Adding and Subtracting Positive and Negative Numbers

1. Adding Positive and Negative Numbers

The ability to work comfortably with negative numbers is essential for success in algebra. For this reason, we will do a quick review of adding and subtracting positive and negative integers. Integers are all the positive whole numbers, zero, and their opposites (negative numbers).

Before we go directly into the rules for adding integers, consider this situation:

A friend gives you 8 chocolate bars, then gives you 4 more. This means your friend gave you 12 chocolate bars

Another way to look at this situation is from your friend's perspective, meaning they lost 8 chocolate bars, then they lost 4 more. Using negative numbers to represent the losses, “-8” can be used to represent the first loss, “-4” can be used to represent the second loss, and “-12” can be used to represent the total loss.

This means we can also write

big idea

When adding integers that have the same sign, add the unsigned numbers first, then keep the sign of the numbers for the final answer.

This is illustrated in the following examples:

	Same sign; just add.
	Our solution

	Same sign; add 5 + 3 and keep the negative.
	Our solution

	Same sign; add 7 + 5 and keep the negative.
	Our solution

Now let’s imagine a scenario where you spent $10 on lottery tickets, but you won $3 back.

You lost $10 and got $3 back, which means overall, you lost $7. This is written

The store gained $10 but then lost $3, which means overall, they gained $7. This is written

big idea

When one integer is positive and one integer is negative, subtract the numbers (as if they were all positive) and then use the sign from the larger number. This means if the larger number is positive, the answer is positive; if the larger number is negative, the answer is negative.

This is shown in the following examples:

	Different signs; subtract and use sign from bigger number (negative).
	Our solution

	Different signs; subtract and use sign from bigger number (positive).
	Our solution

	Different signs; subtract and use sign from bigger number (positive).
	Our solution

	Different signs; subtract and use sign from bigger number (negative).
	Our solution

big idea

When adding two numbers with matching signs, add the two numbers (as if they are positive) and keep the sign. When adding two numbers with opposite signs, subtract the smaller number from the larger number (as if they are positive), and keep the sign of the larger number.

2. Subtracting Positive and Negative Numbers

Consider a situation where you have $12 in your pocket but had to spend $8 for lunch. How much is left? We can look at this problem using two representations.

Mathematical Representation	Explanation
	You started with $12, then spent $8, leaving you with $4.
	You started with $12, then added a loss of $8, leaving you with $4.

big idea

To subtract two numbers, we change the problem to an addition problem, which can then be solved using the addition methods from earlier. The way we change subtraction to addition is to add the opposite of the number after the subtraction sign. Often this method is referred to as “adding the opposite.”

This is illustrated in the following examples:

	Add the opposite of 3.
	Different signs; subtract and use sign from bigger number (negative).
	Our solution

	Add the opposite of 6.
	Same sign; add and keep the negative.
	Our solution

	Add the opposite of -4.
	Same sign; add and keep the positive.
	Our solution

	Add the opposite of -2.
	Different signs; subtract and use sign from bigger number (negative).
	Our solution