Operations with Integers

🌡️ Operations with Integers

Part 1 of 5 — Meet the Integers

Topics in This Part

🔑 Key Concept: Integers are the whole numbers and their negatives: $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$ — no fractions, no decimals. Before we add, subtract, multiply, or divide them, we need to picture them on a number line.

What Is an Integer?

An integer is any number from this list:

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

That includes:

• the positive whole numbers: $1, 2, 3, \ldots$
• zero: $0$ (it is neither positive nor negative)
• the negative whole numbers: $-1, -2, -3, \ldots$

Integers show up everywhere in real life:

⚠️ Not integers: $\frac{1}{2}$, $0.75$, and $-2.5$ are not integers because they aren't whole numbers. Integers never have a fraction or decimal part.

Concept Check 🎯

The Number Line

A number line puts the integers in order. Negative numbers sit to the left of zero; positive numbers sit to the right.

   ←  -5  -4  -3  -2  -1   0   1   2   3   4   5  →

Two big ideas live on the number line:

1. Farther right = bigger. So $3 > 1$, and $-1 > -4$ (because $-1$ is to the right of $-4$).
2. Opposites are the same distance from zero, on opposite sides. The opposite of $4$ is $-4$; the opposite of $-6$ is $6$.

💡 Surprising but true: $-1$ is greater than $-10$. On the number line, $-1$ is closer to the positive side. With negatives, the number that looks smaller (like $10$ vs $1$) is actually the least.

Compare on the Number Line 🎯

Absolute Value

The absolute value of a number is its distance from zero on the number line. Distance is never negative, so absolute value is always $0$ or positive.

We write it with two bars: $|{-7}| = 7$ and $|7| = 7$.

🔑 Shortcut: To take an absolute value, just drop the sign — the answer is how far the number is from zero, never which direction. Absolute value will be the engine behind every addition rule in Part 2.

Absolute Value & Ordering 🧮

Find each value. (Type whole numbers only.)

1) $|{-12}| = \,?$ 2) $|6| = \,?$ 3) Of the three numbers $-2,\; -20,\; -9$, which is the least? $\,?$

🌡️ Operations with Integers

Part 2 of 5 — Adding Integers

🔑 The Two Rules: Adding integers comes down to checking the signs. Same signs? Add and keep the sign. Different signs? Subtract and keep the sign of the bigger-distance number.

Rule 1 — Same Signs: Add and Keep the Sign

When both numbers have the same sign, add their absolute values and keep that common sign.

Worked Example: $(-4) + (-3)$

Both are negative. Add the distances $4 + 3 = 7$, then keep the negative sign:

$(-4) + (-3) = -7$

Think of it as owing $4 and then owing $3 more — you owe $7.

Worked Example: $5 + 9$

Both positive. Add $5 + 9 = 14$, keep positive:

$5 + 9 = 14$

💡 Picture it: On the number line, adding a positive moves you right; adding a negative moves you left. Two negatives means two moves left, so you land further left (more negative).

Same-Sign Sums 🧮

Add. Include the negative sign when needed (for example, type -11).

1) $(-6) + (-5) = \,?$ 2) $8 + 7 = \,?$ 3) $(-10) + (-10) = \,?$

Rule 2 — Different Signs: Subtract and Keep the Winner's Sign

When the signs are different, subtract the smaller absolute value from the larger one. The answer takes the sign of the number with the larger absolute value.

Worked Example: $(-9) + 4$

Distances are $9$ and $4$. Subtract: $9 - 4 = 5$. The $-9$ has the larger distance, so the answer is negative:

$(-9) + 4 = -5$

Worked Example: $12 + (-7)$

Distances $12$ and $7$. Subtract: $12 - 7 = 5$. The $12$ (positive) has the larger distance, so the answer is positive:

$12 + (-7) = +5$

⚠️ Common mistake: Don't just keep the sign of the first number. Keep the sign of whichever number is farther from zero.

Different-Sign Sums 🎯

Which Rule, Then What Sign? 🔽

For each sum, pick the rule and the sign of the answer.

Mixed Addition 🧮

Add. Use a negative sign when needed.

1) $(-13) + 5 = \,?$ 2) $20 + (-8) = \,?$ 3) $(-6) + (-14) = \,?$

🌡️ Operations with Integers

Part 3 of 5 — Subtracting Integers

🔑 The Big Trick: You never really subtract integers — you turn every subtraction into an addition problem, then use the Part 2 rules. The motto is "Keep, Change, Change."

Subtracting = Adding the Opposite

To subtract an integer, add its opposite:

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

The fastest way to remember this is Keep · Change · Change:

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

Worked Example: $5 - 8$

Keep $5$, change $-$ to $+$, change $8$ to $-8$:

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

(Now it's a different-signs addition: $8 - 5 = 3$, and $-8$ is farther from zero, so $-3$.)

💡 Why it works: Subtracting $8$ means moving $8$ steps left on the number line — exactly the same as adding $-8$.

The Double-Negative Case

The trick really shines when you subtract a negative number. The opposite of a negative is a positive.

Worked Example: $4 - (-6)$

Keep $4$, change $-$ to $+$, change $-6$ to $+6$:

$4 - (-6) = 4 + 6 = 10$

Worked Example: $(-3) - (-9)$

Keep $-3$, change $-$ to $+$, change $-9$ to $+9$:

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

⚠️ Watch the two negatives: $4 - (-6)$ is not $4 - 6$. The "minus a negative" becomes a plus, so it grows to $10$. Subtracting a negative makes a number bigger.

Keep · Change · Change 🔽

Rewrite $2 - (-5)$ step by step.

Subtraction Check 🎯

Subtraction Drill 🧮

Rewrite with Keep-Change-Change, then solve. Use a negative sign when needed.

1) $3 - 10 = \,?$ 2) $(-5) - (-12) = \,?$ 3) $(-8) - 8 = \,?$

🌡️ Operations with Integers

Part 4 of 5 — Multiplying & Dividing Integers

🔑 One Sign Rule for Both: Multiplication and division share the exact same sign rule. Same signs → positive. Different signs → negative. Multiply or divide the numbers as usual, then attach the sign.

The Sign Rule

First find the answer as if both numbers were positive. Then set the sign:

In one sentence: like signs give a positive; unlike signs give a negative.

Worked Examples

$(-4) \times 3 = -12 \quad(\text{different signs} \to \text{negative})$ $(-5) \times (-6) = 30 \quad(\text{same signs} \to \text{positive})$ $(-20) \div 4 = -5 \quad(\text{different signs} \to \text{negative})$ $(-18) \div (-3) = 6 \quad(\text{same signs} \to \text{positive})$

💡 Memory hook: "Two negatives make a positive" — a negative times a negative flips back to positive.

Sign Rule Check 🎯

Pick the Sign 🔽

For each product or quotient, choose the sign of the answer (you don't need to compute the number yet).

Multiplying Several Numbers — Count the Negatives

When you multiply a string of numbers, you can just count the negative signs:

• An even number of negatives → the answer is positive.
• An odd number of negatives → the answer is negative.

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

Multiply the values: $2 \times 3 \times 1 = 6$. There are three negatives (odd), so the answer is negative:

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

Worked Example: $(-2) \times (-5)$

$2 \times 5 = 10$, and there are two negatives (even), so the answer is positive: $10$.

⚠️ Zero wins: If any factor is $0$, the whole product is $0$ — sign rules don't matter.

Multiply & Divide 🧮

Compute each. Include a negative sign when needed.

1) $(-9) \times 4 = \,?$ 2) $(-48) \div (-6) = \,?$ 3) $(-2) \times (-3) \times (-2) = \,?$

🌡️ Operations with Integers

Part 5 of 5 — Order of Operations, Word Problems & Mastery

You can now add, subtract, multiply, and divide integers. The last skill is doing them in the right order and using them in real situations.

Order of Operations (PEMDAS) with Integers

The order is the same as always, but now watch the signs:

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

Multiply before you add:

$-3 + 4 \times (-2) = -3 + (-8) = -11$

Worked Example: $(-6 + 2) \times 5$

Do the parentheses first:

$(-6 + 2) \times 5 = (-4) \times 5 = -20$

⚠️ Don't add first! In $-3 + 4 \times (-2)$, you must multiply $4 \times (-2)$ before adding. Doing $-3 + 4$ first gives the wrong answer.

Order of Operations 🧮

Follow PEMDAS. Use a negative sign when needed.

1) $5 + (-3) \times 2 = \,?$ 2) $(-8 + 3) \times (-2) = \,?$ 3) $-10 \div 2 + (-4) = \,?$

Integers in the Real World

Integers model gains and losses, ups and downs. Translate the words into a signed number, then compute.

Worked Example — Temperature

At 6 a.m. the temperature was $-4^\circ$F. By noon it rose $11^\circ$. What is the new temperature?

$-4 + 11 = 7 \quad\Rightarrow\quad 7^\circ\text{F}$

Worked Example — Money

A bank account has $30. You spend $12 three times (three withdrawals). What is the new balance?

$30 + 3 \times (-12) = 30 + (-36) = -6 \quad\Rightarrow\quad \text{a balance of } -\$6 \text{ (overdrawn)}$

💡 Translate first: "rose / gained / above" → positive; "fell / lost / below / spent / owe" → negative.

Word-Problem Check 🎯

Quick Reference

🔑 You're ready for the Exit Quiz. Take your time and check each sign.

Exit Quiz ✅

Answer all three to finish the lesson.