Negative Numbers and Integer Operations

What you will learn

• Place negative numbers on a number line and compare them.
• Add and subtract integers by moving along the line.
• Multiply and divide integers using sign rules.

Before you start: you should be able to compare numbers and use the order of operations. See Comparing and Ordering.

The idea

An integer is a whole number, positive, negative or zero. Negative numbers are numbers less than zero. We meet them in temperatures below freezing, bank balances that are overdrawn and floors below ground level.

The number line

Zero sits in the middle. Positive numbers go to the right, and negative numbers go to the left. The further left, the smaller the number.

So $-5 < -2 < 0 < 3$. A negative number with a big digit part is smaller: $-7 < -2$.

Adding and subtracting

Adding means moving right and subtracting means moving left.

Subtracting a negative number is the same as adding its positive: $5 - (-2) = 5 + 2 = 7$. Adding a negative number is the same as subtracting: $5 + (-2) = 3$.

Multiplying and dividing

Multiplying (or dividing) two numbers with the same sign gives a positive. With different signs, the answer is negative.

• $(+) \times (+) = (+)$ and $(-) \times (-) = (+)$.
• $(+) \times (-) = (-)$ and $(-) \times (+) = (-)$.

Think of $(-3) \times 4$ as four jumps of $-3$: $-3 - 3 - 3 - 3 = -12$. The same rules work for division: $-12 \div 4 = -3$ and $-12 \div (-4) = 3$.

Worked examples

Example 1: Crossing zero

Work out $-2 - 4 + 7$.

1. $-2 - 4 = -6$.
2. $-6 + 7 = 1$.

Answer: $1$.

Example 2: Subtracting a negative

Work out $3 - (-5)$.

1. Subtracting a negative is the same as adding: $3 + 5$.
2. $3 + 5 = 8$.

Answer: $8$.

Example 3: Temperature change

At night the temperature is $-4\,^\circ\text{C}$. By noon it has risen by $9$ degrees. What is the temperature at noon?

1. A rise means add: $-4 + 9$.
2. $-4 + 9 = 5$.

Answer: $5\,^\circ\text{C}$.

A minus sign can mean “subtract” or “negative”. In $5 - (-2)$ the first is subtract and the second is negative. Brackets help.

Why it matters

Negative numbers measure debts, temperatures, depths and changes. Algebra, graphs and coordinates all rely on them, and the sign rules you learn here are used in every later lesson.