Using the Distributive Property

Date: 03/06/2002 at 12:23:14
From: Melissa
Subject: Integers

This question is actually from my chemistry class:

   -2.4222 x 10^-19 - (-6.0556 x 10^-20)

I tried typing it in as is on the calculator. As well I have tried 

   -2.18 x 10^-18 + .06556 x 10^-18

The teacher gave the answer as +1.81664 x 10^-19. I keep getting the 
opposite "sign" on my answers.

Date: 03/06/2002 at 12:44:45
From: Doctor Ian
Subject: Re: Integers

Hi Melissa,

The idea behind adding numbers like this is that if you have the same 
exponent, you can use the distributive property:

  2.13 * 10^-5 + 3.46 * 10^-5 = (2.13 + 3.46) * 10^-5

                              = 5.59 * 10^-5

It's the same as if you had a variable instead of an exponent:

  2.13x + 3.46x = (2.13 + 3.46)x

                = 5.59x

When you start with different exponents, you have to pick a common 
exponent - usually one of the exponents you already have - and convert 
both numbers to it:

    2.13 * 10^-5 + 3.46 * 10^-6 

  = 21.3 * 10^-6 + 3.46 * 10^-6

  = (21.3 + 3.46) * 10^-6

  = 24.76 * 10^-6

Changing the exponent doesn't change the sign:

    -2.13 * 10^-5 + 3.46 * 10^-6 

  = -21.3 * 10^-6 + 3.46 * 10^-6

  = (-21.3 + 3.46) * 10^-6

  = -(21.3 - 3.46) * 10^-6

  = -17.84 * 10^-6

When you use the distributive property explicitly, as I have here, it 
can help you avoid making sign errors.  (And you don't even have to 
enter the exponents in the calculator.)

For your first problem, I get

    -2.4222 * 10^-19 - (-6.0556 * 10^-20)

  = -2.4222 * 10^-19 - (-0.60556 * 10^-19)

  = (-2.4222 - -0.60556) * 10^-19

  = (-2.4222 + 0.60556) * 10^-19

  = -(2.4222 - 0.60556) * 10^-19

  = 1.81664 * 10^-19

This is the answer your teacher got. Can you see now why it's correct?

Note that writing (or typing) numbers with lots of decimals is a good 
way to make careless transcription errors. This is one of the reasons 
that variables were invented. I would do the problem this way:

    -2.4222 * 10^-19 - (-6.0556 * 10^-20)

  = -a * 10^-19 - (-b * 10^-20)

  = -a * 10^-19 - (-b/10 * 10^-19)

  = (-a - -b/10) * 10^-19

  = (-a + b/10) * 10^-19

  = -(a - b/10) * 10^-19

  = -(2.4222 - 0.60556) * 10^-19

  = 1.81664 * 10^-19

This makes it much easier to check that you haven't messed up a sign, 
and it's a lot easier on your fingers.

Does this help?

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

Date: 03/06/2002 at 18:08:13
From: Melissa
Subject: Integers

Thanks for getting back to me. Still having difficulty. I understand 
everything up until here:

 = -(2.4222 - 0.60556) * 10^-19

  = 1.81664 * 10^-19

Where did that negative sign in front of the brackets come from? 
Please refresh my mind on this procedure as it has really been about 
10 years since I learned all this the first time!

Date: 03/06/2002 at 18:37:33
From: Doctor Ian
Subject: Re: Integers

Hi Melissa,

That's a shortcut for a particular use of the distributive property,

  a(b + c) = ab + ac

The longer version looks like this:

    -2.422 + 0.60556

  = (-1 * 2.422) + (-1 * -0.60556)    Each term is -1 times something.

  = -1 * (2.422 + -0.60556)           Distributive property.

  = -1 * (2.422 - 0.60556)            To add a negative, subtract
                                      a positive.

  = -(2.422 - 0.60556)                Keep the sign. 

In practice, when you have something like

  (a - b + c - d + e ...)

changing the sign of the whole thing also changes the signs of the 
individual variables:

      (a - b + c - d + e ...)

  = -(-a + b - c + d - e ...) 

Again, the distributive property explains why this works; but in 
practice, you just use the shortcut.  
 
Does this help? 

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