Integer Operations

Integers are the whole numbers and their opposites: $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$ The sign tells you which side of zero you’re on, so integers are perfect for temperatures below zero, depths below sea level, money you owe, and changes that go up or down. In this lesson you’ll learn the rules for calculating with them, and why the rules work.

Key ideas

Integers describe location, direction and change

Zero is a reference point, and the sign tells you which side of it you’re on:

Situation	Positive	Zero	Negative
temperature	above $0\,^\circ\text{C}$	freezing point of water	below $0\,^\circ\text{C}$
elevation	above sea level	sea level	below sea level
bank balance	money in the account	empty	overdrawn (money owed)
change	goes up, gains, deposits	no change	goes down, losses, withdrawals

For example, a temperature drop of $8$ degrees is a change of $-8\,^\circ\text{C}$, and a diver $12$ m below the surface is at $-12$ m.

Two numbers like $5$ and $-5$ are opposites: they’re the same distance from zero, on opposite sides.

Adding integers

On a number line, start at the first number. Adding a positive number moves you right; adding a negative number moves you left.

Adding a positive moves right; adding a negative moves left.

You can also think of zero pairs: a $+1$ and a $-1$ cancel to make $0$. For $-5 + 3$, three of the five negatives cancel with the three positives, leaving $-2$.

Shortcuts:

• Same signs: add the sizes and keep the sign. $-6 + (-9) = -15$.
• Different signs: subtract the smaller size from the bigger one, and keep the sign of the number with the bigger size. $-8 + 5 = -3$.

Subtracting integers: add the opposite

Subtracting a number is the same as adding its opposite:

$$a - b = a + (-b)$$

So $4 - 9 = 4 + (-9) = -5$, and $4 - (-3) = 4 + 3 = 7$.

Why does subtracting a negative make the answer bigger? Think of a bank balance. Taking away a $30 debt leaves you $30 better off.

Subtraction also measures change or distance:

$$\text{change} = \text{final value} - \text{starting value}$$

Multiplying and dividing integers

Multiplying a negative by a positive is repeated addition: $3 \times (-2) = (-2) + (-2) + (-2) = -6$.

For a negative times a negative, look at the pattern. Each time the first number goes down by $1$, the answer goes up by $2$:

$$3(-2) = -6, \quad 2(-2) = -4, \quad 1(-2) = -2, \quad 0(-2) = 0, \quad (-1)(-2) = 2, \quad (-2)(-2) = 4$$

So a negative times a negative is positive. Division follows the same sign rules, because division undoes multiplication.

Signs	Product or quotient	Examples
same signs	positive	$(-4)(-5) = 20$, $\quad (-12) \div (-3) = 4$
different signs	negative	$(-4)(5) = -20$, $\quad 12 \div (-3) = -4$

With more than two factors, count the negatives. An even number of negative factors gives a positive answer; an odd number gives a negative answer. For example, $(-1)(-2)(-3) = -6$ (three negatives).

Signs in fractions, decimals, ratios and rates

A fraction is a division, so the sign rules apply. One negative sign makes the whole fraction negative, wherever you put it:

$$-\frac{3}{4} = \frac{-3}{4} = \frac{3}{-4} = -0.75 \qquad \text{but} \qquad \frac{-3}{-4} = \frac{3}{4} = 0.75$$

The same goes for rates. A rate compares a change in one quantity to a change in another. If the temperature changes by $-12\,^\circ\text{C}$ over $4$ hours, the rate is

$$\frac{-12\,^\circ\text{C}}{4 \text{ h}} = -3\,^\circ\text{C per hour}$$

The negative sign tells you the temperature is falling. A positive rate means the quantity is increasing.

Order of operations

When an expression has several operations, use BEDMAS:

1. Brackets
2. Exponents
3. Division and Multiplication, from left to right
4. Addition and Subtraction, from left to right

Be careful with exponents and negatives: $(-2)^2 = (-2)(-2) = 4$, but $-2^2 = -(2 \times 2) = -4$. The exponent only applies to what it’s touching. (There’s more on this in exponent laws.)

Worked examples

Example 1: Temperature changes

In Thunder Bay, the temperature at 7 a.m. was $-14\,^\circ\text{C}$. By 3 p.m. it was $3\,^\circ\text{C}$.

• (a) Find the change in temperature.
• (b) By 10 p.m. the temperature had dropped $9$ degrees from its 3 p.m. value. What was the temperature at 10 p.m.?

Solution.

(a) Change is final minus starting:

$$3 - (-14) = 3 + 14 = 17$$

The temperature rose $17\,^\circ\text{C}$. Check on a number line: from $-14$ to $0$ is $14$ steps, and from $0$ to $3$ is $3$ more, for $17$. ✓

(b) A drop of $9$ is a change of $-9$:

$$3 + (-9) = -6$$

It was $-6\,^\circ\text{C}$ at 10 p.m.

Example 2: Adding and subtracting

Evaluate each expression.

• (a) $-8 + 5$
• (b) $-6 - 9$
• (c) $7 - (-4)$
• (d) $-12 - (-20)$

Solution.

(a) Different signs: $8 - 5 = 3$, and the bigger size ($8$) is negative. So $-8 + 5 = -3$.

(b) Add the opposite: $-6 - 9 = -6 + (-9) = -15$.

(c) Add the opposite: $7 - (-4) = 7 + 4 = 11$.

(d) Add the opposite: $-12 - (-20) = -12 + 20 = 8$.

Example 3: Multiplying, dividing and rates

• (a) Evaluate $(-6)(7)$, $\ (-48) \div (-8)$ and $(-2)(-3)(-5)$.
• (b) A diver goes from $-20$ m to $-110$ m in $6$ minutes. Find her rate of change of depth.

Solution.

(a)

• $(-6)(7) = -42$ (different signs, so negative)
• $(-48) \div (-8) = 6$ (same signs, so positive)
• $(-2)(-3)(-5) = (6)(-5) = -30$ (three negatives, so negative)

(b) First find the change in elevation, then divide by the time:

$$\text{rate} = \frac{-110 - (-20)}{6} = \frac{-110 + 20}{6} = \frac{-90}{6} = -15$$

Her rate is $-15$ m per minute: she goes $15$ m deeper each minute.

Example 4: Order of operations

Evaluate $-3(4 - 9) + (-2)^3 \div 4$.

Solution.

$$\begin{aligned} -3(4 - 9) + (-2)^3 \div 4 &= -3(-5) + (-2)^3 \div 4 && \text{brackets} \\ &= -3(-5) + (-8) \div 4 && \text{exponent: } (-2)(-2)(-2) = -8 \\ &= 15 + (-2) && \text{multiply and divide} \\ &= 13 && \text{add} \end{aligned}$$

Common mistakes

Adding the sizes when the signs are different. $-8 + 5$ is not $-13$. On a number line, start at $-8$ and move $5$ to the right: you land on $-3$.

Forgetting to change both signs when subtracting a negative. $7 - (-4)$ becomes $7 + 4 = 11$, not $3$. Rewrite every subtraction as “add the opposite” before you calculate.

Using “two negatives make a positive” for addition. That rule is for multiplying and dividing. When you add two negatives, the answer is more negative: $-3 + (-4) = -7$.

Mixing up $-2^2$ and $(-2)^2$. $(-2)^2 = 4$, but $-2^2 = -4$. Without brackets, the exponent only applies to the $2$.

Doing multiplication before division. Multiplication and division have the same rank, so go left to right: $12 \div (-3) \times 2 = (-4) \times 2 = -8$, not $12 \div (-6) = -2$.

Finding change the wrong way round. Change is final minus starting. Going from $-14\,^\circ\text{C}$ to $3\,^\circ\text{C}$ is a change of $+17$, not $-17$.

Practice

1. (Warm-up) Write an integer for each situation.

• (a) $15$ m below sea level
• (b) a deposit of $40
• (c) the temperature falls $6$ degrees
• (d) $3$ floors below ground level

Solution

(a) $-15$ m (b) $+40$ dollars (c) $-6\,^\circ\text{C}$ (d) $-3$

2. (Warm-up) Evaluate.

• (a) $-9 + 4$
• (b) $-3 + (-11)$
• (c) $6 - 15$
• (d) $-2 - (-10)$

Solution

(a) $-9 + 4 = -5$

(b) $-3 + (-11) = -14$

(c) $6 - 15 = 6 + (-15) = -9$

(d) $-2 - (-10) = -2 + 10 = 8$

3. (Warm-up) Evaluate.

• (a) $(-7)(-8)$
• (b) $63 \div (-9)$
• (c) $(-1)(4)(-5)(-2)$
• (d) $(-36) \div (-4) \div (-3)$

Solution

(a) Same signs: $(-7)(-8) = 56$.

(b) Different signs: $63 \div (-9) = -7$.

(c) Three negative factors, so the answer is negative: $1 \times 4 \times 5 \times 2 = 40$, so the product is $-40$.

(d) Left to right: $(-36) \div (-4) = 9$, then $9 \div (-3) = -3$.

4. (Core) Maya has $65 in her bank account. She buys a video game for $89, then deposits $30, then is charged a $12 fee. What is her balance now? What does the sign mean?

Solution

Purchases and fees are negative changes; deposits are positive:

$$65 - 89 + 30 - 12 = -24 + 30 - 12 = 6 - 12 = -6$$

Her balance is $-6$ dollars. The negative sign means her account is overdrawn: she owes the bank $6.

5. (Core) Which of these are equal to $-\dfrac{2}{5}$? Explain.

• (a) $\dfrac{2}{-5}$
• (b) $\dfrac{-2}{-5}$
• (c) $-0.4$
• (d) $-\left(\dfrac{-2}{5}\right)$

Solution

(a) Equal. One negative sign makes the fraction negative.

(b) Not equal. Two negative signs: $\dfrac{-2}{-5} = \dfrac{2}{5}$.

(c) Equal. $2 \div 5 = 0.4$, so $-\dfrac{2}{5} = -0.4$.

(d) Not equal. The opposite of $-\dfrac{2}{5}$ is $\dfrac{2}{5}$.

6. (Core) Evaluate using the order of operations.

• (a) $5 - 3(-4)$
• (b) $(-6)^2 - 4(-2)(3)$
• (c) $-20 \div (2 - 7) \times (-3)$

Solution

(a) Multiply first: $5 - 3(-4) = 5 - (-12) = 5 + 12 = 17$.

(b) Exponent first, then multiply:

$$(-6)^2 - 4(-2)(3) = 36 - (-24) = 36 + 24 = 60$$

(c) Brackets first, then left to right:

$$-20 \div (2 - 7) \times (-3) = -20 \div (-5) \times (-3) = 4 \times (-3) = -12$$

7. (Core) In Winnipeg, the temperature was $4\,^\circ\text{C}$ at 6 p.m. and fell steadily to $-11\,^\circ\text{C}$ by 11 p.m.

• (a) Find the rate of change of the temperature, in degrees per hour.
• (b) If it keeps falling at the same rate, what will the temperature be at 1 a.m.?

Solution

(a) From 6 p.m. to 11 p.m. is $5$ hours.

$$\text{rate} = \frac{-11 - 4}{5} = \frac{-15}{5} = -3$$

The temperature changed by $-3\,^\circ\text{C}$ per hour (falling $3$ degrees each hour).

(b) From 11 p.m. to 1 a.m. is $2$ hours: $-11 + 2(-3) = -11 - 6 = -17$. It will be $-17\,^\circ\text{C}$.

8. (Challenge) Find two integers whose product is $-24$ and whose sum is $2$.

Solution

The product is negative, so one integer is positive and one is negative. List the pairs that multiply to $24$ and give one a negative sign, then check the sum:

Pair	Sum
$-1$ and $24$	$23$
$-2$ and $12$	$10$
$-3$ and $8$	$5$
$-4$ and $6$	$2$ ✓

The integers are $-4$ and $6$. Check: $(-4)(6) = -24$ and $-4 + 6 = 2$. ✓ (Pairs like $4$ and $-6$ have sum $-2$, so they don’t work.) You’ll use this kind of search in Grade 10 to factor expressions.

9. (Challenge) Without a calculator, decide whether each answer is positive or negative. Then evaluate it.

• (a) $(-3)^5$
• (b) $(-1)(-2)(-3)(-4)(-5)(-6)$

Solution

(a) $(-3)^5$ means five factors of $-3$. Five is odd, so the answer is negative: $3^5 = 243$, so $(-3)^5 = -243$.

(b) Six negative factors is an even number, so the answer is positive:

$$1 \times 2 \times 3 \times 4 \times 5 \times 6 = 720$$

So the product is $720$.