How do you multiply expressions that have more than 3 variables?
For example, how would you multiply:
- $a \cdot b \cdot c \cdot d $
if associativity only defines the multiplication of 3 variables? ie. $(ab)c = a(bc)$
The same goes for matrices. How would you multiply:
- $A \cdot B \cdot C \cdot D$
If associativity only allows for the multiplication of 3 matrices? ie. $(AB)C = A(BC)$
This is something that I had never thought of (I am currently in high school) until recently and it is really problematic for me. Any insight into this would be appreciated.
We want to show $a(bcd)=(abc)d$. Notice that, if I were to write $bc=e$, then we can reduce it to associativity of $3$ variables as follows:
$a(bcd)=a(ed)=(ae)d$ (by associativity on $3$ variables).
But then putting back $bc=e$, we get $a(bcd)=(abc)d$. Thus we have shown associativity on $4$ variables.
Do you see how a similar argument would give us associativity on $5, 6, 7, \dots$ variables?
Basically, if multiplication of something (whether that's real numbers, matrices, or anything else you might encounter) is shown to be associative on $3$ variables, then you deduce that is is on any string of variables. We can thus forget about brackets, and write, for example, $abcde$ for any of
$$a(bcde) =ab(cd)e = a(bc)(de) = \dots$$
since we know these will just all be the same.
Note that there are weirder mathematical objects out there where associativity does not hold, and brackets are important. Nothing you'd meet in a high school course would have this property though, so don't worry about associativity for now.