23:31 This might be interesting:
"There are 18 inequivalent ways to define an associative operation on three elements: while there are, altogether, a total of 3^9 = 19683 different binary operations that can be defined, only 113 of these are associative, and many of these are isomorphic or anti-isomorphic so that there are essentially only 18 possibilities."
7:43 I think mathematicians should deprecate the use of × for multiplication, and use asterisk * instead. The symbol × can easily be confused with x, the letter. I like how you're x is curvy so it looks distinguishable.
Arya Pourtabatabaie It’s a valid complaint mate. This guy is helping us by teaching us, but it’s really infuriating to have every other word be that damn “O-Kay!” That he loves spewing out all the time, which is only made worse by his aggravatingly annoying voice. Reading such a complaint is literally supposed to help the content creator become better, so I don’t think saying “Don’t be a dick” is really helping in the long run.
Yes, because function composition is one possible example of this composition. However, we find out later (Cayley's theorem) that this composition actually *is* (can always be viewed as) a function composition! The group elements can be identified with permutations (bijective functions) of a set and the composition operation of the group can then be identified with the usual composition of permutations.
2:00 But abstract algebra *is* useful for the real world. That might not be the point of it (as far as pure mathematicians are concerned) but time and again it has proved it usefulness; for example in group theory which has proved very useful in describing the symmetries of particle physics. :)
An excellent series. Start at part 1, keep going through them all and you'll have a head start on taking a mathematics degree at college. I'm an engineer with an engineer's understanding of field theory to get by in crypto and comms coding. This series has really helped me back fill with the group theory they didn't teach me at college.
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