Using Parentheses

Date: 11/16/98 at 21:58:05
From: Anna Martinez
Subject: Using Parenthesis in Math problems

Dear Dr. Math,

I don't understand how to use parentheses. Can you explain it step by 
step so I can get it? 

Thanks,
Anna

Date: 11/17/98 at 12:27:18
From: Doctor Peterson
Subject: Re: Using Parenthesis in Math problems

Hi, Anna. 

Did you ever see a Christmas package labelled "Open me first?" That's 
sort of what parentheses do. They wrap up some numbers and symbols, and 
tell us to do whatever is inside the package first, before we do 
anything else with those numbers.

For instance, suppose we have this expression (we write multiplication 
as "*" to avoid confusion with the letter x):

   2 * (3 + 4)

That tells us we first calculate 3 + 4 = 7, and then put that number 
in place of the whole package (3 + 4):

   2 * 7

Now we can do the multiplication, and calculate the answer as 14.

If you like, you can think of parentheses as a funnel that pours some 
ingredients into the calculation at the right place:

       (3 + 4)
        \   /
         \ /
   2 * (  7  )

If we didn't use the parentheses, but had:

   2 * 3 + 4

then the usual rules (called "order of operations") tell us that we 
should multiply 2 by 3 first, getting:

   6 + 4

and then add, so the answer is 10. Why is it different? It's sort of 
like if you got a cat, a mouse, and a cage for Christmas. If they 
weren't wrapped right, you'd end up with an empty cage and a satisfied 
cat. But if you wrapped it like this:

   cat (mouse + cage)

then you could first open the parenthesized package and put the mouse 
in the cage, then end up with a hungry cat and a caged mouse. That's 
probably what you really wanted!

Here's a place in the Dr. Math FAQ to find out more about the order of 
operations:

  http://mathforum.org/dr.math/faq/faq.order.operations.html   

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/