Negative numbers

A negative number is a number below zero. You meet them every winter: when the weather report says −5°, it is five degrees colder than zero. You also meet them in money. If you have $20 in the bank and you spend $30, your balance is −10: you owe the bank $10. People have used negative numbers for a long time. Chinese counting boards about 2,000 years ago used red rods and black rods for the two kinds of numbers. In India in the year 628, the mathematician Brahmagupta wrote down rules for working with "fortunes" and "debts". Those are the same rules you will learn here.

The number line

The best way to see negative numbers is on a number line. This is a straight line with zero in the middle. The positive numbers 1, 2, 3, ... go to the right. The negative numbers −1, −2, −3, ... go to the left. A thermometer is a number line standing up: warmer is up, colder is down.

• Numbers further to the right are bigger. So −2 is bigger than −9, because −2 is warmer than −9.
• The minus sign in front of a number tells you which side of zero it is on. The number without its sign tells you how far from zero it is.
• Zero is neither positive nor negative.

Adding and subtracting with the number line

Every sum is a walk on the number line. Start at the first number. Then look at the sign in front of the second number.

When you take a bigger number away from a smaller one, the answer goes below zero. Here is the plain method.

1. Take the smaller number away from the bigger one.
2. Put a minus sign in front, because you went past zero.

Taking away a negative is adding

What happens when the number you subtract is itself negative, as in 5 − (−3)? The answer is 8. Subtracting a negative number is the same as adding the positive number.

Why? Think of debts. Suppose you have $5, but you also owe your friend $3. That debt is −3, so all together you are worth 5 + (−3) = 2. Now your friend says, "Forget it, you owe me nothing." The debt of 3 is taken away: 2 − (−3). You keep your $5 and owe nothing, so now you are worth 5. Taking away −3 made you 3 richer: 2 − (−3) = 2 + 3 = 5.

Temperatures show it too. In the morning it is −2°. In the afternoon it is 5°. How much warmer is the afternoon? We find the difference: 5 − (−2) = 5 + 2 = 7. On a thermometer you can count it: 2 degrees up to zero, then 5 more, which is 7 degrees.

Multiplying and dividing

For multiplying and dividing, you do not walk on the number line. You work with the numbers as if they had no signs. Then you decide the sign of the answer with one short rule.

Why is a negative times a negative positive? Look at this pattern. Each line multiplies by −2, and the first number goes down by one each time:

• 3 × (−2) = −6
• 2 × (−2) = −4
• 1 × (−2) = −2
• 0 × (−2) = 0
• (−1) × (−2) = 2

The answers go up by 2 on every line: −6, −4, −2, 0. The next answer must be 2 more than 0, which is 2. So (−1) × (−2) is positive. Any other answer would break the pattern.