To learn Negative Numbers quickly and correctly, imagine every number on a number line before memorising sign rules. Positive numbers lie to the right of zero and negative numbers lie to the left. Addition, subtraction, multiplication and brackets describe different actions, so the same minus sign can have different roles. This eduKateSG Quick Notes guide helps students understand signed numbers, avoid common mistakes and prepare for algebra in Secondary Mathematics.
- Locate: put positive and negative values on a number line.
- Add: move right for a positive change and left for a negative change.
- Subtract: add the opposite of the number being subtracted.
- Multiply or divide: equal signs give a positive result; different signs give a negative result.
- Use brackets: separate a negative number from the subtraction operation.
- Check: estimate the direction and compare with the original expression.
First, treat negative numbers as ordinary positions and changes. If the temperature is −3°C and rises by 5°C, the new temperature is 2°C. On a number line, start at −3 and move five spaces right. Therefore, −3 + 5 = 2. If the temperature instead falls by 5°C, move left: −3 − 5 = −8. This model explains addition and subtraction without needing to guess from the number of minus signs.
Second, distinguish a negative value from subtracting a value. In 4 − (−2), the number we subtract is already negative. Subtracting it has the same effect as adding 2, so the expression becomes 4 + 2 = 6. This is a precise rule: a − b = a + (−b). It does not mean that any two minus signs in the same line must magically produce a plus. For example, −2 + (−3) = −5, not +5.
Third, learn the separate rule for multiplication and division. Two positive factors give a positive product; two negative factors also give a positive product; one positive and one negative factor give a negative product. Thus (−3) × (−4) = 12 and (−3) × 4 = −12. This sign rule is not copied into addition. After every calculation, ask whether the result’s sign is reasonable. Keep the number line for adding and subtracting, and the sign table for multiplying and dividing.
Worked example: subtract a negative
Calculate −3 − (−5). Replace the subtraction with addition of the opposite: −3 + 5. Starting at −3, move five places right. You land on 2. Check by comparing the starting and ending positions: 2 is five units greater than −3.
Another example: use brackets with squares
Compare −2² and (−2)². Under normal order of operations, −2² means the negative of 2 squared, so it is −4. But (−2)² means (−2) × (−2), which is +4. Brackets show exactly which quantity is being squared. Write them whenever the meaning might be unclear.
The sign rules you really need
- Add two negatives: combine their magnitudes and keep the negative sign: −2 + (−4) = −6.
- Add opposite signs: compare the magnitudes: −7 + 3 = −4.
- Subtract: change subtraction to addition of the opposite: 5 − (−3) = 8.
- Multiply two negatives: the result is positive: (−2) × (−6) = 12.
- Multiply different signs: the result is negative: (−2) × 6 = −12.
Common wrong turns
The most frequent mistake is to use one blanket sentence — “minus and minus make plus” — in every situation. It is safer to read the whole expression. Ask: Is the sign attached to a number? Is this subtraction? Is it multiplication? Which operations happen first? Then put brackets around negative inputs while practising. Once the operation is clear, the sign becomes predictable.
A 10-minute negative-number workout
- Draw a number line from −10 to 10. Mark −8, −3, 0, 4 and 9.
- Answer two additions using movement on the line.
- Rewrite two subtractions as addition of an opposite.
- Calculate two products with different sign combinations.
- Check one expression containing brackets or a square.
Try these questions
- −8 + 3 = ?
- 6 − (−2) = ?
- (−3) × 4 = ?
- Explain why −2² and (−2)² have different answers.
Answers and explanations
1. −5; from −8, move three spaces right. 2. 8; subtracting −2 adds 2. 3. −12; the factors have different signs. 4. −2² = −4, while (−2)² = 4, because brackets decide what is squared.
When you know you have learned it
You can handle the same idea in a temperature question, a number-line drawing, a written calculation and an algebraic expression. That ability to transfer is the real test. If the rule disappears when brackets appear, practise a matched pair with and without brackets, then try again from memory the next day.
Where to go next
Continue with the related lessons: Mathematics · Fractions · Algebra.
For the longer learning pathway, read Secondary 1 Mathematics Tutor Clementi | Small Groups Tutorials. This Quick Notes lesson is a separate, short learning resource; the reference page remains unchanged.