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Order of Operations and Adding Negative Numbers

Date: 08/08/97 at 16:17:06
From: Dayna Chapman
Subject: Adding negative numbers

Dear Dr. Math,

Can you help me better understand a problem? The problem is:

   16 times 8 + 6 (-9 + -10) = ?

I know the PEMDAS: parenthesis, exponents, multiply, divide, add, 
subtract. I also know that the parenthesis go first, but I am still  
having a little trouble. I just started school Thursday and I'm in 8th 
grade algebra.  	


Date: 08/09/97 at 13:45:10
From: Doctor Mike
Subject: Re: Adding negative numbers

Hi Dayna,   

Okay, you have made a good start with the PEMDAS rule.
Also you are right that it means for expressions inside of the
parentheses to go first.  So, what does (-9 + -10) mean?  -19.
Let's use * for times, so we have  16*8 + 6*(-19).  There are
no more expressions inside parentheses (just the number -19) and
there are no exponents, so next we multiply and divide, left
to right.  That gives 128 + (-114), which equals 128 - 114 = 14.
If there is some part of this you don't understand, write back.
I'll just explain one more thing.  You might get confused about
what happens when 2 negatives are involved.  It is true that -9 
times -10 equals +90 (neg times neg is positive) but -9 plus -10
is -19 (neg plus neg is negative).  For a while you might have to
just remember the multiplication rule, but here's an easy way
to get comfortable with the rule about adding negatives.
Let's say you are measuring distances "up and down" from a spot on
the surface of the earth. One way to do it is to use positive
numbers for everything, like the flagpole is 25 feet high, and the
well is 123 feet deep. Another common way is to use positive numbers
for up and negative for down. That would put the top of the flag
pole at +25 and the bottom of the well at -123.  

Okay, so you've got that, up is plus and down is minus. Now, let's say 
you and a few friends are being paid to dig a hole, and the first day 
you make it to 10 feet deep (bottom of hole now at -10). The next day
you are all really tired from the first day and you only are able
to dig it 9 feet DEEPER (-9 more). You can tell from common sense
that if you dig it 10 feet deep, and then 9 feet deeper, it's going
to be 19 feet deep.  At that time you're going to take your money
and leave, but my point is that "9 feet deeper" and "-9 more" are both
ways of adding -10 and -9 together.  So.... -10 + -9 equals -19.  

This is written in terms of the "-" sign meaning "negative".  It
can also be written in terms of the "-" sign meaning "subtraction"
because : 
           -10 + -9  =  -10 - 9   

That is, in words, "negative ten plus negative nine" is the same as
"negative ten minus 9", and they both equal -19. This may be a little 
more than you wanted to know, but I hope it helps.    
P.S.  If you dig the ten-foot-deep hole in your front yard (better
  ask permission!), then set up a ladder at the bottom of this new
  hole, and then climb up the steps of the ladder until your shoes
  are 15 feet above the bottom of the hole, how far above the level
  of your yard are your shoes?  Right, -10 + 15 = +5 so they are
  5 feet above the level of the yard.  Now fill in the hole!   
-Doctor Mike,  The Math Forum
 Check out our web site!    
Associated Topics:
Middle School Negative Numbers

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