Operations with positive and negative numbers

In this lesson you will learn

Add rational numbers with the same and with different signs, and replace subtraction by adding the opposite
Find the sign of a product or a quotient, evaluate powers of negative numbers and tell (−a)ⁿ from −aⁿ
Use the laws of operations to simplify calculations and evaluate multi-step expressions in the right order
Solve word problems about temperature, depth and money

At 7 a.m. it was −5 °C in Gabala. By noon the temperature had risen by 9 degrees, and in the evening it fell by 6 degrees. Elvin’s phone account was 3 manat in debt (−3 manat); he topped it up with 10 manat and then spent 8.5 manat on calls. What was the temperature in the evening, and what is Elvin’s balance now? To answer such questions you need the four operations with positive and negative numbers, powers, and the order of operations.

In the previous lesson, “Positive and negative numbers, the number line and absolute value”, you met the number line, opposite numbers and absolute value. Here they are the basis of every rule: in addition, absolute values are added or subtracted, and in subtraction the opposite number helps. You already know how to calculate with fractions from the lessons “Operations with fractions” and “Decimal fractions” — now only the sign is added.

Addition: same signs and different signs

Picture addition as moving along the number line. You start at the first addend. Adding a positive number means moving right, adding a negative number means moving left, and the absolute value of the addend tells you how many units to move. For example, 3 + (−5): start at 3, move 5 units to the left and arrive at −2.

A positive addend moves right, a negative addend moves left. The length of each arrow is the absolute value of the addend.

(−a) + (−b) = −(a + b)

where:

a, bpositive numbers — the absolute values of the addends

To add numbers with the same sign, add their absolute values and keep the common sign. On the line you move twice in the same direction; two debts together make a bigger debt: (−3) + (−5) = −8.

Addends with the same sign

Calculate: 1) (−3) + (−5); 2) (−2.4) + (−1.6); 3) (−1/3) + (−1/6); 4) −12 + (−7.5).

Show solution

1) Absolute values: 3 + 5 = 8, sign “−”: −8.
2) 2.4 + 1.6 = 4, so −4.
3) 1/3 + 1/6 = 2/6 + 1/6 = 3/6 = 1/2, so −1/2.
4) 12 + 7.5 = 19.5, so −19.5.

When the signs are different, the two moves work against each other: moving left undoes part of the way we moved right. The result depends on which move is longer — the addend with the larger absolute value “wins”.

a + (−b) = a − b if a ≥ b; a + (−b) = −(b − a) if a < b (a, b > 0)

where:

athe positive addend
−bthe negative addend; b is its absolute value

To add numbers with different signs, subtract the smaller absolute value from the larger one and give the answer the sign of the addend with the larger absolute value. Opposite numbers add up to zero: a + (−a) = 0.

1
Compare the signs
If the signs are the same, go to step 2; if they are different, go to step 3.
2
Same signs
Add the absolute values and keep the common sign: (−4) + (−9) = −13.
3
Different signs
Subtract the smaller absolute value from the larger one: for (−9) + 4, 9 − 4 = 5.
4
Put the sign
Take the sign of the addend with the larger absolute value: −9 has the larger one, so the answer is −5.

Addends with different signs

Calculate: 1) (−9) + 4; 2) 7.5 + (−10); 3) −3/4 + 5/6; 4) 2.8 + (−2.8).

Show solution

1) 9 − 4 = 5, and −9 has the larger absolute value: −5.
2) 10 − 7.5 = 2.5, and −10 has the larger absolute value: −2.5.
3) Common denominator 12: −9/12 + 10/12. 10 − 9 = 1, and the larger one, 10/12, is positive: 1/12.
4) The sum of opposite numbers: 0.

Subtraction means adding the opposite

Definition

Difference

The difference of a and b is the number a − b which, added to b, gives a: (a − b) + b = a. So subtraction is the inverse of addition.

Taking 2 away from 5 is the same as adding −2 to 5: 5 − 2 = 5 + (−2) = 3, because in both cases you move 2 units to the left of 5 on the line. The same idea works when you subtract a negative number: 2 − (−6) = 2 + 6 = 8. Check by the definition: 8 + (−6) = 2. In money terms: if someone cancels your debt of 6 manat, you are 6 manat better off.

a − b = a + (−b)

where:

athe number you subtract from (the minuend)
bthe number being subtracted (the subtrahend)
−bthe opposite of the subtrahend

Subtraction can always be turned into addition: the minuend stays, “−” becomes “+”, and the subtrahend is replaced by its opposite. In particular: a − (−b) = a + b.

Subtraction

Calculate: 1) 2 − (−6); 2) −5 − 3; 3) −4 − (−9); 4) −1.2 − 0.8; 5) 1/3 − 5/6.

Show solution

1) 2 + 6 = 8.
2) −5 + (−3) = −8 (same signs).
3) −4 + 9 = 5 (9 − 4 = 5, and 9, the larger one, is positive).
4) −1.2 + (−0.8) = −2.
5) 2/6 − 5/6 = 2/6 + (−5/6) = −3/6 = −1/2.

An important use of subtraction is the difference between two quantities: a temperature difference, a height difference, the distance between two points on the number line. In the previous lesson we answered such questions by counting units; now one formula does it.

AB = |a − b|

where:

A(a), B(b)two points on the number line
ABthe distance between them

The distance between two points is the absolute value of the difference of their coordinates. If you subtract the smaller coordinate from the larger one, you do not need the absolute value: the result is positive anyway.

Differences and distances

1) It is +6 °C in Baku and −8 °C at Shahdag. What is the temperature difference?
2) Find the distance between the points A(−7) and B(−2).
3) A diver came up from a depth of −18 m onto the deck of a boat 2 m above the water. How many metres did he rise?

Show solution

1) 6 − (−8) = 6 + 8 = 14 °C — the same answer we got in the previous lesson by counting units.
2) AB = |−2 − (−7)| = |−2 + 7| = |5| = 5.
3) Subtract the starting height from the final one: 2 − (−18) = 2 + 18 = 20 m.

Multiplication and division: the sign rules

Multiplying by a positive number is repeated addition: 3 · (−2) = (−2) + (−2) + (−2) = −6 — three debts of 2 manat make a debt of 6 manat. But what is (−1) · (−2)? Look at the pattern: 3 · (−2) = −6, 2 · (−2) = −4, 1 · (−2) = −2, 0 · (−2) = 0. Each time the first factor goes down by 1, the product goes up by 2. For the pattern to continue, the next step must be (−1) · (−2) = 2. So the product of two negative numbers is positive.

(−a) · (−b) = a · b; (−a) · b = a · (−b) = −(a · b)

where:

a, bthe absolute values of the factors (a, b ≥ 0)

The product of two numbers with the same sign is positive, and the product of two numbers with different signs is negative. The absolute value of the product is the product of the absolute values. If one factor is 0, the product is 0.

·	+	−
+	+	−
−	−	+

Table of signs: the row is the sign of the first factor, the column the sign of the second. It is the same for division.

Multiplication

Calculate: 1) (−4) · (−3); 2) (−2.5) · 4; 3) (−3/4) · (−8/9); 4) (−1) · (−1) · (−1) · 5; 5) (−2) · 5 · (−0.5) · (−1).

Show solution

1) Same signs: 4 · 3 = 12.
2) Different signs: 2.5 · 4 = 10, so −10.
3) Same signs; cancel first: 3/4 · 8/9 = (1 · 2)/(1 · 3) = 2/3.
4) From left to right: (−1) · (−1) = 1, 1 · (−1) = −1, (−1) · 5 = −5.
5) (−2) · 5 = −10, (−10) · (−0.5) = 5, 5 · (−1) = −5. There are three negative factors, so the answer is negative.

Division is the inverse of multiplication: (−12) ÷ 3 = −4, because (−4) · 3 = −12. That is why the sign in division follows the same rule. Division by zero is impossible: no number multiplied by 0 gives −12.

(−a) ÷ (−b) = a ÷ b; (−a) ÷ b = a ÷ (−b) = −(a ÷ b), b ≠ 0

where:

athe absolute value of the dividend
bthe absolute value of the divisor (it cannot be zero)

The quotient of numbers with the same sign is positive, and the quotient of numbers with different signs is negative. 0 divided by any non-zero number is 0.

Division

Calculate: 1) (−20) ÷ 5; 2) (−36) ÷ (−4); 3) 1.8 ÷ (−0.6); 4) (−3/5) ÷ (−9/10); 5) 0 ÷ (−7).

Show solution

1) Different signs: 20 ÷ 5 = 4, −4.
2) Same signs: 36 ÷ 4 = 9.
3) 1.8 ÷ 0.6 = 18 ÷ 6 = 3; the signs differ: −3.
4) Same signs: 3/5 · 10/9 = 30/45 = 2/3.
5) 0.

Powers of negative numbers: (−a)ⁿ and −aⁿ

A power is a product of equal factors: (−2)³ = (−2) · (−2) · (−2). There are three negative factors, so the result is negative: −8. In (−2)⁴ there are four negative factors, and the result is positive: 16. The brackets matter a lot: in (−2)⁴ the number −2 is raised to the power, while in −2⁴ only 2 is raised to the power and then the opposite is taken: −2⁴ = −(2 · 2 · 2 · 2) = −16.

(−a)ⁿ = aⁿ if n is even; (−a)ⁿ = −aⁿ if n is odd; −aⁿ = −(aⁿ)

where:

athe absolute value of the base
nthe exponent — a natural number

An even power of a negative number is positive, an odd power is negative. In −aⁿ the power is worked out first and then the opposite is taken: the power comes before the “−” in front of it.

Powers

Calculate: 1) (−2)⁴ and −2⁴; 2) (−3)³; 3) (−1)¹⁰⁰ and (−1)¹⁰¹; 4) (−0.5)²; 5) (−2/3)³.

Show solution

1) (−2)⁴ = 16 (four negative factors); −2⁴ = −(2⁴) = −16.
2) (−3)³ = (−3) · (−3) · (−3) = −27.
3) 100 is even: (−1)¹⁰⁰ = 1; 101 is odd: (−1)¹⁰¹ = −1.
4) (−0.5) · (−0.5) = 0.25.
5) An odd power, so the sign is “−”: 2³/3³ = 8/27, so −8/27.

Laws and order of operations

The laws you know for natural numbers hold for all rational numbers too. They make calculations much easier: you can group the positive numbers and the negative numbers separately, put numbers that make 10, 100 or 0 next to each other, and take a common factor outside the brackets.

a + b = b + a; (a + b) + c = a + (b + c); a · b = b · a; (a · b) · c = a · (b · c); a · (b + c) = a · b + a · c

where:

a, b, cany rational numbers

The commutative, associative and distributive laws also hold for negative numbers, fractions and decimals. Subtraction and division are not commutative.

Clever calculation with the laws

Calculate in a clever way: 1) −17 + 45 + (−23) + 5; 2) (−25) · 7 · (−4); 3) −3.7 · 6 + (−3.7) · 4; 4) (−8) · (1/2 − 3/4).

Show solution

1) Group the negatives and the positives: (−17 + (−23)) + (45 + 5) = −40 + 50 = 10.
2) Swap the factors: (−25) · (−4) · 7 = 100 · 7 = 700.
3) Distributive law: −3.7 · (6 + 4) = −3.7 · 10 = −37.
4) Expand the brackets: (−8) · 1/2 − (−8) · 3/4 = −4 − (−6) = −4 + 6 = 2. Check: 1/2 − 3/4 = −1/4 and (−8) · (−1/4) = 2.

The order of operations is the same as for natural numbers (see the lesson “Natural numbers and the order of operations”); you just have to watch the sign at every step. A negative number written after an operation sign goes in brackets: 5 · (−2), not “5 · −2”.

1
Brackets
First do the operations inside the brackets, starting with the innermost ones.
2
Powers
Work out the powers; do not mix up (−a)ⁿ and −aⁿ.
3
Multiplication and division
From left to right; each time find the sign first, then the absolute value.
4
Addition and subtraction
Replace each subtraction by adding the opposite, then add from left to right or group the positives and the negatives separately.

Multi-step expressions

Evaluate: 1) (−3) · 4 − (−10) ÷ 2 + (−2)²; 2) (−1/2 + 1/3) · (−6) − (−2)³ ÷ 4; 3) −0.5 · (−2.4 + 1.4)² − 3.

Show solution

1) Power: (−2)² = 4. Multiplication: (−3) · 4 = −12. Division: (−10) ÷ 2 = −5.
−12 − (−5) + 4 = −12 + 5 + 4 = −3.
2) Brackets: −3/6 + 2/6 = −1/6. Power: (−2)³ = −8.
(−1/6) · (−6) = 1; (−8) ÷ 4 = −2.
1 − (−2) = 1 + 2 = 3.
3) Brackets: −2.4 + 1.4 = −1. Power: (−1)² = 1.
−0.5 · 1 = −0.5; −0.5 − 3 = −3.5.

Word problems

1) The temperature from the introduction: −5 °C in the morning, then it rose by 9 degrees and fell by 6 degrees.
2) Elvin’s account: it was −3 manat, he added 10 manat and spent 8.5 manat.
3) A submarine was at a depth of −120 m. It rose 45 m and then went down 30 m. At what depth is it now?

Show solution

1) Rising means “+” and falling means “−”: −5 + 9 − 6 = 4 − 6 = −2 °C.
2) −3 + 10 − 8.5 = 7 − 8.5 = −1.5 manat: Elvin still owes 1.5 manat.
3) −120 + 45 − 30 = −75 − 30 = −105 m, that is 105 m below the surface.

Average temperature

For a week, Aysel wrote down the temperature every morning: −3; −5; 0; 2; −1; −4; −3 °C. Find the average morning temperature of the week.

Show solution

Average = sum ÷ number of values.
Group the negatives: (−3) + (−5) + (−1) + (−4) + (−3) = −16; the rest: 0 + 2 = 2.
Sum: −16 + 2 = −14.
Average temperature: (−14) ÷ 7 = −2 °C.

Interactive

Loading simulation…

Put each expression into a group by the sign of its value. Hint: count the negative factors and turn subtraction into addition.

Check yourself

1.(−7) + 3 =
2.−6 − (−10) =
3.(−2.5) · (−4) =
4.(−48) ÷ 6 =
5.(−1)¹⁵ =
6.−5² + (−5)² =

You can now do the four operations and raise to powers with any rational numbers — whole numbers, fractions and decimals, positive and negative. In the next topic, “Algebraic expressions, monomials and polynomials”, letters take the place of numbers: the “−” in front of brackets, negative coefficients and collecting like terms all rely on the sign rules of this lesson.

Key points

To add numbers with the same sign, add the absolute values and keep the sign; with different signs, subtract the smaller absolute value from the larger and take the sign of the number with the larger absolute value.
Subtraction is adding the opposite: a − b = a + (−b), in particular a − (−b) = a + b. The distance between two points is |a − b|.
In multiplication and division, same signs give “+” and different signs give “−”; with many factors, an even number of negative factors gives a positive product. Division by zero is impossible.
(−a)ⁿ is positive for even n and negative for odd n; −aⁿ = −(aⁿ). So (−3)² = 9, but −3² = −9.
The commutative, associative and distributive laws hold for rational numbers; the order is brackets → powers → multiplication and division → addition and subtraction.

Check yourself

12 questions. Every correct answer earns XP.

1 / 12

How can subtraction be replaced by addition?