Multiplication First:
It is an unfortunate truth that multiplication and addition do not get along as well with each other as addition gets along with addition and multiplication gets along with multiplication. Since both are associative 2+3+4 is unambiguously 9, and 2*3*4 is unambiguously 24. However 2+3*4 takes different values depending on if you interpret it as (2+3)*4, which is 20, or 2+(3*4), which is 14. So, to resolve this ambiguity, we decree that multiplication occurs before addition. This is a common convention upon which we agree, rather than a mathematical truth which we discover, but we should stick with it nonetheless. So, things within parenthesis happen before things outside of them and multiplication happens before addition, these are conventions by which we must agree to live.
Distribution
The other interesting way in which addition and multiplication interact is called distribution. Consider 2(3+4), which is 2*7 or 14. This is the same value as 2*3+2*4 which, since we know to multiply before we add, is 6+8 or still 14. This is not a coincidence, when we switch from doing addition first to doing multiplication first, we must make sure that we multiply both values in the sum by 2. Consider if you had two bank accounts, into one you deposited a certain amount of money each month to save up for a vacation, and into the other you deposited a different amount of money to save for a rainy day. If, after three months, you wished to know how much money was in the accounts total, you could find out in two different ways. You could first figure out how much you deposited each month, by adding the amounts going into each account together, then multiply this monthly deposit size by 3, this way you add first. You could also figure out how much money was in each account individually after three months, by multiplying the deposit to each account by 3 separately, then add these account totals together to obtain the overall total, this way you multiply first. Notice that in the second case, the amount of money you put into the first account must be multiplied by 3 and the amount you put into the second account must be multiplied by 3, we say that the 3 is distributed to each of them.
Would you believe that I had to write the section on distribution 3 times because I kept going to make the heading large, forgot to close a quotation mark in my HTML, which Blogger interpreted as my command to irretrievably discard the second heading and everything after it! It took me a while to figure out that this was what was happening, but after I wrote it the third time I made sure to copy it so I could just paste it back after it mysteriously, and frustratingly, disappeared. Just goes to show that a little knowledge can be a dangerous thing.
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