I was watching a khan academy video about division which described division in two different ways and I’m now confused by my interpretation of division as the inverse of multiplication.
I understand why multiplication is commutative: 12 objects could be counted as 3 sets of 4 or counted as 4 sets of 3 with the same result:
My interpretation of how 3x4=12 AND 4x3=12 $$$$ However, when he introduced division he asked ‘what is 8÷2?’, and said it was the answer to the question ‘how many groups of two can be made out of eight?’, and he drew:
4 groups of 2 could be made, so the answer is 4. $$$$ But, he then immediately said ‘Or it’s like the answer to the question - split 8 into 2 equal groups, how many are in each group?’, and he drew:
4 objects were in each group, so the answer is 4. $$$$
I was confused to see two different ways to model division, since there’s only one way to write it.
With multiplication there are two ways to write the expression, and they correspond to two ways to model the situation - and it all demonstrates the commutative nature of multiplication.
Thinking of division as the inverse of multiplication, it’s like the two models of division are recovering the two models of multiplication and it all fits together - the commutativity of multiplication is still ‘in’ division somewhere. However, whereas multiplication has notation for its two commutations, it seems like there’s a discrepancy with the division notation. It’s like there’s information lost somewhere.
It feels like division wants commutativity, it just doesn’t know how.
Q1: What have I misunderstood, how can the same notation ‘a÷b’ describe two different processes?
Q2: Does mathematics talk about information being lost by having something which is noncommutative as the inverse of something which is commutative?
Q3: What area(s) of mathematics have I touched on/where can I go for more information?
Thanks so much for your time!