

Awesome to hear, thanks Santosh.

[@imyame_se]

Thank you so much, I really appreciate the simple yet detailed explanation.

[@huzi37709]

Thank you! Was taught this earlier today and everyone got it n whizzed through it except me. I became the laughing stock of the classroom for struggling with such basic maths. Thanks for this video. I see what was so hilarious now. I don't see any humanity in my peers though...

[@TomRocksMaths]

Glad I could help.

[@lifewith_IFE]

this is very explanatory! I totally understand now! thank you!

[@TomRocksMaths]

Glad it was helpful Simi!

[@FernandoQuevedo]

Thanks Tom. Best explanation of modulo!

[@TomRocksMaths]

Thanks Fernando :)

[@lindenjohnson1709]

best elucidation of this concept I have seen thank you

[@patrickmayer9218]

Incredible explanation! It's awesome that we all use modular arithmetic when doing time math and we don't even realize it! Thanks for the video!

[@fisulohi8707]

The best explanation of modular arithmetic on the internet. Thank you!

[@nihinelizabetho4198]

ive watched a couple videos on this topic but yours is the best.

[@TomRocksMaths]

Thanks Nihin!

[@TheJProducti0ns]

Brilliant video. I'm taking Abstract Algebra atm and I was looking for a video that explains it further. This is it!

[@TomRocksMaths]

Glad it was helpful Jay!

[@yelenalovrenovich3048]

Thank you so much for your simple, clear explanation! The penny has finally dropped!!

[@TomRocksMaths]

Glad it helped!

[@musicislife665]

Thank you. This was really clear!

[@madhureshminoshi4272]

This is the best explanation of modular on youtube. Great work. Thank you.

[@TomRocksMaths]

Thanks - glad I could help.

[@nealsonf]

Thank you so much! It was explained in a way I completely understand! Great job!

[@munahassan9860]

Thank you for using an intuitive example, made the concept easier to grasp!

[@TomRocksMaths]

Glad it was helpful!

[@jasonchia3959]

The best explanation I found online. Bravo

[@TomRocksMaths]

Thanks Jason - glad it was helpful.

[@angelesfigueredo5740]

Marvelous explanation!!

[@duketrain8247]

Your content is really good for abstract algebra!

[@TomRocksMaths]

Awesome - thanks :)

[@MrPopworld]

Thanks bro!! I studied this math for like 4 hours and I was clueless…. But I only watched your video for like 5 mins and I understand everything!!!

[@TomRocksMaths]

happy I could help :)

[@beatricedurosinmi-etti3352]

Thanks for explaining this topic, I really appreciate this, it really helped a lot with my assignment

[@TomRocksMaths]

You're very welcome Beatrice!

[@alpiearts1026]

thank you so much, i was afraid that i would fail since i cant understand a single thing which my teacher thought me , but after watching your video i can understand everything, once again thank you so much sir ....

[@TomRocksMaths]

glad I could help :)

[@Deemjuhh12]

Watching this in 2020! THANK YOU TOM!!!!

[@TomRocksMaths]

YOU'RE VERY WELCOME SAAR!!!! loving your enthusiasm :)

[@yesterdayssjamm]

Thank you so much for this video the way you explained it was very clear and I understood everything very quickly

[@TomRocksMaths]

Awesome, thanks!!

[@vaishnaveesuriyanarayanan4503]

its so mind blowing the way concepts are explained using simple examples

[@TomRocksMaths]

Happy to help.

[@prishabhalla4658]

Loved this. Thank you so much!!! (I also love your hair) 💕😜

[@TomRocksMaths]

😊 thank you

[@kyanbaghdasaryan1332]

Brilliant. Thanks

[@TomRocksMaths]

You're very welcome.

[@iLinked]

First explanation i really understood, the clock analogy was neat :D ty

[@VladTepesh409]

Best explanation of modular arithmetic.

[@28maitreyagupta21]

I swear this guy makes things easier. Thanks man.

[@TomRocksMaths]

happy to help :)

[@emmanuelletshilumba8957]

My lecturer spent 2 hours explaining this I didn't get it. after 9 minutes of watching this video, I understood very well. Thank you so much.

[@TomRocksMaths]

You're very welcome Emmanuelle :)

[@nuwanudara6343]

Finally. Good tutorial. Thank you. 😊

[@TomRocksMaths]

You're welcome Nuwan!

[@yohomiedude]

This was very helpful and clearly explained, thank you.

[@TomRocksMaths]

glad it was useful!

[@marielangelieamodia4087]

I easily understood the concept which is a great achievement for me thank you so much!

[@TomRocksMaths]

Glad it was helpful Mariel!

[@treyflorence9632]

This just helped a ridiculous amount thank you!

[@miriamanikor9308]

I really understand your teaching.THANK YOU SO MUCH SIR

[@TomRocksMaths]

You are most welcome

[@trevorwinger1700]

Tom, you rocked this maths. Thank you

[@TomRocksMaths]

Glad you enjoyed it Trevor!

[@cindyconner7604]

Thank you!!! This helped me so much!!!

[@TomRocksMaths]

You're very welcome Cindy!

[@joshjames8290]

this was really helpful!

[@TomRocksMaths]

Awesome - glad you found it helpful Josh.

[@silvadossantos6803]

While i found hard to have another chanel on m already full yt playlist I'm following you through your other social media, thk a lot

[@TomRocksMaths]

Awesome :)

[@just_matt3937]

Big brain! Man, you actually know how to teach math in English and I love it! Some folks out here sound like they are from another planet with their explanations lol. Appreciate it.

[@TomRocksMaths]

happy to help :)

[@chromerims]

Perfect intro to modular arithmetic for decimal number system 👍
Please do the equivalent presentations for hexadecimal and octal 😀
Kindest regards, friends and neighbours.

[@ewafront]

love your video, your are so passionate about math!

[@TomRocksMaths]

Thanks - and you know it.

[@akumaram]

What's the difference between this and the modulo used in computing? For example here, we say 34 = 10 mod 12. But in computing, it would be 34 mod 12 = 10. Why is the notation different?

[@sharlesy2241]

You're so good at this. Thank you very much. :)

[@TomRocksMaths]

you're very welcome :)

[@vuyanijaka9507]

Wow, amazing explanation

[@osmanmusse9432]

Great Explanation

[@TomRocksMaths]

Glad it was helpful!

[@areelkhan4004]

*MIND BLOWN*

[@TomRocksMaths]

[@johnpowell3500]

Thankyou so much I'm studying and this completely went over my head clock was a good example.

[@TomRocksMaths]

Glad it was helpful!

[@rawanstech402]

Well done Tom! Thanks

[@TomRocksMaths]

Thanks for watching Rawan!

[@iirazor1086]

Thanks bro this was helpful

[@priyavartrajain2166]

Good job!

[@TomRocksMaths]

Thanks Priyavart.

[@johnoommen8383]

good vid, greatly explained, would like to say that it might be confusing for some that you did not use paranthesis, since not using them is a different notation to what you are explaining. (denominator and remainder switch places)

[@blossombliss8093]

Thanks it's very fascinating

[@grapy83]

Great explanation mate 👊

[@TomRocksMaths]

Glad it helped

[@snake5457]

Great Video :D!

[@TomRocksMaths]

Thanks Snake!

[@avezl]

Thank you so much!!!

[@katherine54122]

You explained like it's so easy to do😂 and yes! It makes me do it easily! Thank you

[@TomRocksMaths]

Glad it helped Katherine :)

[@katherine54122]

@@TomRocksMaths all thanks to you😊

[@sharvilpatel1651]

Hey what is the angle between two hands of a clock at 2:12 ?

[@jejefred2812]

Nice one sir. Lov the explanation.

[@TomRocksMaths]

Glad you liked it!

[@dumpster1947]

Great presentation
Thanks very much

[@TomRocksMaths]

You're very welcome :)

[@Ron_DeForest]

I’m Canadian. I took all advanced math through high school, never learned mod math. Even took the pre university math courses. Have to wonder why. All these concepts that help you manipulate numbers are useful to know if you’re going to take math further.

[@yunafix19]

Very helpful explanation, thank you! So can we say that in a week it's a modular 7 arithmetic? :D

[@TomRocksMaths]

Yes, absolutely :)

[@468_manavgupta8]

best explanation

[@nemaahmed5660]

Thank you very much, you are amazing 🙌

[@TomRocksMaths]

You're welcome 😊

[@jimmysaxblack]

thank you so much !!!

[@TomRocksMaths]

You're welcome!

[@JorgeNizamaPlus]

Thks brot! you are clear!

[@iamsand1952]

Love your channel so much . Now I can tell my friend to stop bragging of how smart he is, because I understand this 😂 😌

[@TomRocksMaths]

Awesome - happy to help!

[@dakoni184]

Excellent video Tom, pretty good 👍. Greetings for Mexico

[@TomRocksMaths]

Thanks Emiliano - and hello Mexico! Y'all stay safe.

[@dakoni184]

@@TomRocksMaths hahaha Tom that sounds like The Purge. But thanks u to over there. Stay with the t-shirt on.

[@rejuke868]

Thank You so much!

[@TomRocksMaths]

You're welcome!

[@nimeshjeyarajasingam7791]

Can you do some follow up videos on modular arithmetic (don't forget to mention some interesting and challenging problems please!)

[@davidbodunwa4618]

Thanks alot

[@quran5905]

very helpful

[@MichaelSalo]

Would modular arithmetic be better named as cyclical arithmetic?

[@TomRocksMaths]

It certainly works based on the concept, yes!

[@justinsankar1164]

What an absolute legend

[@TomRocksMaths]

Thanks Justin!

[@rhaphell483]

Damn I love this

[@TomRocksMaths]

Thanks Raphael!

[@AzharKhan-fb6qv]

Hi tom, i understood this but how does this relate to congruency?if a is equivalent b in modulo n, is this equivalent to saying a is congruent to b in modolu n

[@TomRocksMaths]

yes they're the same thing!

[@zeenuexe8362]

thanks. I have a cryptography exam tomorrow and I forgot what this was.

[@TomRocksMaths]

Glad it helped - hope it went well!

[@jvenkat2841]

Thank you., sir

[@TomRocksMaths]

You're very welcome :)

[@topilinkala1594]

9:20 Little correction: You only do division properly mod p if p is a prime number. Non-prime modular arithmetics do not have all inverses under multiplication. You can see this easily if you think of any divisors of the mod. Let's say we use the 12. Because both 2 & 6 divide twelve 2*6 = 12 which is 0 mod 12 and thus neither 2 or 6 can have multiplicative inverses mod 12. Which implies that you can't divide all numbers by 2 in mod 12 arithmetic.

[@jaydeestrada5965]

how would this apply for much larger modulos like 3 mod 645 ?

[@TomRocksMaths]

That would still be 3. Any number less than the modulo is unchanged.

[@yw5833]

Dude is the bob Ross of mathematics!

[@moadan4934]

After doing 3years of maths did understood now.

[@peachyspalace]

THANK YOU ah I was ready to give up. Thanks for being so clear, you've saved my grade!

[@TomRocksMaths]

You're very welcome.

[@hastinmy]

thanks

[@hastinmy]

ok

[@TomRocksMaths]

You're very welcome Hastin.

[@jwm239]

....the clock is always an elegant, effective illustration of very critical aspect of modular math; C Y C L E S !

[@TomRocksMaths]

Very true.

[@Overlord_4900]

Wow this guy is great

[@ireneakabane4396]

Thank you so much! I am now able to understand it. Very impressive.

[@TomRocksMaths]

Glad it helped Irene!