Negative Numbers

Study Sheet

Negative Numbers

Numbers below zero

Both Sides of Zero

Concept
The Big Idea
−6−5−4−3−2−10123456−444 units4 units
Opposites: −4 and 4 are the same distance from 0, on opposite sides.

Negative numbers continue the number line to the left of zero. Adding still means moving right, and subtracting means moving left — the rule never changes.

Negative numbers extend the number line to the left of zero. Adding a negative moves left; subtracting a negative moves right, which is why 5−(−3)=85 - (-3) = 8. Multiplying two negatives gives a positive — not by decree, but because the rules of arithmetic force it.

Example
Finding −6+4-6 + 4
−8−7−6−5−4−3−2−10123+4
Start at −6 and move 4 to the right: −2.

Start at −6-6 and move 44 to the right, landing on −2-2.

Example
Comparing −7-7 and −2-2
−8−7−6−5−4−3−2−101−7−2
Further right is greater, even among negatives.

On the number line −2-2 is to the right of −7-7, so −2>−7-2 > -7. Being farther from zero on the negative side means being smaller: owing $7\$7 is worse than owing $2\$2.

Side note
Where it goes wrong

Believing "two negatives make a positive" applies to addition. −3+(−5)=−8-3 + (-5) = -8: adding two debts makes a bigger debt. The sign rule for two negatives is about multiplication and division only.

Try it
Try it: put them in order

Order from least to greatest: −4,3,−1,0,−6-4, 3, -1, 0, -6.

Show the answer

Place them on a number line and read from left to right: −6,−4,−1,0,3\mathbf{-6, -4, -1, 0, 3}.

In Plain Terms & More Examples

Concept
In plain terms

Negatives are the numbers left of zero. Adding still moves right on the number line; subtracting still moves left.

Example
Finding −3+7-3 + 7
−4−3−2−10123456+7
Seven steps right from −3 cross zero and land on 4.

Start at −3-3 and move 77 right, landing on 44.

Example
Finding 2−52 - 5
−5−4−3−2−101234−5
Five steps left from 2 cross zero and land on −3.

Start at 22 and move 55 left, landing on −3-3.

Side note
Where it goes wrong

Moving the wrong way. Subtracting a positive number moves left, whatever the starting point: 2−52 - 5 starts at 22 and goes 55 left to −3-3, not right to 77.

Try it
Try it: two moves

Find −4+9-4 + 9 and 1−61 - 6.

Show the answer

−4+9-4 + 9: start at −4-4, move 99 right, land on 5\mathbf{5}. 1−61 - 6: start at 11, move 66 left, land on −5\mathbf{-5}.

Multiplying and Dividing

Concept
The sign rule
productvalue3 × (−2)−62 × (−2)−41 × (−2)−20 × (−2)0(−1) × (−2)2(−2) × (−2)4up 2each row
Each row the first factor drops by 1 and the product rises by 2, so the pattern forces (−1) × (−2) = 2.

Same signs give a positive; different signs give a negative. (−3)(−4)=12(-3)(-4) = 12, (−3)(4)=−12(-3)(4) = -12, −12÷(−3)=4-12 \div (-3) = 4.

Example
Three factors

(−2)(−3)(−1)(-2)(-3)(-1): the first two give 66, then 6×(−1)=−66\times(-1) = -6. An odd number of negative factors gives a negative.

Example
Why negative times positive is negative
−12−8−40−4−4−4
Three jumps of −4 from 0.

3×(−4)3 \times (-4) means three jumps of −4-4: (−4)+(−4)+(−4)=−12(-4) + (-4) + (-4) = -12. Starting at 00 and moving 44 left three times lands on −12-12.

Side note
Where it goes wrong

Getting the sign of a quotient wrong. Division follows the same rule as multiplication: −12÷3=−4-12 \div 3 = -4 (different signs) and −12÷(−3)=4-12 \div (-3) = 4 (same signs). Check by multiplying back: 3×(−4)=−123 \times (-4) = -12.

Try it
Try it: evaluate

Compute −8−(−5)-8 - (-5), (−6)(7)(-6)(7), and −45÷(−9)-45 \div (-9).

Show the answer

−8+5=−3-8 + 5 = \mathbf{-3}; −42\mathbf{-42}; 5\mathbf{5}.

Word Problems

Concept
Negatives in the world

Below zero, below sea level, owing money, losing points: each has a zero and two directions. Choose which direction is positive, and the story becomes adding and subtracting on a number line.

A thermometer is a vertical number line and a bank balance is a horizontal one. Once the story is on a line, "rises 55" means +5+5 and "falls 99" means −9-9, and the arithmetic follows the rules you already have.

Example
Temperature drop
−6−4−20246falls 9°start 4°
A thermometer is a number line standing up: 9 degrees down from 4 is −5.

The temperature was 4∘4^\circ and fell 9∘9^\circ: 4−9=−5∘4 - 9 = -5^\circ.

Example
A diver

A diver is at −12-12 m, twelve metres below the surface, and rises 55 m. Rising moves up, toward zero: −12+5=−7-12 + 5 = -7 m.

Side note
Where it goes wrong

Moving away from zero when "rising". Writing −12−5=−17-12 - 5 = -17 sends the diver deeper. Rising adds, so the position gets closer to 00: −7-7 m.

Try it
Try it: the elevator

An elevator starts at floor −2-2 (a basement level), goes up 77 floors, then down 1010. Where is it?

Show the answer

−2+7−10=−5-2 + 7 - 10 = \mathbf{-5}, five levels below ground.

Subtracting a Negative

Concept
Take away a debt

Subtracting a negative is the same as adding its opposite: a−(−b)=a+ba - (-b) = a + b. On a number line, a−ba - b is the trip from bb to aa: how far, and which way.

Think of a score that includes penalty cards worth −3-3 each. Taking a penalty card away raises the score by 33. Removing a negative is adding a positive, which is why 5−(−3)=5+3=85 - (-3) = 5 + 3 = 8.

Example
Finding 5−(−3)5 - (-3)
−4−3−2−10123456−358 steps right
From −3 up to 5 is 8 steps.

Rewrite as adding the opposite: 5+3=85 + 3 = 8. On the number line, the trip from −3-3 to 55 is 88 steps to the right, so the difference is +8+8.

Example
Finding −2−(−6)-2 - (-6)

Rewrite: −2+6=4-2 + 6 = 4. The trip from −6-6 to −2-2 is 44 steps to the right.

Example
Finding −4−3-4 - 3

Subtracting a positive: start at −4-4 and move 33 left, to −7-7. So −4−3=−7-4 - 3 = -7.

Side note
Where it goes wrong

Making both signs negative. −2−(−6)-2 - (-6) is not −8-8; that would be −2+(−6)-2 + (-6). Subtracting −6-6 adds 66: −2+6=4-2 + 6 = 4.

Try it
Try it: three subtractions

Find 7−(−2)7 - (-2), −5−(−5)-5 - (-5) and −1−4-1 - 4.

Show the answer

7+2=97 + 2 = \mathbf{9}. −5+5=0-5 + 5 = \mathbf{0}. For −1−4-1 - 4, move 44 left from −1-1, to −5\mathbf{-5}.

Why It Works

Why is (−1)×(−1)=1(-1)\times(-1) = 1? Start from (−1)×0=0(-1)\times0 = 0 and write 0=1+(−1)0 = 1 + (-1): then (−1)×1+(−1)×(−1)=0(-1)\times1 + (-1)\times(-1) = 0, so −1+(−1)(−1)=0-1 + (-1)(-1) = 0, hence (−1)(−1)=1(-1)(-1) = 1. Nothing was decreed; keeping the distributive property true for negatives leaves no other choice. Subtracting a negative is adding its opposite for the same reason — the opposite of −3-3 is 33.

Tip
Reading a proof

Do not read a proof the way you read a rule. Cover the last line, ask what the previous line forces, and only then uncover it. If a step feels like magic, that is the step to work by hand with a concrete number.