Operations with Integers

➖➕ Operations with Integers

Part 1 of 5 — Integers & the Number Line

Topics in This Part

🔑 Key Concept: Integers are the whole numbers and their negatives: $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$ Master the number line first, and every operation that follows becomes a short walk in one direction.

What Is an Integer?

An integer is a number with no fractional or decimal part — the counting numbers, their negatives, and zero.

$\dots,\; -3,\; -2,\; -1,\; 0,\; 1,\; 2,\; 3,\; \dots$

• Positive integers: $1, 2, 3, \dots$ (often written without a sign)
• Negative integers: $-1, -2, -3, \dots$ (always written with a $-$ sign)
• Zero: $0$ is an integer, but it is neither positive nor negative.

These are NOT integers

💡 Where they show up: temperatures below zero ($-8^\circ$), elevation below sea level ($-200$ ft), money you owe (a -$50 balance), and yardage lost in football.

The Number Line & Opposites

Picture every integer as a position on a horizontal line. Negatives sit to the left of $0$; positives sit to the right.

$\underset{\textstyle -3}{\bullet}\;\;\underset{\textstyle -2}{\bullet}\;\;\underset{\textstyle -1}{\bullet}\;\;\underset{\textstyle 0}{\bullet}\;\;\underset{\textstyle 1}{\bullet}\;\;\underset{\textstyle 2}{\bullet}\;\;\underset{\textstyle 3}{\bullet}$

The further right you go, the bigger the number; the further left, the smaller.

Opposites

Two integers are opposites if they are the same distance from $0$ but on opposite sides. The opposite of a number is found by flipping its sign.

🔑 Key Idea: The opposite of the opposite brings you back: $-(-7) = 7$. Two negative signs cancel.

Concept Check 🎯

Absolute Value

The absolute value of a number is its distance from $0$ on the number line — and distance is never negative. We write it with two vertical bars:

$\lvert -7 \rvert = 7 \qquad \lvert 4 \rvert = 4 \qquad \lvert 0 \rvert = 0$

Both $-7$ and $7$ are a distance of $7$ from zero, so both have absolute value $7$.

⚠️ Watch out: Absolute value asks "how far," not "which side." The answer is always positive or zero — never negative. $\lvert -15 \rvert = 15$, not $-15$.

We will use absolute value as a tool in Part 2 to add and subtract integers cleanly.

Absolute Value Drill 🧮

Evaluate each. (Distance from $0$ is never negative.)

1) $\lvert -6 \rvert = \,?$ 2) $\lvert 10 \rvert = \,?$ 3) $-\lvert -4 \rvert = \,?$ (careful — the minus sign is OUTSIDE the bars)

Order on the Number Line 🔽

Fill in each comparison with the correct symbol or value.

➖➕ Operations with Integers

Part 2 of 5 — Adding Integers

🔑 The Big Idea: Adding an integer means moving along the number line. Add a positive → move right. Add a negative → move left.

Rule 1 — Same Signs: Add, Keep the Sign

When two numbers have the same sign, add their absolute values and keep that shared sign.

Why it works: Two negatives both push you left. Start at $-5$, move $3$ more to the left, and you land on $-8$.

💡 Money analogy: Owing $5 and then owing $3 more means you owe $8 total: $-5 + (-3) = -8$.

Rule 2 — Different Signs: Subtract, Keep the Bigger Sign

When the signs are different, subtract the smaller absolute value from the larger, and keep the sign of the number with the larger absolute value.

Worked Example: $-8 + 3$

$\lvert -8 \rvert = 8, \quad \lvert 3 \rvert = 3 \;\Rightarrow\; 8 - 3 = 5$

Since $-8$ has the larger absolute value, the answer takes the negative sign: $-8 + 3 = -5$.

Worked Example: $9 + (-4)$

$\lvert 9 \rvert = 9, \quad \lvert -4 \rvert = 4 \;\Rightarrow\; 9 - 4 = 5$

Since $9$ has the larger absolute value, the answer is positive: $9 + (-4) = 5$.

🔑 The two-step recipe: (1) subtract the absolute values, (2) take the sign of whichever number was "bigger" (further from $0$).

Concept Check 🎯

Adding Drill 🧮

Find each sum. Watch the signs!

1) $-9 + (-6) = \,?$ 2) $12 + (-5) = \,?$ 3) $-15 + 8 = \,?$ 4) $-4 + 4 = \,?$

Pick the Rule, Then the Sign 🔽

For each sum, choose whether you ADD or SUBTRACT the absolute values, and the sign of the result.

➖➕ Operations with Integers

Part 3 of 5 — Subtracting Integers

🔑 The One Trick You Need: Subtraction is just "adding the opposite." Change the subtraction to addition, flip the sign of the second number, then use your Part 2 addition rules.

Keep–Change–Change

Every subtraction problem can be rewritten as an addition problem:

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

The memory hook is Keep–Change–Change:

1. Keep the first number the same.
2. Change the subtraction sign to addition.
3. Change the sign of the second number to its opposite.

Worked Example: $7 - 10$

$7 - 10 \;=\; 7 + (-10)$

Now it's an addition problem: different signs, $10 - 7 = 3$, and the larger value ($10$) is negative → $\boxed{-3}$.

Worked Example: $4 - (-6)$

$4 - (-6) \;=\; 4 + (+6) \;=\; 4 + 6 = \boxed{10}$

⚠️ The classic trap: Subtracting a negative adds. The two minus signs in $4 - (-6)$ become a plus: $4 + 6 = 10$. "Minus a minus is a plus."

Concept Check 🎯

Walk Through Keep–Change–Change 🔽

You're simplifying $-7 - (-4)$. Choose what happens at each step.

Subtracting Drill 🧮

Use Keep–Change–Change on each.

1) $3 - 9 = \,?$ 2) $-6 - 5 = \,?$ 3) $8 - (-2) = \,?$ 4) $-4 - (-10) = \,?$

Why Subtraction = Distance

Subtraction also answers "how far apart?" The temperature drops from $5^\circ$ to $-3^\circ$. How big was the change?

$-3 - 5 = -3 + (-5) = -8$

The temperature changed by $8$ degrees (downward). The size of the gap is $\lvert -8 \rvert = 8$.

💡 To find the distance between two integers on the number line, subtract them and take the absolute value: $\text{distance} = \lvert a - b \rvert$. For $5$ and $-3$: $\lvert 5 - (-3) \rvert = \lvert 8 \rvert = 8$.

➖➕ Operations with Integers

Part 4 of 5 — Multiplying & Dividing Integers

🔑 The Sign Rule (the same for × and ÷): Same signs → positive. Different signs → negative. The numbers multiply or divide exactly as usual; only the sign needs the rule.

The Sign Rules

For both multiplication and division, look only at the two signs:

The same table works for division:

$\frac{-12}{-3} = 4 \qquad \frac{-12}{3} = -4 \qquad \frac{12}{-3} = -4$

🔑 Quick saying: "Same signs, positive. Different signs, negative." It holds for × and ÷ alike.

⚠️ Don't confuse this with adding! For addition, $(-)+(-)$ stays negative. But for multiplying, $(-)\times(-)$ becomes positive. Two different rule sets — keep them straight.

Concept Check 🎯

Multiply & Divide Drill 🧮

Apply the sign rule, then compute.

1) $(-9)(4) = \,?$ 2) $(-7)(-6) = \,?$ 3) $\dfrac{-40}{-8} = \,?$ 4) $\dfrac{36}{-9} = \,?$

Counting Negative Signs

When you multiply three or more numbers, count how many are negative:

• An even number of negative factors → the product is positive.
• An odd number of negative factors → the product is negative.

Worked Example: $(-2)(-3)(-1)$

There are three negative factors (odd), so the product is negative. The size is $2 \cdot 3 \cdot 1 = 6$:

$(-2)(-3)(-1) = -6$

Worked Example: $(-2)(-5)(4)$

There are two negative factors (even), so the product is positive. The size is $2 \cdot 5 \cdot 4 = 40$:

$(-2)(-5)(4) = 40$

💡 Pair the negatives off: each pair of negatives makes a positive. One left over (odd) keeps the whole thing negative.

Predict the Sign 🔽

For each expression, choose the sign of the final answer (don't compute the digits — just the sign).

➖➕ Operations with Integers

Part 5 of 5 — Order of Operations & Mastery Check

You can now add, subtract, multiply, and divide integers. The last skill is combining them in one expression — and getting the order right.

Order of Operations (PEMDAS)

When an expression mixes operations, follow this order:

Worked Example: $-4 + 3 \cdot (-2)$

Multiply first, then add:

$-4 + 3 \cdot (-2) = -4 + (-6) = -10$

Worked Example: $(-2)^2 - 5$

Exponent first. Note $(-2)^2 = (-2)(-2) = 4$ (same signs → positive):

$(-2)^2 - 5 = 4 - 5 = -1$

⚠️ A famous trap: $(-2)^2 = 4$, but $-2^2 = -4$. With no parentheses, the exponent attaches only to the $2$, and the minus sign is applied last: $-2^2 = -(2^2) = -4$.

Order It Correctly 🔽

Simplify $10 - 2 \cdot (-3)$ one step at a time.

Mixed Operations Drill 🧮

Use PEMDAS. Take it one step at a time.

1) $-3 + 4 \cdot (-2) = \,?$ 2) $(-5)^2 - 30 = \,?$ 3) $\dfrac{-18}{3} + (-2) = \,?$ 4) $8 - 2(3 - 7) = \,?$

Quick Reference

🔑 The two ideas to never mix up: adding two negatives stays negative ($-3 + (-3) = -6$), but multiplying two negatives turns positive ($(-3)(-3) = 9$).

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.