ARITHMETIC WITH NEGATIVE NUMBERS

• Absolute value
• Addition
• Double negatives
• Multiplication and Division
• Go to the number line/negative numbers home page
• contact me

The absolute value will leave positve numbers alone, but it will change a negative number into it's positve counterpart.
Thus |8| = 8 and |-7| = 7.

Before starting our study of addition with negative numbers, practice using absolute values.

• Adding a positve number to a positive number
• Adding a negative number to a negative number
• Adding a positve number to a negative number

We'll start with how to write a problem such as this.

Suppose I wanted to add -2 to -3. I could write it in three different ways

• -2 + -3
• -2 - 3
• (-2) + (-3)

Adding two negative number's is really quite simple: simply add the absolute values of the two numbers, and place a negative sign right before this sum.

Thus (-2) + (-3) would equal -5

|-2|=2 and |-3|=3.
The sum of these absolute values is 5.
Placing a negative sign before 5 yields -5.
Thus -2 - 3 = -5.

Here is another way to think of this problem:
Remember when I advised you to think of negative numbers as owing money?
-2 would mean owing $2 and -3 would mean owing an additional $3
Putting these two numbers together would mean that you now owe $5.
Again -2 - 3 = -5.

Before going on to the third case scenario (adding a negative number to a positve number), let's work out a few more problems.

Example 1
-21.45 - 31.23

Now we know that -21.45 - 31.23 means the same thing as adding -21.45 to -31.23.
We know 21.45 + 31.23 = 52.68 so
-21.45 - 31.23 = (-21.45) + (-31.23) = -52.68

Example 2
-2/5 - 1/15

Now 2/5 + 1/15 = 6/25 + 1/15 = 7/15
So -2/5 - 1/15 = -7/15.

As with the above scenario, adding a negative and positve number can be written in a few different ways.
Suppose I wanted to add -4 and 1. This could be written as

• -4 + 1
• 1 - 4
• (-4) + (1)

Let's see how this problem, and ones like it, can be accomplished.

Now if this were 4 - 1 (a positive 4 plus a negative 1) then there would be no problem. 4 - 1 = 3.
However here |-4| > |1| (Note the absolute values).
Since the negative value is greater then simply change the 3 into -3.
Thus 1 - 4 = -3.

Using our monetary analogy you could think of this as having $1 (the positve 1) and owing $4 (the negative 4).
In the long run you would still owe $3 (negative 3).

In general when you have a negative and positive number, perform the calculation as you would if it were a subtraction problem and the absolute value of the positve number were greater than the absolute value of the negative number. (in other words thing of 1 - 4 as 4 - 1)
Once you have the sum (difference) give it the sign of the number with the larger absolute value.

Let's look at two more examples.

Example 1
212 - 431

Now |-431| > |212| (431 > 212) so we know that the answer will be a negative number.
Now 431 - 212 would be 219
so
212 - 431 = -219.

Example 2
(-1/14) + (2/7)

First change 2/7 into 4/14.
The problem is now (-1/14) + (4/14) which is the same as writing 4/14 - 1/14.
4/14 - 1/14 = 3/14.
This number stays positive since |2/7| > |-1/14|.
(-1/14) + (2/7) = 3/14

Next we'll explore numbers with two negative signs in front of them, but first practice what you just learned.

Well this is the case when it comes to a negative of a negative number.
Thus -(-1) = 1 ; -(-2.3) = 2.3 ; -(-1/2)=1/2 ; etc.

What does this mean to our current discussion regarding adding numbers, you may ask.
Well let's look at a few problems and find out.

Example 1
4.3 -(-1.23)
Now 4.3 -(-1.23) = (4.3) + (-(-1.23)) and -(-1.23) = 1.23.
So 4.3 -(-1.23) = 4.3 + 1.23 = 5.53.

Example 2
-3.5 -(-0.3)
-3.5 -(-0.3) = -3.5 + 0.3 = -3.2

Next we'll explore multiplication and division, but first practice what you just learned.

1. First multiply (or divide) the absolute values of your numbers.
2. Assign a sign to your answer according to the following rule:
   • Two positive numbers will yield a positive answer
   • Two negative numbers will yield a positive answer (This is an extension of the above double negative rule)
   • A positive and a negative number will yield a negative answer

Example 1
-1 2/3 * -2 3/4

First of all consider the two factors to be 1 2/3 and 2 3/4.
1 2/3 = 5/3 and 2 3/4 = 11/4.
5/3 * 11/4 = 55/12 = 4 7/12

Now we have to assign a sign to the answer
We have two negatives which according to the rule will yield a positive number
Thus -1 2/3 * -2 3/4 = 4 7/12

Example 2
-2.13 * 1.5

First of all consider the factiors to be 2.13 and 1.5
2.13 * 1.5 = 3.195

Now to assign a sign , refer to the rules. A negative and a positive number yield a negative number.
Thus -2.13 * 1.5 = -3.195

Example 3
Divide 3150 by -126.

First of all consider the 3150 divided by 126.
3150/126 = 25

Now to assign a sign , refer to the rules. A negative and a positive number yield a negative number.
Thus 3150/(-126) = -25

Example 4
Divide -3/4 by -1 1/8

First of all consider the dividing 3/4 by 1 1/8
1 1/8 = 9/8 3/4 ¸ 9/8 = 3/4 * 8/9 = 3/1 * 2/9 = 1/1 * 2/3 = 2/3

Now to assign a sign , refer to the rules. A negative and a negative number yield a positive number.
Thus -3/4 ¸ -1 1/8 = 2/3

Now it's time for your practice session again. We'll start with multiplication.

• Review what you learned in this unit with a few more problems
• Go back to the number line/negative main page
• Go back to the tutorial main page
• Read my closing remarks