Positive & negative numbers

1Summary 重點摘要

Negative numbers show opposite direction: below zero, debt, below sea level. Number line: right is bigger. Subtract negative = add positive. Multiply/divide: same sign = positive, different = negative.

2Vocabulary cards Vocabulary

3Step by step Steps

1. 1. Why negative numbers?Some quantities have an 'opposite direction' that positive numbers alone cannot express. Examples: a winter temperature of 5 below zero is written −5°C; owing 20 dollars can be written −20; a diver 8 metres underwater is written −8. Once opposite directions carry a minus sign, calculation becomes uniform. A common mistake is thinking negative numbers are 'not real numbers'; they are as real as positives, only opposite in direction — +5 and −5 are a pair of opposites, earning 5 dollars and owing 5 dollars.
2. 2. Number line: right is biggerA number line is a straight line with 0 in the middle, positives on the right and negatives on the left; on a number line, the further right, the bigger. Examples: −3 sits to the right of −8, so −3 > −8; likewise −5 > −10. How to judge: a negative with a bigger-looking digit is actually smaller — −20 is smaller than −3 because it is further from 0. A common error is seeing that 20 beats 3 and concluding −20 > −3; sketch the line, place both numbers, and compare.
3. 3. Opposite numbersTwo numbers differing only in sign are opposites: the opposite of 3 is −3; the opposite of −5 is 5. They lie at the same distance from 0 in opposite directions, and their sum is always 0, e.g. 3 + (−3) = 0. On the number line they are symmetric about 0. A common error is claiming −7 has no opposite; the correct move is to flip the sign, so the opposite of −7 is 7. Remember: a number plus its opposite always gives 0.
4. 4. Addition: same vs different signsAdding the same signs: keep the sign and add the numbers — (−3) + (−5) = −8; (−7) + (−2) = −9. Adding different signs: take the sign of the larger number and subtract the smaller from the larger — (−3) + 5 = 2 because 5 beats 3, so positive; (−7) + 4 = −3 because 7 beats 4, so negative. A life example: owing 3 dollars then earning 5 leaves a net gain of 2. A common error is adding the numbers for different signs, e.g. computing (−3) + 5 as −8; remember, different signs mean 'subtract the smaller from the larger', the sign following the bigger side.
5. 5. Subtraction: subtract negative = add positiveSubtracting a negative equals adding its opposite: 5 − (−3) = 5 + 3 = 8; (−5) − (−3) = (−5) + 3 = −2. A life example: a debt of 5 dollars reduced by 3 leaves 2 owed, so (−5) − (−3) = −2 makes sense. The chant: 'subtracting a negative becomes adding a positive'. A common error is computing 5 − (−3) as 5 − 3 = 2, forgetting to flip the sign; whenever two minus signs meet, turn them into a plus first and the calculation stays safe.
6. 6. Multiplication: same = positive, different = negativeMultiplication's sign rule: same signs give positive, different signs give negative. Examples: 3 × 4 = 12 (positive times positive); (−3) × (−4) = 12 (negative times negative); 3 × (−4) = −12 (positive times negative); (−3) × 4 = −12 (negative times positive). One way to see it: (−3) × 4 is three copies of −4 added, which is −12. A common error is expecting a negative times a negative to be negative; remember, the two minuses cancel and the result is positive — (−2) × (−5) = 10, not −10.
7. 7. Division: same as multiplicationDivision follows the same sign rule as multiplication: same signs positive, different signs negative. Examples: 12 ÷ 3 = 4; (−12) ÷ (−3) = 4; (−12) ÷ 3 = −4; 12 ÷ (−3) = −4. The order of work: divide the number parts first, then settle the sign by the rule. As a full check: decide the sign first, then the size — two steps. A common error is minding the numbers but missing the sign, e.g. writing (−12) ÷ 3 as 4; the habit of 'sign first, value second' keeps all four operations safe.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Negative = opposite direction
• Number line: right is bigger
• Subtract negative = add positive
• Same sign = positive, different = negative
• Absolute value: ignore sign