Understanding Negative Numbers: Adding, Subtracting, Multiplying

Negative numbers are one of those topics where students often learn the rules without really understanding what is going on. They can recite "a negative times a negative is a positive" but cannot explain why, and when problems get more complex, the rules start to blur together.

This guide walks through the core operations with negative numbers, with worked examples and the reasoning behind each rule. If you are a student working through this topic or a parent trying to help at home, this should give you a solid foundation.

What negative numbers actually mean

A negative number is simply a value less than zero. Think of a number line: zero sits in the middle, positive numbers stretch to the right, and negative numbers stretch to the left. The further left you go, the smaller the number. So -5 is less than -2, even though 5 is bigger than 2.

Real world examples help make this concrete. A temperature of -3 degrees is three degrees below zero. A bank balance of -$50 means you owe $50. An elevator at level -2 is two floors below ground.

---

Adding negative numbers

Adding a negative number is the same as subtracting its positive version. Think of it as moving left on the number line.

Example 1: 7 + (-3)

Start at 7 on the number line. Adding -3 means moving 3 steps to the left. You land on 4.

So 7 + (-3) = 4.

Example 2: -4 + (-5)

Start at -4. Adding -5 means moving 5 more steps to the left. You land on -9.

So -4 + (-5) = -9.

Example 3: -6 + 10

Start at -6. Adding 10 means moving 10 steps to the right. You land on 4.

So -6 + 10 = 4.

The key idea: adding a negative number always makes the result smaller. Adding a positive number always makes it larger.

---

Subtracting negative numbers

This is where students often get confused. Subtracting a negative is the same as adding its positive version. It sounds strange at first, but it makes sense on the number line: if adding a negative means going left, then subtracting a negative means going right.

Example 1: 5 - (-3)

This becomes 5 + 3 = 8.

Example 2: -2 - (-7)

This becomes -2 + 7 = 5.

Example 3: -4 - 6

Start at -4 and move 6 to the left: -4 - 6 = -10.

A helpful way to remember: two negatives side by side (the subtraction sign and the negative sign) turn into a plus. Some students find it useful to think of it as "subtracting a debt is the same as gaining money."

---

Multiplying negative numbers

The rules for multiplication are straightforward once you see the pattern:

• Positive × Positive = Positive. 3 × 4 = 12
• Positive × Negative = Negative. 3 × (-4) = -12
• Negative × Positive = Negative. (-3) × 4 = -12
• Negative × Negative = Positive. (-3) × (-4) = 12

The simple version: if the signs are the same, the answer is positive. If the signs are different, the answer is negative.

Why does negative times negative equal positive? Think of it in terms of patterns. We know that:

(-3) × 3 = -9

(-3) × 2 = -6

(-3) × 1 = -3

(-3) × 0 = 0

Each time we reduce the second number by 1, the answer increases by 3. Following that pattern:

(-3) × (-1) = 3

(-3) × (-2) = 6

The pattern demands it. That is not just a rule someone invented; it is a consequence of how multiplication works.

---

Dividing negative numbers

Division follows exactly the same sign rules as multiplication:

• Same signs = positive result. (-12) / (-3) = 4
• Different signs = negative result. (-12) / 3 = -4

Example: (-20) / (-5) = 4 (both negative, so the answer is positive).

Example: 18 / (-6) = -3 (different signs, so the answer is negative).

---

Common mistakes to watch for

• Confusing -3 - 5 with -3 - (-5). The first equals -8 (moving further left). The second equals 2 (subtracting a negative means adding).
• Forgetting signs when multiplying several numbers. If you multiply three negative numbers together, the result is negative (odd number of negatives). Four negative numbers gives a positive result (even number of negatives). Count the negatives.
• Thinking -3 squared is -9. Be careful: (-3) squared means (-3) × (-3) = 9. But -3 squared without brackets means -(3 × 3) = -9. The brackets matter.

---

Putting it into practice

The best way to build confidence with negative numbers is to practise with the number line until the operations feel natural. Once you can picture what is happening, the rules stop being something you memorise and become something you understand.

If your child is working through negative numbers in the Australian Curriculum and needs some guided practice, imSteyn covers this topic in detail. It walks students through each operation step by step, asking questions rather than giving answers, so they build genuine understanding rather than just following procedures.

---

Quick reference

Operation	Rule	Example
Adding a negative	Move left (subtract)	5 + (-3) = 2
Subtracting a negative	Move right (add)	5 - (-3) = 8
Same signs multiply/divide	Positive result	(-4) × (-3) = 12
Different signs multiply/divide	Negative result	(-4) × 3 = -12