Algebraic Structures: Groups, Rings, and Fields

[@mikegoldsmith3600]

Extremely clear and covers all the basics. The best gentle introduction to algebraic structures I've been able to find!

[@arify7344]

Good luck in your Algebra exams, fellow students

[@ArifYunando]

Hello Arif

[@its_roggy]

Thanks, we have one tomorrow

[@imcloudy1909]

Y am I doing this in year 9...?

[@daytonrowen4515]

i know it is quite off topic but does anyone know a good site to stream newly released movies online?

[@supongmenwalling5318]

I failed...ooooooo😭🤣

[@nadaabdulla7556]

I finished the course 2 years ago, I didn't understand it then, but now I'm interested and regret that I didn't do my best :(

[@hermanaksom5303]

No doubts, it is the best introduction to the subject I ve ever seen

[@johntryl8009]

The example of Z mod n (when n=prime) being a field and not a ring is the coolest thing ever. Furthermore, your explanation of why complex numbers are a vector space made things finally click ... it has scalar multiplication and it has addition, but it just has even more properties. This was so helpful. Thank you for being super approachable and clear!

[@shivanya816]

heyy how come... Z mod n is a field but not a ring? isnt a field a commutative ring? please clarify 😊

[@plaustrarius]

cannot thank you enough for this video!!

[@mcmoodoo]

Absolute Gold! Thank you, sir!

[@orang1921]

Imagine a student in Algebra I or Pre-Algebra finding this video ... RIP Also great video

[@lamalamalex]

I do agree with you that you built up according to complexity of the structures. With vector spaces appropriately at the end. So that’s why I find it very strange that that’s where we start students at. Linear algebra being such an early class students takes. It can even be taken before a multi variable calculus course.

[@anantrelan4071]

@12:01 Field is a ring with two operations . @18:12 F is a Field under (only) Multiplication . Q. Why is there only 1 operation for the field F at @18:12 ? Thanks

[@HamblinMath]

A field always has two operations, addition and multiplication. I'm distinguishing the field multiplication (scalar times scalar) from "scalar multiplication" (scalar times vector).

[@anantrelan4071]

@@HamblinMath oh ok Thanks for the Reply !!

[@leylaalkan6630]

Thanks for these amazing clarifications.

[@rithikseth1404]

Thanks very helpful for my engineering studies ....

[@clu5ter892]

Thanks for this vid from Russia)

[@LucyMuthoni]

Thank you very much for this video. Be blessed!

[@aritraroygosthipaty3662]

A very helpful lecture.

[@eset3649]

Crystal clear, thank you sir.

[@c0t556]

Very good explanation! Thank you so much!

[@MunkyChunk]

You are a... GENIUS!!! Thank you!!

[@Ivane.h]

Single-handedly getting me trough ADM mit Gittenberger...

[@sunildhull8878]

at 5:43 set of integers mod n became non negative integer which not follow inverse property over addition so it not supposed to be grp i.e. |-3|+|3|=6 not 0 plssss reply

[@HamblinMath]

Sunil, in arithmetic mod n, you take the remainder when the number is divided by n. For example, in arithmetic mod 7, the inverse of 3 is 4 because 3+4 = 0.

[@DavidVonR]

Too cool! I love group and ring theory :)

[@katelikesrectangles]

That was really helpful, thank you!

[@asmamokr1345]

thk'x a lot but i have a question ... for groups the first example for the inverse (-a) don't belong to Z ( but in the rule it should belong ) ...i am confusing 😣😣

[@MathMaster19]

-a belongs to Z, it doesn't belong to N

[@shrimp8594]

Thank you:) Really helpful video.

[@TheHuggableEmpire]

So addition and multiplication in rings doesn't necessarily mean the usual sum and product?

[@janoprivracki1992]

Correct, these are abstractions. Don't mind me commenting a year later... rofl. Hopefully it's helpful to someone in the future

[@evrenunal3644]

@@janoprivracki1992 it indeed helped me, thanks

[@AkamiChannel]

This was great. I just wish you had gone into what an algebra is. I'm on a mission to understand that, but google and youtube search results are completely worthless to me because they're full of content explaining ordinary algebra.

[@HamblinMath]

An "algebra" is a vector space over a field that has multiplication of vectors. Complex numbers are an example of an algebra.

[@AkamiChannel]

@@HamblinMath Yes, I had realized as much. Was thinking of a more formal explanation like one often sees for vector spaces. I did find one on youtube yesterday. It seemed to me, though, that the formal definition of an algebra is so general that just having a vague idea of it is enough.

[@jasminefitzsimons896]

ok I literally love you

[@fraktallyfractals2083]

About Z as a group, at the beginning of the video, does that mean that zero is its own opposite?

[@josvandeneynde5849]

Great video! I thank you!

[@pragyapathak8660]

Nice vedio sir but in group definition closure property is not mentioned

[@HamblinMath]

Closure is typically understood to be part of what you mean when you say that the operation is "on" the set G.

[@AyushSharma-ux4fk]

can you please share the slides that you are using to teach?

[@Caleb-qr6lo]

lol I see xor symbol and get really confused.

[@rushikeshkavar6128]

Nice video. But is Ring Definition correct? According to Wikipedia, There should be additive identity and additive inverse. Am i wrong? Please clarify.

[@HamblinMath]

Kavar Rushikesh R being a commutative group under addition includes those properties.

[@gunjanrathore9337]

Is division for any R' N' Q' is made an algebraic structure??? R set of real no N set of natural no (1, 2,3...) Q set of rational no

[@monsieurfrog]

Wouldn't Monoids be the simplest algebraic structure? When defining a group (M, #), it must first be a monoid, in addition for each element having an inverse.

[@arnabdasphysics]

Superb!

[@SzechSauce]

Great explnantions thanks!

[@roysarit]

One more property of Groups - Closure property. If A , B belong to G, then if A ⊕ B = C, C also belongs to G.

[@HamblinMath]

Using your notation, if "A ⊕ B" didn't belong to G, what are we even talking about? This is often rolled into the definition of what it means for "⊕" to be an operation on the set G.

[@roysarit]

​@@HamblinMath I guess you are right, but making it explicit may help beginners, so thought of mentioning here.

[@ebrimagajaga4639]

Please help me answer this question… Is (N, +) a group ? N is a set of natural numbers…

[@HamblinMath]

No, because not every element of N has an inverse.

[@tunistick8044]

​@@HamblinMath so in Z it's true?

[@HamblinMath]

@@tunistick8044 The set of integers is a group under the addition operation.

[@lamalamalex]

I don’t understand why y’all want to hide the operations of ➕ and ✖️ and then just talk about those. I mean, what else is there? Why the back and forth?

[@privateaccount1266]

Because the operation symbol could be either + or x. For example when you saw the properties that characterise a group there was a symbol. And something could be a group with + or a group with x. You use + or x depending on the question.

[@poomalaip2620]

Give each definition examples

[@osebrainquestfoundation9631]

It knowledgeable. Thanks

[@footage6402]

How does a group differ from a field?

[@MagikarpKano]

a field has 2 operations and an inverse. A ring does not always have an inverse and a group only has 1 operation.

[@harirao12345]

very clear ... thank you!

[@DantalionNl]

Why is the property A * B = B * A called identity and not commutativity ?

[@HamblinMath]

Commutativity says "for all A and B in the set, A*B=B*A." It's not called identity. Read the "identity" property carefully.

[@andreiparaschiv3257]

great video one of my concerns is that people could get the idea that you can prove a property by trying out random examples, as you did with the multiplicative inverse over Q[radical 2] by choosing a=3 and b=4. it has to be generalised, and that means not assigning specific values. that could have been made a little clearer in the clip.

[@HamblinMath]

I do specifically say "this isn't a proof"...

[@Okapi000]

Why don't you include closure as a necessary property to be a group?

[@HamblinMath]

"Closure" is typically assumed when we say that "+ is an operation on G."

[@shiina_mahiru_9067]

I dont think R is a field since 0 has no multiplicative inverse, but R* would be

[@Enriquecav]

The multiplicative inverse is defined for all numbers except 0, so R is a field

[@safofoh]

Thanks, it's very useful

[@joeflaubert5597]

Thank you

[@Ambagaye]

Closure

[@alex-my8hp]

you missed out closure

[@HamblinMath]

While "closure" is sometimes included as a group/ring axiom, it's not really necessary, since for the operation on two elements to make any sense, the result of the operation must be in the set you're talking about.

[@alex-my8hp]

@@HamblinMath oh, fair enough

[@48_subhambanerjee22]

love it

[@AlessandroZir]

❤️❤️❤️🙏🙌

[@haentertain8383]

Please send to me solved ring examples all

[@felixhsu9583]

祝我信安数基能及格🌝

[@a_ghoul]

i am only in the 2nd grade how do I find x 2x+2=6

[@qusai05]

subtract 2 from both sides, u get 2x=4, then divide by 2 on both sides, u get x=2

[@soseipirialadick-iruenaber7273]

You're talking too fast and I'm not understanding 😔

[@yasser-l3x]

Zb ybki

[@joeyquiet4020]

thank you