Opposites

Stand on a line on the floor. Take 3 steps forward. Now take 3 steps back. Where are you? Right where you started. Forward and back cancel each other out. Numbers can do the same thing.

Mirror images

In the Below Zero lesson we saw that -3 is as far to the left of zero as 3 is to the right. Numbers like these, the same distance from zero on opposite sides, are called opposites:

• the opposite of 3 is -3;
• the opposite of -7 is 7;
• the opposite of 0 is 0 itself: it is no distance from zero at all.

If you folded the number line at zero, each number would land on its opposite, like a reflection in a mirror.

How far a number is from zero - leaving aside which side - is sometimes called its size. 3 and -3 have the same size, three, but they are not the same number.

Together they make zero

Here is what makes opposites special. A number plus its opposite is zero.

• -4 + 4 = 0. You owe 4 coins (that is -4). You earn 4 coins. You give them back, and now you have nothing: 0.
• On the number line: start at -4 and take 4 steps to the right. You land on 0.

This is the forward-and-back from the start of the lesson. Three steps forward and three steps back cancel out.

Going past zero

Opposites also answer a question that has no answer with counting numbers alone. What is 3 take away 5?

If you have 3 coins, you can't hand over 5. But on the number line you can always keep walking. Start at 3 and take 5 steps to the left: 3 steps bring you to 0, and 2 more take you to -2. So 3 - 5 = -2.

In money words: you had 3 coins and needed to pay 5. You pay the 3 and still owe 2. That is -2 coins.

This is one big reason negative numbers were invented: with them, taking away always has an answer, however much you take away.

Try it at home

1. Draw a number line from -6 to 6 with a ruler. Put a button on 2 and walk it 5 steps to the left. Where is it? Answer below.
2. What is the opposite of 9? Of -1? What do you get when you add each number to its opposite?

Answers: -3. The opposites are -9 and 1, and each pair adds up to 0.

Words to know

• opposites - two numbers the same distance from zero on opposite sides, like 3 and -3
• size of a number - how far it is from zero, whichever side it is on

Big idea

Every number has an opposite on the other side of zero, and a number plus its opposite is zero, like steps forward and back. Walking past zero means taking away always has an answer: 3 - 5 = -2.