abstract algebra in under 15 seconds

In this #short, I explain the essence of abstract algebra in under 15 seconds. This is inspired by a TikTok post created by @tibees

[Risuchan]

If you can rescale, it's a vector space. If you can measure distances, it's a metric space. If you can measure angles, it's an inner product space.

[qschroed]

can you do a series on abstract algebra in the same style as the one you did on analysis? I think that would be very interesting

[helloitsme]

Except for dividing by 0 ofc, that's only allowed with trivial fields

[patipateeke]

Great! We always learned it in a much more difficult manner and then lost the understanding of it. Thanks for clarifying it in such an easy way, dr Peyam!

[iabervon]

I liked the set of integers 0 through 84 inclusive, but I didn't know what operations I needed to define. Thanks for clarifying Beyonce's advice!

[nicolastorres]

If you can add and subtract; and you can multiply by elements of another set, it’s a vector space.

[Uni-Coder]

If you do these things modulo 2^N, it's programming (C/C++/C#/Java).

[CharlesPanigeo]

What about a non-commutative division ring! That has addition, subtraction, multiplication, and division but it's not a field since the multiplication isn't commutative. The quaternions are an example.

[mudkip_btw]

Oh... I didn't know that was the meaning of ring, field etc. in maths. Really useful 14 seconds hahahaha

[BlackEyedGhost]

Being abstract algebra, the operations are defined by these properties rather than by any particular manipulation of numbers:
Addition: associative, identity (0), commutative (not required for monoid or group)
Subtraction: inverse under addition
Multiplication: associative, identity (1, where 0≠1 for a field), distributive over addition, commutative (only required for fields)
Division: inverse under multiplication

[laurenzhartmann]

Not true. For example, there are division rings that are not fields due to a lack of commutativity, e.g. the quaternions.

[randomvideos]

Most illuminating 13 seconds one can have...

[sumeshrajurkar]

You make it so simple to understand. 👍👍

[sergioconfero]

If you can add, subtract but you love SU(N) it is quantum field theory. If you can add, subtrack with diffeomorphism, Reiman tensor and SL(2, C) goes brr it is string theory.

[nasekiller]

honestly, i cant think of a single group, that has an operation called addition and is neither a ring, nor a vectorspace/module.
and in group theory you use the same notation as if you were using multiplication.
and if an operation is referred to as addition, i expect it to be commutative, which is obviously not true for groups in general.

[jordanweir]

*drinks from flask*
i left that life behind

[tricky]

Is {+, -} different from {+, invert} ?

[brendanlawlor]

simplicity is clear thinking evidence of talent & benefits students, wish my academics were like Peyam 😀

[cbbuntz]

Who needs classes when you can just watch a 15 second video by Dr Peyam?

[lucadr]

Think I have to watch it again and again and ...