Suppose you want to solve the equation X² – 4=0. In regular algebra you would factor the left-hand

side giving you (x-2)(x+2)=0. Here the product of two terms is zero,

so one of the terms must be zero. If x–2=0, then X=2. And if x +2=0, then X=-2. So this equation has two solutions: 2 and -2. This seems straightforward, right? But to solve this equation we used a BIG assumption: that if the product of two terms is zero,

then one of the terms must be zero. In Abstract Algebra, this isn’t always true. Let’s see an example where this technique

of solving equations does NOT work. We’ll solve the equation x² + 5x + 6=0

in the integers mod 12. If you factor the left-hand side, you get

(x+2)*(x+3) is congruent to 0 mod 12. If you set each term to 0 mod 12,

you get x=10 or x=9. Are these all the solutions? Since there are only 12 numbers in the ring

of integers mod 12, let’s go ahead and check all possibilities. We’ll plug in 0 through 11 into the polynomial

x² + 5x + 6 and see what we get. If you plug in 0 you get 6, so x=0 is not a solution. If you plug in 1 you get 12 which is congruent

to 0 mod 12, so 1 IS a solution. This is interesting. Here’s a solution to the equation that we

did NOT find by factoring. Plugging in 2 gives us 8. 3 gives you 6. And let’s quickly fill out the table for

the remaining numbers. We see this equation has not two, but FOUR

solutions: 1, 6, 9 and 10. Factoring only identified the solutions 9 and 10. So what went wrong? To see the problem, let’s take a closer

look at the equation. After factoring, we have

(x+2)*(x+3) is congruent to 0 mod 12. Setting each term to zero gave us two of the

solutions: 9 and 10. But why did we miss the other two? To see why, look what happens when you plug in 1. This gives you 3 × 4, which is 0 mod 12. And if you plug in 6, you get 8 × 9

which is also 0 mod 12. We now see the culprit. Working mod 12, it’s possible to multiply

two non-zero numbers together and get 0. We say the integers mod 12 have “zero divisors,” and it’s the zero divisors which

make solving equations more difficult. With real and complex numbers, this doesn’t happen. These fields do not have zero divisors. That’s why in regular algebra, you learn

to solve many equations by factoring and setting the terms to zero. But in abstract algebra,

this technique is not good enough. I’d like to take a moment and talk about

the term “zero divisors.” This name was chosen because division can

be defined in terms of multiplication. For example, when working with integers,

we say B divides A if B × C=A for some integer C. From this definition,

the term “zero divisors” makes sense. In the integers mod 12, 3 × 4=0,

so both 3 and 4 divide 0. They really are “zero divisors.” Let’s return to the equation X² + 5X + 6=0, except this time we’ll work mod 11 instead of mod 12. The integers mod 11 do NOT have any zero divisors. This is because 11 is a prime number. The only way the product of two numbers

is a multiple of 11 is if one of the numbers is divisible by 11. And none of the integers 1 through 10 is a multiple of 11, so this ring does not have any zero divisors. Like before, we can factor the left-hand side giving us (X + 2)*(X + 3) is congruent to 0 mod 11. Setting each term to zero gives us x=9 and x=8. Let’s see if there are any other solutions

by plugging in all the integers mod 11. When x=0, x² + 5x + 6 is equal to 6. Plugging in x=1 gives us 1. x=2 gives us 9. Let’s go ahead and fill out the rest of the table. We see that the only solutions are 8 and 9,

the two solutions we found by factoring. This is a bit of good news. When a ring R does NOT have any zero divisors,

the traditional technique of solving an equation by factoring and setting the terms to zero DOES work. For this reason, there’s a term for such

rings: integral domains. Here’s the complete definition: an integral

domain is a commutative ring R with a multiplicative identity 1 and NO zero divisors. A natural question is why do we require an

integral domain to be commutative? After all, it seems that the most important

thing, at least for solving equations, is there are no zero divisors. One reason, is when the idea of an integral

domain was developed, much of the focus was on generalizing the integers for use in number theory. Hence the word “integral” in “integral domain.” There is, however, a term for an arbitrary

ring with no zero divisors: a DOMAIN. But you’ll encounter integral domains more frequently. Integral domains have another useful property:

the cancellation property. Suppose you want to find the solutions to

the equation 2X=6Y. Your instinct might be to first simplify this

by dividing both sides by 2, giving you x=3y. But when working with a ring, you may not

be able to divide. So a different approach would be to factor

both sides, then cancel the 2s, giving you x=3y. Factoring and cancelling does not require division. But can you safely cancel in any ring? Unfortunately, no. Look at the equation 3x=6y in the ring of

integers mod 12. This equation has multiple solutions, but

one solution is x=6 and y=1. This is because 18 is congruent to 6 mod 12. But look what happens if you factor out 3

then cancel. We get x=2y. Now if you plug in x=6 and y=1 you get

6 is congruent to 2 mod 12, which is false. So in the original equation, 6 and 1 is a

solution, but after cancelling, it is not. So, when can you safely cancel?? To find out, suppose a*x=a*y

in some ring R, and ‘a’ is not 0. If we get everything on one side, we can factor

out ‘a’ using the distributive property. If the ring R does NOT have any zero divisors,

then we know that x – y must be 0 because ‘a’ isn’t 0. This implies that x=y. In other words, we can cancel ‘a’ from both sides. So another benefit of an integral domain is

the cancellation property. To recap, an integral domain is a commutative

ring R with 1, that has no zero divisors. The absence of zero divisors means you can

use the cancellation property. It also means you can solve an equation by

factoring and setting the terms to zero. Here’s a puzzle for you to discuss: how

many different quadratic equations are there mod 12? And do any of them have exactly two solutions? And what’s the deal with integral domains? (laughter) They’re neither an integral, nor a function’s domain? (laughter) I mean, come on, you know? And how about those viewers who haven’t subscribed? I mean it’s free, it’s easy? What’s stopping them?? (laughter)

## 100 Comments

## crapper

There are 11*12*12 different quadratics, if you mean just different coefficients and not the same functions

x^2 + 2x + 4 is an example of a quadratic with only 2 solutions mod 12 (2 and 8)

## MJ Kluck

These people are GREAT! Love the vids.

## Raul Kaztrö

se que va en contra de todo, pero cásate conmigo!

## ZOUHAIR ZG

I'm subscribed why K/0=F.I K in R and 0/0=f.i

## ejecutor35

the lady needs to go more basic, didn't got anything what she is talk8ng about!

## modi yadav

nice explanation mam…..i have studied before but never understood……this example makes me clear all doubt about integral domain ……thnxx mam

## MuffinsAPlenty

We generally also require an integral domain's 1 element to be distinct from 0. In other words, we don't want the 0 ring to be classified as an integral domain, since we want an ideal I to be prime if and only if R/I is an integral domain.

## miguel aphan

Beauty, you and your team are doing a pretty nice job, my hope is, that you will go deep in this subject, be keen and fun…student use to scare in this topic….good luke

## usman jan official

Its very very good

## Preme

Your videos are very well done and helpful. Keep it up!

## Bryan Villegas

Very informative and easy to follow. 9/10.

## Igor Vinícius

I thought I was the only who asked why Integral Domains are called integral domains u.u

## An Jin Geon

I really liked the video, but it would be much better if it was little bit slower 🙂 I had to stop n play again lots of times cuz it was too fast for my brain lol

## TheTuntunpa

Hate algebra, pero qué guapa es esta mujer 😅

## JFSHAZAM!

what a beautiful example of Mediterranean beauty

## Ismael Monsegur

Good videos. Can you upload Topology videos?

## The Walking Crow

Integral Domains were interesting to me because it's the minimum requirement for greater than or less than to be defined. Beneath that everything gets wibbly wobbly and is just there. Like colors, there's no way to define which one is greater or less than, but once you define color by it's frequency of light, you get the ROYGBIV we're familiar with red being the lower end and violet being the upper.

## Edward Teixeira de Albergaria

Representando Brasil

## kirby march barcena

Uhh,I don't know what to comment…

## Lycheeee11

Love your channel! Glad to know you are uploading new videos. Could you please talk more about commutative rings?

## veronicats100

MOD 12 ????

## 發阿

I am stuck with the puzzle she asked, there should be infinitely many quadratic equation right? right?

## Tural SADIGOV

Just great! I just discovered your videos, and it is fun to watch them!

## Sam Dies

I totally falling in love with the girl.

## Rahul Sharma

A beautiful channel based on knowledge.. Thanks +Socratica

## Gregory Fenn

Fields (and integral domains) have one (and only one) zero-divisor: zero. The video is good but misleading as says that domains (like Z_{11}) have no zero divisors, which is false.

## priya kataria

these videos are very helpful,May I get more videos in ring theory by socratica,these are pretty particular abt topic..i need them

## Łîvîň ĖĽf

..You are so beautiful …

## shivani dixit

awesm video

## Heisenberg

Thank you so much.

## TheSidyoshi

It seems to me that the cancellation property should hold for non-integral domains. It doesn't require commutation to work, just the lack of zero-divisors… or maybe I'm missing something.

## Ihsan waseem

Plzzz ma'am tell me how to find zero divisor and unit in Qotient ring e.g R/i

## Ihsan waseem

Plzz ans me

## Divinceo Cristao

She's so pretty

## Michl !

Thank you

## Hrishi

Calm down sweetie! I'll subscribe.

## Zenene1507

(5:42) Why doesn't the factoring and cancelling method require division? Isn't cancelling a number in both sides of an equation the product of doing multiplication of the original number by it's multiplicative inverse (and isn't this the essence of division in the first place)? Edit: Never mind, the last part of the video solved my doubts. I didn't realize that, being a common property of all rings, distribution could be used to cancel a common factor from both sides of any equation.

## Gaurav Sinha

Great Video 👍👍

## Tom Hu

Really best video for abstract algebra: the instructor explains clearly in reasoning

## Algirdas K

Too far from the true algebra . In an integral domain, first 1 <>0 , i. e. it's not a zero ring. In any nonzero commutative ring the set of zero divisors is a union of prime ideals – so, to have many highly important, classical theorems of such type we SHOULD include 0 into zero divisors. Other way we will not become a specialists in ALGEBRA. Don't listen such sort of videos

## poomalai p

Good teaching. Every one need example better

## Whoo

I'm impressed

## dar hilal

is ring of even integers i.d

## لآ تبكي يآ صغيري

😋🇸🇦

## 馬陸

What are the tatoos in her left forearm?

They look like something for passing a examination.

## gundamlh

Thank you! Could anyone be kind to tell me what's the solution to the 2 puzzles? My answers correct(?: *FALSE*): there exist (12^2-12)/2 + 12 = 78 quadratic equations; the number of those who have only 2 solutions is 78 – 4*12 = 30, where 12 eqns out of 30 are in the form of (x-a)^2 = 0 (that is a = b), while for the other 18 eqns, (x-a)*(x-b) = 0 (where a!=b). Could anyone tell me the correct answer? (I'm *WRONG*)

## Dengdai Geduo

keep'em coming beautiful!

## Randall Prather

Center

## monian

awesome video

## vishnu parihar

mam please explain what would be approach to think how many equation have exactly two solutions?

## Jiansen Zheng

Hahaha.

## Kassa Adane

Always I am interested when I am watching videos u present!

I like it.

## krong krong

1. change the question to "how many unique solutions". And we do not have to worry about the "zero divisors" because it's quadratic. zero divisors like accidental solutions. So it gives 12C1 + 12C2 = 78

there should be 78 sets of different solutions, which gives 78 unique equations.

2. 12 = 2x2x3

2×6, 3×4, 3×8, 4×6, 4×9, 6×8, 6×10 <== the differences of two numbers are 1, 2, 4, 5.

(x+a)(x+b) where a-b is not 1 or 2 or 4 or 5 will give two solutions exactly.

## p Upadhyay

what a nice explanation ma'am. Lucid one ..

## Carrie Ganote

I was lost in the first 10 seconds. Why factor and not just move the -4 to the other side? x^2 = 4?

## Adolf Ninh

ok from now my brain starts burning !

## SheMo Mousa

I love your charisma ❤️

## Hajira Ali

Wowwww thanks mam 😘😘😘😘your speaking is very nice and you are also cute

## Veglia Borletti

intensity of hotness is too much for me i just cant consume you liliana

## Aakash LonE

Shit…why not you making full vedio ,,,plz one plx make more nd more vedio about algebra….you are so smart in Algebra….u hv creative mind….

## Aakash LonE

It ix humble request to u ….plx make vedio 1 hour ….even I can watch your your vedio 10 sec even 10 hours also….best of best explanation…

## hamad mohammad

7:17 lol

## Now Or Never

Thanks

## Guido Feliz

Do you have a math degree, lady?

## Maciek300

So what's the answer to the first puzzle? I only could come up with an upper bound of 11*12*12.

## iKnney

Integral domain: If ab=0, then either a or b is 0 where a,b in R.

## Lekha Pratap

Chapelle wannabe.

Dank comedy!

## Kai Hiwatari

Maths teacher we all want❣️

## thorcuntdestroyess

loving the videos here — are there any notable groups/rings/etc with zero divisors other than groups of modular arithmetic? it seems that most of the caveat examples in this series' videos up to now come from modular arithmetic

## Wentworth Miller

The music at 0:30! Somebody please share where to find it!

## Maverick

Your videos are superb! Interesting to learn. Please make more videos on Rings – subrings and ideals, homomorphisms; principal ideal domains, Euclidean domains and unique factorization domains.

Thanks a lot!

## BlacKNyT

After seeing your presentation I feel like this is more of a Sarcatica than socratica.

## Ron McKay

best last 15 seconds

## DamnStupidOldIdiot

I came here thinking it is about domains of functions you integrate. I learned something new instead.

## chandrusekar 95

Hi

## samuil marshak

Бляяяя ну почему у меня не преподавала такая математичка!???

## Haim Ben Avraham

Sleeves rolled up, she means business.

## Sarosh Zaman

wat is mod 12

## Guy

Standup made me sub. I thought I had already clicked that, but, you know, it made me check again. And sub again. ^^

## mike lutta

Doggone it! I actually learned something…

Bet you don't know what Riemannian geometry is…

## Kris Kupiec

I think about math but also about her perfect mathematical figures;)

## mixbaal0

Thank you. You helped me giving sense to such a kind of concept. I am a self learning wanna be mathematician.

## Razor M

damn it, i lost it at the mod part.

## Diktakt

6 : 38

## Macam2macam

My teachers didn t teach me this. 🙁

## The Luminous One

This is not very advanced innerlectually…I've done all sorts…integration and disintegration…quadruple integrals for 4 variables and 4 unknowns …impartial differential equations…eigenscalars and eigencross product vectors with multi-domain tensor quadratics and that's just scratching the surface

## Win Ro

He he

## Mario Boley

Good stuff. A bit too fast. At this pace it's hard to even stop video at the right moment to do some thinking

## Brian Meyrick

Best stand up Maths I ever understood! 🙂

## tushar Joshi

ring (z = {0,1,2,….(p-1)},+,×) is an integral domain iff p is prime pls solve pls..

## Y N

She is a alien.

Inglês perfeito.

## Liam Martin 2: Electric Boogaloo

This lecture is so easy to understand, thanks Michael Jackson!

## ReviewTube

Why isn’t my math teacher this attractive ?

## Joja

If my math teacher is only half attractive as you i can make pluto a planet

## Jerry Gundecker

An equation with two answers. Isn't this how they found out about the existence of antimatter?

## Jonggi Lumbantobing

I don't understand anything she's saying but i watch it till finish, then comment and sub because she's so good looking and her voice makes me sleepy

## Balaporte Jean

thk you

Jesus loves you

Believe in him and repent

## Ademilson Silva

Love her voice…

## Shane Landy

This is so much better!

## Alejandro Elías Ruiz

Why not call it Integer Domain instead