1.3 Add and subtract integers  (Page 6/10)

Subtract: ⓐ $\mathrm{-3}-1$ ⓑ $3-\left(\mathrm{-1}\right).$

Solution

ⓐ

Take 1 positive from the one added neutral pair.

−3 − 1 −4

ⓑ

Take 1 negative from the one added neutral pair.

3 − (−1) 4

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Have you noticed that subtraction of signed numbers can be done by adding the opposite ? In [link] , $\mathrm{-3}-1$ is the same as $\mathrm{-3}+\left(\mathrm{-1}\right)$ and $3-\left(\mathrm{-1}\right)$ is the same as $3+1.$ You will often see this idea, the subtraction property , written as follows:

Look at these two examples.

Of course, when you have a subtraction problem that has only positive numbers, like $6-4,$ you just do the subtraction. You already knew how to subtract $6-4$ long ago. But knowing that $6-4$ gives the same answer as $6+\left(\mathrm{-4}\right)$ helps when you are subtracting negative numbers. Make sure that you understand how $6-4$ and $6+\left(\mathrm{-4}\right)$ give the same results!

Look at what happens when we subtract a negative.

Subtracting a negative number is like adding a positive!

You will often see this written as $a-(\text{−}b)=a+b.$

Does that work for other numbers, too? Let’s do the following example and see.

Let’s look again at the results of subtracting the different combinations of $5,\mathrm{-5}$ and $3,\mathrm{-3}.$

What happens when there are more than three integers? We just use the order of operations as usual.

Key concepts

• Addition of Positive and Negative Integers
  $\begin{array}{cccc}5+3\hfill & & & -5+\left(\mathrm{-3}\right)\hfill \\ 8\hfill & & & \mathrm{-8}\hfill \\ \text{both positive,}\hfill & & & \text{both negative,}\hfill \\ \text{sum positive}\hfill & & & \text{sum negative}\hfill \\ \\ \\ \mathrm{-5}+3\hfill & & & 5+\left(\mathrm{-3}\right)\hfill \\ \mathrm{-2}\hfill & & & 2\hfill \\ \text{different signs,}\hfill & & & \text{different signs,}\hfill \\ \text{more negatives}\hfill & & & \text{more positives}\hfill \\ \text{sum negative}\hfill & & & \text{sum positive}\hfill \end{array}$
• Property of Absolute Value : $\left|n\right|\ge 0$ for all numbers. Absolute values are always greater than or equal to zero!
• Subtraction of Integers
  $\begin{array}{cccc}5-3\hfill & & & \mathrm{-5}-\left(\mathrm{-3}\right)\hfill \\ 2\hfill & & & \mathrm{-2}\hfill \\ 5\text{positives}\hfill & & & 5\text{negatives}\hfill \\ \text{take away}3\text{positives}\hfill & & & \text{take away}3\text{negatives}\hfill \\ \text{2 positives}\hfill & & & \text{2 negatives}\hfill \\ \\ \\ \mathrm{-5}-3\hfill & & & 5-\left(\mathrm{-3}\right)\hfill \\ \mathrm{-8}\hfill & & & 8\hfill \\ 5\text{negatives, want to}\hfill & & & 5\text{positives, want to}\hfill \\ \text{subtract}3\text{positives}\hfill & & & \text{subtract}3\text{negatives}\hfill \\ \text{need neutral pairs}\hfill & & & \text{need neutral pairs}\hfill \end{array}$
• Subtraction Property: Subtracting a number is the same as adding its opposite.