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Evaluating Parentheses, Brackets, Braces and Grouping Symbols

Date: 01/20/2007 at 16:58:10
From: Nelson
Subject: Pre-algebra math (parentheses, brackets and braces)

(18/2){[(9 x 9 - 1)/ 2]-[5 x 20 - (7 x 9 - 2)]}

Parentheses, brackets and braces are very confusing to me.  For
instance, the mathematical sentence above displays more brackets,
parentheses and braces than numbers.

According to the mathematical rules in place today, what am I 
supposed to know in order to see this problem and quickly know where 
to start and where to follow next in order to ensure that I will 
always get to the right answer?

Even though I know that I am supposed to start with the innermost 
parentheses and then follow with the [ ], and then the { }, I still 
feel very uncomfortable with these figures mixed with the numbers.

(18/2){[(9 x 9 - 1)/ 2]-[5 x 20 - (7 x 9 - 2)]}

Here, I start by: 9 x 9 = 81 - 1 = 80.  But then what do I do?  Keep 
going through the ( ) and divide 80 by 2 = 40?  If that's the right 
course of action, then I'd follow with: 7 x 9 = 63 - 2 = 61, then I 
would come back and do: 5 x 20 = 100 - 61 = 39.  And then, last, I'd 
divide 18 by 2 = 9.  Then, 40 - 39 = 1; then 9 x 1 = 9?

Obviously, I am going more by instincts than by logic.  What do I need
to know in order to laugh about problems like the one above instead of
worrying about them?



Date: 01/20/2007 at 23:50:33
From: Doctor Peterson
Subject: Re: Pre-algebra math (parentheses, brackets and braces)

Hi, Nelson.

The first thing to do is to start with something not quite so
complicated, so you can get used to the ideas first.  But let's go
ahead with this one, to check your work.

Here's one way I write these to demonstrate how to think it through:

  (18/2){[(9 x 9 - 1)/ 2]-[5 x 20 - (7 x 9 - 2)]}
  \____/  \_________/               \_________/
     9  {[    80     / 2]-[5 x 20 -     61     ]}
         \______________/ \____________________/
     9  {       40       -           39         }
        \_______________________________________/
     9 *                  1
     \____________________/
                9

So you're correct, and your work was fine.  The only thing I did
differently was to evaluate 18/2 earlier, because nothing stood in its
way; I could have waited as you did.

What parentheses do is to contain a subexpression that has to be fully
evaluated before it can be used in any containing expression.  That's
why you work from the inside out: you can't use what's inside until
you evaluate it all, so you might as well start there.  But if you
forgot to, you'd still have a reminder.  Here's an example:

  2[(3 + 7)(3 - 2) - 3(2 + 2)]

If I didn't bother with the inside-out "rule", I  might just start
trying to evaluate at the left (paying attention to the order of
operations, of course): 2 times ... what?  Well, the second number in
that multiplication is the whole thing inside [...], so I have to put
it on hold until I do that.  So I focus on

  (3 + 7)(3 - 2) - 3(2 + 2)

Now I start that.  The first piece is (3 + 7), so I evaluate that
whole thing and get 10.  Now I have to multiply it by (3 - 2), so I
stop and evaluate that, which gives 1.  Now I can multiply 10 by 1 and
get 10.  So I keep going; I have to subtract something from that, but
since the next bit is a product, I have to do that first.  I'll have 3
times the next parenthesis; that's 3 times 4, so I have 12.  The
subtraction I put off is 10 - 12 = -2.

Now, this is what the whole [...] is, so I go back and do that last
multiplication:

  2*(-2) = -4

That was pretty ugly, wasn't it?  That's how a computer would evaluate
it (more or less); it just keeps putting something on hold and coming
back to it when it's ready, because it has a really good memory and
won't forget to do it!  We don't recommend this method for humans, but
it can be done.  Here's how I'd usually do it:

  2[(3 + 7)(3 - 2) - 3(2 + 2)]
    \_____/\_____/    \_____/
  2[   10  *  1    - 3*  4   ]
       \______/      \___/
  2[      10       -   12    ]
   \_________________________/
  2*            -2
  \______________/
        -4

Again, I just evaluated ANY parenthesized subexpression that didn't
have anything standing in its way (that is, ALL the innermost ones);
then I went through it again evaluating everything that was left the
same way.

In reality, on paper, my work might look like this, just jotting down
the value of each parenthesis as I came to it, but doing most of the
work in my head:

  2[(3 + 7)(3 - 2) - 3(2 + 2)] = 2[10 - 12] = -4
     10     1          4

My point is that inside-out is not a magic formula; it's just a 
natural result of what parentheses do.  Wherever you find a 
parenthesis, you make sure it gets evaluated before you use that
particular result in any further operations.  So, for example, that
(18/2) in your example could be done at any time, because it wasn't
needed until the end anyway.

I suspect your "instinct" is to do exactly this, and it fits with the
logic perfectly (when you let it operate).

If you have any further questions, feel free to write back.  If I've
totally confused you, let me know and I'll try again from a different
perspective.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
Elementary Addition
Elementary Division
Elementary Multiplication
Elementary Subtraction
Middle School Division

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