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The associative property for multiplication states that:

$(A \times B) \times C = A \times (B \times C)$

What happens in the case where there is an expression with more than $3$ factors?

For example, $(A \times B) \times C \times D$, since the associative property for multiplication only gives us a 'rule' for $3$ factors.

Would it be accurate to define a number, $E = C D$

and then if :

$(A \times B) \times C \times D$

$= (A \times B ) \times E$

$= A \times ( B \times E)$

$= A \times (B \times (C \times D) )$

$= A \times (B \times C) \times D$

which would show that the order of multiplication of any of those factors doesn't matter.

Can you use this logic for a larger amount of factors to conclude that rearranging the parenthesis in a multiplication won't change the result?

FishDrowned
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Kuskuba
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    That is a correct argument. You can prove by induction that you can put the parentheses anywhere in a product of $n$ factors. (Note that the "order of the factors" can matter in some situations. What doesn't matter is the order in which you group the multiplications.) – Ethan Bolker Nov 20 '23 at 01:38
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    @BenjaminWang The linked duplicate is a pretty heavy lift for this OP. – Ethan Bolker Nov 20 '23 at 01:41
  • yes you are correct in your reasoning – RyRy the Fly Guy Nov 20 '23 at 01:49
  • @EthanBolker when does the order of factors matter? Are you referring to matrix multiplication? Also, I know one Associativity also holds for matrix multiplication. Can I use the same argument that I used above to show that associativity also holds when more than 3 matrices are multiplied together? – Kuskuba Nov 20 '23 at 09:29
  • @Kuskuba Yes. That associativity argument works whether or not multiplication is commutative. – Ethan Bolker Nov 20 '23 at 12:11
  • @Ethan Bolker Thank you! Can you please show me how you would prove by induction that you can put parenthesis in a product of n factors? I am not sure what proof by induction is. – Kuskuba Nov 21 '23 at 23:26
  • There is a proof at this link https://math.stackexchange.com/questions/21581/how-does-one-actually-show-from-associativity-that-one-can-drop-parentheses , which @ArturoMagidin posts in his vote to close the question as a duplicate. – Ethan Bolker Nov 22 '23 at 00:33

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