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Mom's Basement University

Integers and Signed Numbers

What You’ll Learn

Section titled “What You’ll Learn”

In this lesson you’ll learn how positive and negative numbers (signed integers) work on a number line and the rules for operating with them.

The Concept

Section titled “The Concept”

Integers are whole numbers including negatives, zero, and positives: …, −3,−2,−1,0,1,2,3-3, -2, -1, 0, 1, 2, 3…

Signed numbers show direction or change (e.g., −5 dollars = debt, +5 dollars = gain).

Rules:

Addition/Subtraction (use number line or rules):

  • Same signs: Add absolute values, keep sign.
5+3=8(−5)+(−3)=−8\begin{aligned} 5 + 3 &= 8 \\ ({-}5) + ({-}3) &= {-}8 \end{aligned}
  • Different signs: Subtract absolute values, keep sign of larger number.
7+(−4)=3(−7)+4=−3\begin{aligned} 7 + ({-}4) &= 3 \\ ({-}7) + 4 &= {-}3 \end{aligned}

Multiplication/Division:

  • Same signs → positive result
6×4=24(−6)×(−4)=24\begin{aligned} 6 \times 4 &= 24 \\ ({-}6) \times ({-}4) &= 24 \end{aligned}
  • Different signs → negative result
6×(−4)=−24(−6)÷3=−2\begin{aligned} 6 \times ({-}4) &= {-}24 \\ ({-}6) \div 3 &= {-}2 \end{aligned}

Zero rules: Any number × 0 = 0, division by zero is undefined.

Worked Examples

Section titled “Worked Examples”

1. −8+12-8 + 12

−8+12=4(different signs: subtract 8 from 12, positive)\begin{aligned} {-}8 + 12 &= 4 \quad \text{(different signs: subtract 8 from 12, positive)} \end{aligned}

2. −15−(−7)-15 - (-7)

−15−(−7)=−15+7=−8(subtracting negative = adding)\begin{aligned} {-}15 - ({-}7) &= {-}15 + 7 \\ &= {-}8 \quad \text{(subtracting negative = adding)} \end{aligned}

3. (−9)×(−4)(-9) \times (-4)

(−9)×(−4)=36(same signs → positive)({-}9) \times ({-}4) = 36 \quad \text{(same signs → positive)}

4. 24÷(−6)24 \div (-6)

24÷(−6)=−4(different signs → negative)24 \div ({-}6) = {-}4 \quad \text{(different signs → negative)}

Check: Use the number line or opposite operation (e.g., −8 + 12 = 4 → 4 − 12 = −8, correct).

Real-World Applications

Section titled “Real-World Applications”

Signed numbers appear in banking (overdraft = negative balance), temperature (−10°C), elevation (below sea level), debt/credit, stock changes, or sports scores.

Example: Account starts at 200 dollars, you spend 350 dollars:

200+(−350)=−150 dollars (overdrawn by 150 dollars)200 + ({-}350) = {-}150 \text{ dollars (overdrawn by 150 dollars)}

Or temperature drops from 5°C to −3°C:

−3−5=−8°C change{-}3 - 5 = {-}8\text{°C change}

Quiz

Section titled “Quiz”
What is $-12 + 18$?
Calculate $-7 - (-3)$.
What is $(-8) \times (-5)$?
A temperature drops from 8°C to −5°C. What is the change?
What is $(-6) \div (-2)$?