Negative numbers without the guesswork
Most marks lost in Grade 9 are not about the new idea in a question. They are a sign that went missing on the way. The sign rules look like five rules taught in five units; they are one idea — a negative is an opposite — turning up in rational numbers, powers, polynomials and inequalities.
Why do negative numbers need rules at all?
Negative numbers were invented for quantities with a direction: below zero, owed rather than owned, down rather than up. Chinese mathematicians calculated with them about two thousand years ago, using red counting rods for one kind and black for the other. In India in 628, Brahmagupta wrote rules for them as fortunes and debts, including that the product of two debts is a fortune. In the 1500s some European mathematicians were still calling them absurd.
What a negative sign really does is reverse a direction. Subtracting 5 moves left on the number line; subtracting −5 reverses that and moves right. Multiplying by −1 flips a number to the other side of zero, and flipping twice brings it back — which is all “a negative times a negative is a positive” means.
The arithmetic is almost never the hard part. What costs marks is a minus sign that changes meaning halfway along a line of working, and on Part A of the PAT there is no calculator to catch it.
Where it turns up
- Weather — a chinook that takes Calgary from −18 °C to 4 °C is a rise of 4 − (−18) = 22 degrees, not 14
- Money owed — a card balance of −$40 plus a $15 charge is −$55, and a $15 payment takes it back to −$40
- Science 9 — an ion's charge is protons minus electrons, so chlorine's 17 protons and 18 electrons give chloride its −1
- Golf — scores are counted against par, so −3 beats −1, and ordering negatives is ordering the leaderboard
- 1. The words, first
- 2. Subtracting is adding the opposite
- 3. Multiplying and dividing: count the negatives
- 4. Powers: the brackets decide
- 5. Subtracting a polynomial
- 6. Reversing an inequality
The words, first
The idea: One symbol, two jobs. Most sign errors come from reading a minus as the wrong one.
| Word | What it means |
|---|---|
| Opposite | The number the same distance from zero on the other side. The opposite of 7 is −7, and the opposite of −7 is 7. |
| Negative sign, subtraction sign | The same symbol doing two jobs. In −5 it says which side of zero; in 8 − 5 it says take away. |
| Factor | One of the numbers being multiplied. (−2)(−3)(−1) has three factors, all negative. |
| Sign rule | For multiplying and dividing: same signs give a positive answer, different signs a negative one. |
| Base and exponent | In (−2)⁴ the base is −2, and the exponent 4 says how many copies of it are multiplied. |
| Polynomial | An expression made of terms like 3x², −5x and 7, added together. |
| Inequality | A comparison using <, >, ≤ or ≥. |
Subtracting is adding the opposite
The idea: Every subtraction can be rewritten as adding the opposite. After that there is only addition to think about.
8 − 5 = 8 + (−5) = 3 · 3 − (−5) = 3 + 5 = 8 · −4 − 6 = −4 + (−6) = −10
Why subtracting a negative adds. Taking away a debt leaves you better off: if a $5 debt is cancelled, your balance goes up by $5.
Adding: the signs are directions. −3 + (−5) = −8: start 3 to the left of zero and go 5 further left. “Two negatives make a positive” is a rule for multiplying; used for adding, it gives +8.
With fractions, find the common denominator, then think about direction.
−2/3 − (−1/4) = −2/3 + 1/4 = −8/12 + 3/12 = −5/12
Sanity check. You started at about −0.67 and moved a quarter to the right, so the answer should be negative and a little closer to zero. −5/12 is about −0.42 ✓.
Multiplying and dividing: count the negatives
The idea: An even number of negative factors gives a positive answer; an odd number gives a negative one. Decide the sign first, then do the arithmetic with positive numbers.
(−2)(−3) = 6 · (−2)(−3)(−1) = −6 · (−12) ÷ (−4) = 3 · 12 ÷ (−4) = −3
Worked example. (−3/4) × (−2/3) × (−2). Three negative factors is an odd number, so the answer is negative. The size, with everything positive, is 3/4 × 2/3 × 2 = 12/12 = 1. So the answer is −1.
Settling the sign before touching the numbers means the arithmetic is all positive, which is where far fewer slips happen.
Powers: the brackets decide
The idea: An exponent applies only to what it touches. (−2)⁴ multiplies four copies of −2; −2⁴ multiplies four copies of 2 and puts a negative in front.
| Expression | Means | Value |
|---|---|---|
| (−2)⁴ | (−2)(−2)(−2)(−2): four negatives | 16 |
| −2⁴ | −(2 × 2 × 2 × 2) | −16 |
| (−2)³ | (−2)(−2)(−2): three negatives | −8 |
| (−5)⁰ | any base except 0, to the power 0 | 1 |
| −5⁰ | −(5⁰) | −1 |
In the order of operations, the power comes before the negative in front, because that negative is really multiplying by −1. So −3² + 4 = −9 + 4 = −5, while (−3)² + 4 = 9 + 4 = 13.
Substituting a negative: always use brackets. For x = −3, x² − 2x becomes (−3)² − 2(−3) = 9 + 6 = 15. Without brackets, x² turns into −3² = −9, and the answer comes out as −3.
On a calculator — Part B allows one — typing −3² on most scientific calculators gives −9, for the same reason. If you mean (−3)², type the brackets.
Subtracting a polynomial
The idea: A minus sign in front of a bracket is multiplying by −1, so it reaches every term inside. Change every sign, not just the first.
Worked example. (5x² + x − 3) − (2x² − 3x + 4) = 5x² + x − 3 − 2x² + 3x − 4 = 3x² + 4x − 7.
Check with x = 2. The original: (20 + 2 − 3) − (8 − 6 + 4) = 19 − 6 = 13. The answer: 12 + 8 − 7 = 13 ✓. Changing only the first sign would have given 3x² − 2x + 1, which is 9 at x = 2, and the check would catch it.
A negative monomial works the same way. −2x(3x − 4) = −6x² + 8x: the −2x multiplies both terms, and −2x times −4 is positive.
Reversing an inequality
The idea: Multiplying or dividing both sides by a negative number reverses the order of everything, so the inequality sign has to turn round to stay true.
Why. 2 < 5. Multiplying by −1 reflects every number to the other side of zero, and a reflection swaps left and right: −5 is now the one further left, so −2 > −5.
Worked example. −4x − 3 > 9.
- Add 3 to both sides: −4x > 12. Adding reverses nothing.
- Divide both sides by −4 and reverse the sign: x < −3.
- Check the boundary: x = −3 gives 12 − 3 = 9, exactly equal ✓. Check the direction with x = −5: 20 − 3 = 17, and 17 > 9 ✓.
Only multiplying or dividing by a negative reverses it. Adding −3 does not. Dividing by positive 4 does not, even when the answer comes out negative. Math 20-1 has a whole unit on inequalities, and this is still the rule that costs marks there.
One habit covers all five sections: predict the sign of the answer before you finish. Warming 12 degrees from −8.5 °C must end above zero. If your answer does not, you know where to look.