Integer Operations

Up to this point, most of your math work has used positive numbers — counts, measurements, and prices that are always zero or above. But the real world has temperatures below zero, bank accounts overdrawn, elevations below sea level, and electrical charges that flow in opposite directions. Integers are the number set that includes positive numbers, negative numbers, and zero, giving you the language to describe all of these situations with a single system.

What Are Integers?

Integers are the set of whole numbers and their negatives:

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

Key facts about integers:

• Positive integers ($1, 2, 3, \ldots$) represent quantities above zero
• Negative integers ($-1, -2, -3, \ldots$) represent quantities below zero
• Zero ($0$) is an integer that is neither positive nor negative
• Integers do not include fractions or decimals — $-2.5$ and $\frac{3}{4}$ are not integers

On a number line, negative integers appear to the left of zero and positive integers appear to the right. The further left a number is, the smaller its value: $-7 < -3 < 0 < 4 < 10$.

Adding Integers

There are two cases to consider when adding integers.

Case 1: Same Sign

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

$5 + 3 = 8 \qquad \text{(both positive, answer is positive)}$

$(-4) + (-6) = -10 \qquad \text{(both negative, answer is negative)}$

Rule: Same signs — add the absolute values, keep the sign.

Case 2: Different Signs

When the numbers have different signs, subtract the smaller absolute value from the larger absolute value, then take the sign of the number with the larger absolute value.

$7 + (-3) = 4 \qquad \text{(|7| > |-3|, answer is positive)}$

$(-9) + 5 = -4 \qquad \text{(|-9| > |5|, answer is negative)}$

Rule: Different signs — subtract the absolute values, take the sign of the larger.

Example 1: Adding Integers

Compute $(-8) + (-5)$.

Both numbers are negative (same sign).

$|-8| + |-5| = 8 + 5 = 13$

Keep the negative sign:

$(-8) + (-5) = -13$

Example 2: Adding Integers with Different Signs

Compute $(-14) + 9$.

The signs are different. Find the absolute values: $|-14| = 14$, $|9| = 9$.

$14 - 9 = 5$

The number with the larger absolute value is $-14$ (negative), so the answer is negative:

$(-14) + 9 = -5$

Subtracting Integers

Subtraction of integers follows one key rule: subtracting a number is the same as adding its opposite (also called its additive inverse).

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

This rule converts every subtraction problem into an addition problem, and then you apply the addition rules above.

Example 3: Subtracting a Positive Integer

Compute $3 - 10$.

Rewrite as addition:

$3 - 10 = 3 + (-10)$

Different signs: $|{-10}| > |3|$, so the answer is negative.

$10 - 3 = 7 \implies 3 + (-10) = -7$

Example 4: Subtracting a Negative Integer

Compute $(-6) - (-4)$.

Rewrite: subtracting $-4$ means adding $+4$.

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

Different signs: $|-6| = 6 > |4| = 4$, answer is negative.

$6 - 4 = 2 \implies (-6) + 4 = -2$

The “Keep-Change-Change” Shortcut

Many students memorize this pattern for subtraction:

1. Keep the first number as is
2. Change the subtraction sign to addition
3. Change the sign of the second number (flip it)

Then add using the addition rules.

Multiplying Integers

Multiplication follows simple sign rules:

Summary:

• Same signs $\rightarrow$ positive product
• Different signs $\rightarrow$ negative product

Example 5: Multiplying Integers

Compute $(-7) \times (-8)$.

Both factors are negative (same sign), so the product is positive:

$(-7) \times (-8) = 56$

Compute $(-5) \times 6$.

Different signs (negative times positive), so the product is negative:

$(-5) \times 6 = -30$

Multiplying More Than Two Integers

When multiplying several integers, count the negative factors:

• Even number of negative factors $\rightarrow$ positive product
• Odd number of negative factors $\rightarrow$ negative product

Example 6: Three Factors

Compute $(-2) \times 3 \times (-4)$.

Multiply the absolute values: $2 \times 3 \times 4 = 24$.

Count negatives: two negative factors (even), so the product is positive.

$(-2) \times 3 \times (-4) = 24$

Dividing Integers

Division follows the same sign rules as multiplication:

• Same signs $\rightarrow$ positive quotient
• Different signs $\rightarrow$ negative quotient

Example 7: Dividing Integers

Compute $(-36) \div (-9)$.

Same signs (both negative), so the quotient is positive:

$(-36) \div (-9) = 4$

Compute $42 \div (-7)$.

Different signs, so the quotient is negative:

$42 \div (-7) = -6$

Important Note: Division by Zero

Division by zero is undefined — it is not allowed. There is no number that, when multiplied by zero, gives a nonzero result.

$\text{Any number} \div 0 = \text{undefined}$

Real-World Application: HVAC — Temperature Changes

An HVAC technician is monitoring a walk-in freezer. At 6:00 AM, the temperature reads $-12\degree$F. Over the next two hours, the compressor drops the temperature by $7\degree$F. Then a door is left open and the temperature rises $15\degree$F.

Step 1: Starting temperature: $-12\degree$F

Step 2: Drop by $7\degree$F (temperature decreases, so subtract):

$-12 - 7 = -12 + (-7) = -19\degree\text{F}$

Step 3: Rise by $15\degree$F (temperature increases, so add):

$-19 + 15 = -4\degree\text{F}$

The freezer is at $-4\degree$F — still below zero, but warmer than the target of $-19\degree$F. The technician knows the door must be shut and the unit needs time to recover.

Real-World Application: Electrician — Voltage Readings

When an electrician tests circuits, voltage readings can be positive or negative depending on the probe orientation. If a multimeter reads $-24$ volts across one component and $-24$ volts across a second identical component in series, the total voltage drop is:

$(-24) + (-24) = -48 \text{ volts}$

The negative sign simply indicates the direction of the voltage drop relative to the probe placement. Understanding integer addition prevents misreading the total as $0$ (a common mistake when someone thinks the negatives “cancel”).

Common Mistakes to Avoid

1. Confusing subtraction with negative signs. In $5 - (-3)$, the first minus is subtraction and the second is the sign of the number. Rewrite as $5 + 3 = 8$.
2. Forgetting that subtracting a negative means adding. $(-4) - (-9) = (-4) + 9 = 5$, not $-13$.
3. Getting sign rules backward for multiplication. Two negatives make a positive: $(-3) \times (-5) = 15$, not $-15$.
4. Ignoring zero’s special role. $0$ times any integer is $0$, and $0$ divided by any nonzero integer is $0$. But dividing by $0$ is undefined.
5. Applying multiplication sign rules to addition. Adding two negatives gives a negative (not positive). The “same sign gives positive” rule only applies to multiplication and division.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Integers include all positive whole numbers, their negatives, and zero
• Adding same signs: add absolute values, keep the sign
• Adding different signs: subtract absolute values, take the sign of the larger
• Subtracting is the same as adding the opposite — $a - b = a + (-b)$
• Multiplying/dividing: same signs give a positive result, different signs give a negative result
• When multiplying multiple integers, count the negative factors — even count means positive, odd count means negative
• Division by zero is undefined
• Negative numbers model real-world quantities like temperature drops, debt, below-sea-level elevation, and voltage drops

Return to Pre-Algebra for more topics in this section.