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Maths 1A Ch2: Directed Numbers

Key points + Practice

Chapter 2, Directed Numbers, is a key topic in the UT1 paper and one of the most frequently tested chapters in S1 Mathematics. Questions include calculations, the number line and word problems.

📏 1. What are Directed Numbers?

Directed numbers are numbers with a positive or negative sign, used to show quantities in opposite directions.
The number line:
←──┼──┼──┼──┼──┼──┼──┼──→
−3 −2 −1 0 1 2 3
On a number line, a number further to the right is always greater:
−3 < −2 < −1 < 0 < 1 < 2 < 3
So −5 < −1, and −1 < 1.
Absolute value: the distance of a number from 0; it is never negative.
|−7| = 7, |7| = 7, |0| = 0
Opposite numbers: same value, opposite signs. The opposite of −5 is 5.
本節生詞：zero

➕ 2. Addition and Subtraction

Addition:
1. Same signs: keep the sign and add the absolute values.
(+3) + (+5) = +8
(−3) + (−5) = −8
2. Different signs: take the sign of the number with the larger absolute value, and subtract the smaller absolute value from it.
(+7) + (−3) = +4
(−7) + (+3) = −4
Subtraction: subtracting a number is the same as adding its opposite.
a − b = a + (−b)
Examples:
(−3) − (+5) = (−3) + (−5) = −8
(−3) − (−5) = (−3) + (+5) = +2
(+3) − (−5) = (+3) + (+5) = +8
Remember: subtracting a negative is the same as adding a positive.

✖️ 3. Multiplication and Division

Rules for multiplication:
(+) × (+) = (+)
(−) × (−) = (+)
(+) × (−) = (−)
(−) × (+) = (−)
Rule of thumb: same signs give a positive answer; different signs give a negative answer.
Rules for division: the same as multiplication.
(+12) ÷ (+3) = +4
(−12) ÷ (−3) = +4
(+12) ÷ (−3) = −4
(−12) ÷ (+3) = −4
Multiplying several numbers: count the negative signs.
An even number of negative signs gives a positive answer.
An odd number of negative signs gives a negative answer.
Example: (−1) × (−2) × (−3) = −6 (three negative signs: odd)
Example: (−1) × (−2) × (−3) × (−4) = +24 (four negative signs: even)
本節生詞：productquotient

🧮 4. Mixed Operations and Applications

Mixed operations follow the usual order: brackets → powers → multiplication and division → addition and subtraction.
Example 1: (−6) + 10 ÷ 2 = (−6) + 5 = −1
Example 2: (−2) × [(−3) + 7] = (−2) × 4 = −8
Everyday applications:
1. Temperature: from −3 °C to 5 °C, the rise is 5 − (−3) = 8 °C.
2. Altitude: above sea level is positive, below is negative.
3. Money: income is positive, spending is negative. Earning $200 then spending $350 gives (+200) + (−350) = −$150.
4. Lift: from 2 floors below ground (−2) up to floor 6 (+6) is 6 − (−2) = 8 floors.
本節生詞：owenet value

📝 例題

例題 1：Work out (−8) + (+3) − (−5).
例題 2：Work out (−3) × (−4) ÷ (+6).
例題 3：The temperature in the morning is −4 °C and rises to 3 °C in the afternoon. By how many degrees did the temperature rise?
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