Math 9 · Rational numbers
Fractions and decimals, on both sides of zero
Rational numbers are the fractions and decimals you already know, now allowed to be negative. Nothing about the arithmetic is new except the signs — which is exactly where the marks go, and exactly what the no-calculator Part A of the PAT tests.
- 1. The words, first
- 2. Comparing and ordering
- 3. The four operations
- 4. Rational numbers in problems
- 5. What costs marks
The words, first
The idea: A rational number is anything that can be written as a fraction. Every rule on this page is a fraction rule plus a sign rule.
| Word | What it means |
|---|---|
| Integer | A whole number, positive, negative or zero: …, −2, −1, 0, 1, 2, … |
| Rational number | Any number that can be written as a fraction a/b of two integers, with b not 0. −3/4, 2.5 (5/2), −7 (−7/1) and 0.333… (1/3) are all rational. |
| Opposite | The number the same distance from zero on the other side: the opposite of 2/3 is −2/3. |
| Reciprocal | The fraction turned upside down: the reciprocal of −3/4 is −4/3. A number times its reciprocal is 1. |
| Lowest common denominator | The smallest number both denominators divide into. For 4 and 6 it is 12. |
| Mixed number | A whole number and a fraction together. −1½ means −(1 + ½), which is −3/2. |
Comparing and ordering
The idea: On a number line, further left is smaller. For negatives, that means the one that looks bigger is the smaller number.
Why negatives feel backwards. −5 is further below zero than −2, so −5 < −2, even though 5 > 2. −5 °C is colder than −2 °C.
Worked example. Order −0.6, −2/3, −5/8 and 0.3 from least to greatest.
- Make them all decimals: −2/3 = −0.666… and −5/8 = −0.625.
- Place them. The positive one is greatest. Among the negatives, the one furthest from zero is least: −0.666…, then −0.625, then −0.6.
- So: −2/3, −5/8, −0.6, 0.3.
Or use a common denominator. To compare −3/4 and −5/6, write both in twelfths: −9/12 and −10/12. −10/12 is further from zero, so −5/6 < −3/4.
A number in between. There is always another rational number between two others. Halfway between −0.5 and −0.4 is −0.45.
One number, three ways to write it. −3/4 = (−3)/4 = 3/(−4). One negative sign makes the value negative wherever it sits.
The four operations
The idea: Adding and subtracting need a common denominator and care with direction. Multiplying and dividing need the sign rule: same signs give a positive, different signs a negative.
Adding. −3/4 + 5/6: in twelfths, −9/12 + 10/12 = 1/12. Start 9 twelfths below zero and go 10 twelfths up.
Subtracting means adding the opposite.
−2/3 − (−1/4) = −2/3 + 1/4 = −8/12 + 3/12 = −5/12
Multiplying. Decide the sign first. (−3/5) × (10/9): the signs differ, so the answer is negative. Simplify across — 10 and 5 share a 5, 3 and 9 share a 3 — leaving 2/3. The answer is −2/3.
Dividing: multiply by the reciprocal of the second number.
(−3/4) ÷ (−9/8) = (−3/4) × (−8/9) = 24/36 = 2/3
Mixed numbers: convert first, and keep the sign on the whole thing. −1½ × 2⅔ = (−3/2) × (8/3) = −4. Writing −1½ as −1 + ½ = −½ is a very common slip.
Decimals follow the same rules. −2.5 × 0.4 = −1. −1.2 − 3.5 = −4.7. −7.2 ÷ (−0.9) = 8.
Rational numbers in problems
The idea: Write one expression for the whole situation, then follow the order of operations. Choosing the signs is part of setting it up.
Worked example. At 6 a.m. it was −8.5 °C in Grande Prairie. The temperature rose 2.4 °C an hour for 5 hours, then fell 3.25 °C when cloud moved in. What was it then?
- Signs: rising is positive and falling is negative: −8.5 + 5 × 2.4 − 3.25.
- Multiply first: 5 × 2.4 = 12.
- Then left to right: −8.5 + 12 = 3.5, and 3.5 − 3.25 = 0.25 °C.
Is it sensible? The air warmed 12 degrees from −8.5 and then cooled a little, so just above zero fits. An answer like −23.75 would mean the rise had been treated as a fall — visible without redoing the arithmetic.
A mean with negatives. Five overnight lows: −4.5, −2.0, 1.5, −3.5 and −1.5 °C. The negatives add to −11.5, adding the 1.5 gives −10.0, and dividing by 5 gives a mean of −2.0 °C.
(−1/2)² − 3/4 ÷ (−3) = 1/4 − (−1/4) = 1/4 + 1/4 = 1/2
The square comes first and is positive. The division comes next and is negative. Subtracting that negative then adds.
What costs marks
The idea: Every one of these is a sign, and none of them is caught by a calculator on Part A.
- “Two negatives make a positive” used for adding. −3 + (−5) = −8. The rule is for multiplying and dividing.
- Ordering negatives by their size. −0.7 is less than −0.2.
- Subtracting a negative. 4 − (−2) = 6, not 2.
- Dividing without the reciprocal, or turning over the first fraction instead of the second.
- Splitting a negative mixed number. −2¼ = −9/4, not −2 + ¼.