This video covers the definitions for some basic algebraic structures, including groups and rings. I give examples of each and discuss how to verify the properties for each type of structure.
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!
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.
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 😣😣
Nice video. But is Ring Definition correct? According to Wikipedia, There should be additive identity and additive inverse. Am i wrong? Please clarify.
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 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.
@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
A field always has two operations, addition and multiplication. I'm distinguishing the field multiplication (scalar times scalar) from "scalar multiplication" (scalar times vector).
Why in rings case addition should be commutative? What if one operation is commutative, but it is distributive over second which is not? Can this structure be considered ring? What if none of the operation is commutative but one of them distributes over another? What is the point of distributive properity? Why is it even introduced? Is it states superiority of one operation over another or what? What if both operations are distributive over each other, like conjuction and disjunction? Should all those cases be considered as rings?
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
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.
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.
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.
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.
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?
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.
