# Negative Numbers

## Introduction to negative numbers

We have been picturing subtraction problems using green squares and gray squares. You can also use pink squares for an amount to subtract. For example, there are \$5\$ green squares and \$3\$ pink squares on the grid to the left, representing \$5-3\$. If we combine \$3\$ green squares and \$3\$ pink squares to form \$3\$ gray squares, only \$2\$ green squares would remain. So this is a picture of the subtraction problem \$5-3=2\$.

For each row in the table below, how many green squares remain after combining with the pink squares? Click the button to move the pink squares on top of the green squares.

Click to display the subtraction problem \$5-6\$. If we combine the green and pink squares, we are now left with one pink square rather than any number of green squares. This means the answer is the negative number \$-1\$.

By thinking of pink squares as negative numbers, you can do addition problems involving negative and positive numbers.

Solve the addition problem in each row of the table below, counting pink squares as \$-1\$ each and green squares as \$+1\$ each.

Solve the addition problems in the table below.

## Opposites

We call two numbers opposites if they’re the same size, but one is positive and the other is negative.

What are the opposites of these numbers?
NumberOpposite

Notice that opposites can be drawn on the grid in exactly the same way, except that the positive number uses green squares and the negative number uses pink squares.

Solve each subtraction and addition problem in the table below, as shown on the grid.

 How would you describe the answers to the subtraction and addition problems in the first row of subIsAdd? Are they the same or are they different?
 What about the answers to the two problems in the second row? Are they the same or are they different?

This tells you another way to do subtraction. You already know how to do addition; to subtract a number, just add its opposite!

## Subtracting negative numbers

Let’s now see why the “add its opposite” rule still works when the number you’re subtracting is negative. We’ll look at another way of thinking about subtracting a negative number, by closely examining the subtraction problem \$3-(-2)\$.

 How many green squares are there on the grid to the left?

We want to subtract \$-2\$ from \$3\$. Think of this as removing some squares on the grid with value \$-2\$: that is, removing \$2\$ pink squares. Click to see a way of writing \$3\$ that uses \$2\$ pink squares.

 In addition to the two pink squares on the grid to the left, how many green squares are there?
 Click to remove the two pink squares. What number is now shown on the grid?
 What is \$3-(-2)\$?

In each row of this table, click the button to split the number up into some green squares and some pink squares. Then click the button to remove some of those pink squares. What is the total value of the squares that remain on the grid after removing these pink squares?

Notice that your answers in Questions subNegQ and addPosQ are the same. This is because subtracting a negative number means removing pink squares, which has the same effect as adding green squares. So this is another way to see that subtracting a number means the same thing as adding its opposite.

## Subtraction with negative numbers

For each subtraction problem shown, turn it into an addition problem and then solve it.

Solve each subtraction problem.

## Debt

One common reason to add and subtract negative numbers is to solve problems related to debt. Owing somebody money is the same thing as having a negative amount of money. Some specific ways this works include:

Debt concept Arithmetic translation
owing money a negative number (amount of money)
owing money, but also having some money adding a negative and a positive number
owing money to two different people adding two negative numbers
removing a debt subtracting a negative number from another number

Translate each word problem into an arithmetic problem, and solve it.