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Parentheses from the Inside Out

Date: 03/03/2003 at 19:26:51
From: Kaila
Subject: I don't understand evaluating expressions

Here is one of the problems.

   2a + b - 1 = 

The things that I find most difficult are following PEMDAS and 
multiplying the fractions by the whole numbers without using a 
calculator.

Here's a problem from my homework. 

2b divided by a+2 to the second power. b = 1.5, a = 3 
  1. 2(1.5)divided by 3 + 2 to the second power.
  2. 3.5 divided by3 +2 to the 2 power
  3. 1.5+4
  4. 5.5 is this right?


Date: 03/04/2003 at 13:26:30
From: Doctor Ian
Subject: Re: I don't understand evaluating expressions

Hi Kaila, 

Let's look at an expression like 

  2 + 3 * (4 - 5 * (6 / 3)) - 4

The parentheses are like an envelope that we have to open to find out
what's inside.  The first parentheses we run into are

  (4 - 5 * (6 / 3))

so we forget about everything else while we work this out:

   4 - 5 * (6 / 3)

Oops!  We have more parentheses.  So let's forget about everything
else until we work out

  6 / 3

Okay, there's just one operation, so we do it: 6/3 = 2, so we can go
back and replace the parentheses (and everything inside) with the 
result:

   4 - 5 * 2

Now we have no parentheses, so we look for multiplications and
divisions.  There is just one multiplication, so we do it:

  4 - 10

And now we just have a subtraction, so we do that:

  -6

So now -6 replaces what was in the original parentheses:

  2 + 3 * -6 - 4

Again, we have just one multiplication, so we do it:

  2 + -18 - 4

And now we have additions and subtractions, so we do them left to 
right:

  2 + -18 - 4

   -16 - 4

   -20

And that's our result. 

Note that doing operations right to left (instead of left to right)
can give you the wrong answer.  For example, consider

  72 / 6 / 3

Doing the operations right to left would give us

  72 / 2

which is 36; while doing them left to right gives us

  12 / 3

which is 4.  Also, if we do 

  10 - 5 - 3

from right to left, we get

  10 - 2

which is 8; but if we do it from left to right, we get

  5 - 3

which is 2. 

Now, is there anything magic about these rules?  No, it's just that we
like to be able to write expressions as simply as possible, and 
writing

  3 + 4 * 5 - 6 * 7 / 3 + 8

is simpler than writing

  ((3 + (4 * 5)) - ((6 * 7) / 3)) + 8

If we all follow the PEMDAS conventions, though, the two expressions
have exactly the same meaning. So one the one hand, we all have to
memorize a small set of rules. But on the other hand, over the course
of a lifetime, we don't have to write thousands or millions of extra
parentheses. Doesn't that seem like a reasonable trade-off? 

Now, let's stop a moment and look at what I did there.  How did I get
from the simple expression to the one with all the parentheses?  I
started with 

  3 + 4 * 5 - 6 * 7 / 3 + 8

and I looked for each multiplication or division, from left to right.
And I put them in parentheses:

  3 + 4 * 5 - 6 * 7 / 3 + 8
      ------

  3 + (4 * 5) - 6 * 7 / 3+ 8
                -----

  3 + (4 * 5) - (6 * 7) / 3 + 8
                ----------- 

  3 + (4 * 5) - ((6 * 7) / 3) + 8

Note that once I've got parentheses around some stuff, it's just like
a single number.    

When I run out of those, I do the same thing with additions and
subtractions:

  3 + (4 * 5) - ((6 * 7) / 3) + 8
  -----------

  (3 + (4 * 5)) - ((6 * 7) / 3) + 8
  -----------------------------

  ((3 + (4 * 5)) - ((6 * 7) / 3)) + 8

Now, this is a lot of extra work, but it has this advantage: Once 
you've done it, the parentheses tell you, step by step, what you have
to do. Just look for any set of parentheses with no other parentheses
inside, and do the operation. As long as you do them from the inside
out, you don't even have to pay attention to order:

  ((3 + (4 * 5)) - ((6 * 7) / 3)) + 8
        -------

  ((3 + 20) - ((6 * 7) / 3)) + 8
   --------
        
  (23 - ((6 * 7) / 3)) + 8
         -------

  (23 - (42 / 3)) + 8
        --------

  (23 - 14) + 8
  ---------

  9 + 8
  
  17

So in a sense, these rules are a little like deciding that the letters
of the alphabet will go in a certain order. It's not really all that
important what the actual order is, but it's VERY important that we
all use the same order! Otherwise, we'd never be able to find 
anything. 

> Here's a problem from my homework. 
> 2b divided by a+2 to the second power.

Note that the phrase

  2b divided by a+2 to the second power

could be interpreted in more than one way.  It might mean

                                                 2b   2
  (2b divided by a+2) to the second power -->  ( --- )
                                                 a+2

or

                                                    2b   
  2b divided by ((a+2) to the second power) -->  -------- 
                                                       2
                                                  (a+2)

or

                                                    2b   
  2b divided by (a+(2 to the second power)) -->  ---------- 
                                                        2
                                                  a + (2 )


And in fact, with no parentheses, following the PEMDAS order, the
phrase would be interpreted this way:

  2b divided by a+(2 to the second power)       E

  (2b) divided by a+(2 to the second power)     M or D

  ((2b) divided by a)+(2 to the second power)   M or D

that is, 

    2b      2
   ---- + (2 )
     a

So if they really wrote it in symbols, as 

      o       2
  2b --- a + 2
      o

instead of using words, then this last interpretation is the
conventional one. Note that a fraction bar acts like two pairs of
parentheses with a division sign in between:

  blah blab blah
  -------------- = (blah blah blah) / (yadda yadda)
   yadda yadda


> 2b divided by a+2 to the second power. b = 1.5, a = 3 
>   1. 2(1.5)divided by 3 + 2 to the second power.
>   2. 3.5 divided by3 +2 to the 2 power
>   3. 1.5+4
>   4. 5.5 is this right?

Let's follow the PEMDAS order, and see if we get the same thing. There 
is an exponent, so we want to evaluate that first. Then we want to do 
multiplications and divisions, from left to right. Finally, we want to 
do additions and subtractions from left to right. 

    2 * 1.5 / 3 + 2^2
                  ---           Exponent

    2 * 1.5 / 3 + 4
    -------                     Multiplication           

    3 / 3 + 4
    -------                     Division

    1 + 4
    -----                       Addition


    5                           Result

Why is this different from your answer? For one thing, it looks as if
you multiplied 2 by 1.5 and got 3.5, instead of 3. That may have been
a slip, or it might indicate that you need to work on your
multiplication skills.  

To check, I'd probably think about money. If two of us have $1.50 
each, how much money do we have? Two dollars and two half dollars is 
three dollars, so 2 * 1.50 must be 3. But everyone thinks about math a 
little differently from everyone else, so you have to find out what 
works for you. 

Anyway, I hope this has helped you see why we have these conventions,
and what can happen if you do operations out of order. Write back if
you'd like to talk more about this, or anything else. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
Middle School Algebra

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