A Note on Order

When it is necessary or useful to indicate that something is happening before something else, I shall enclose the things to be done first in parentheses. For example, I might have broken down 3+4 into (1+1+1)+(1+1+1+1), indicating that you can recover 3+4 by adding the first three 1's and the last four 1's together before you finally add their sums. Of course, since each time you add a one you simply move to the next number, no matter what order you add them, seven ones must add up to 7. This is why we may write 1+1+1+1+1+1+1 unambiguously, it does not matter how you add them. Since you can do addition in any order, this means that 1+(2+3)=(1+2)+3. It doesn't matter whether the 2 is grouped, or associated, with the 1 or the 3, the sum is still 6. Because addition works the same no matter how the numbers are associated, we say that addition is associative.