Learning Objectives
Compare positive and negative integers using a number line.
Add, subtract, multiply, and divide signed numbers.
Evaluate expressions using grouping, powers, and operation order.
Prerequisites: whole-number arithmetic and the meaning of multiplication and division. Integers include negative whole numbers, zero, and positive whole numbers. Examples are −4, 0, and 7.
1. Position and Distance Are Different
On a number line, numbers increase as you move right. Therefore −2 is greater than −5: it is farther right, even though the digit 5 is larger than 2. Zero separates the negative and positive numbers.
The absolute value of a number is its distance from zero. Distance is nonnegative, so |−5| = 5 and |5| = 5. Opposite numbers have the same absolute value but lie on different sides of zero, except that zero is its own opposite.
Worked Example: Temperature
The temperature is −3°C and rises by 8°C. Calculate −3 + 8 = 5, so the final temperature is 5°C.
On a number line, start at −3 and move 8 steps right. Three steps reach zero and the remaining five reach 5.
The starting temperature's distance from zero was 3 degrees. That distance does not mean the starting temperature was positive.
2. Subtract by Adding the Opposite
Subtracting b is the same as adding its opposite: a − b = a + (−b). This rule also works when b is negative. For example, 4 − (−3) = 4 + 3 = 7. Parentheses help separate a subtraction operation from the negative sign of a number.
Worked Example: Two Negative Numbers
Evaluate −6 − (−2). Replace subtraction of −2 with addition of 2: −6 + 2 = −4.
Check by reversing the subtraction: −4 + (−2) = −6. The result −4 lies to the right of −6, as expected when adding a positive number.
3. Multiply and Divide with Sign Rules
Signs of two nonzero numbers | Product or quotient sign | Example |
|---|---|---|
Same signs | Positive | (−3)(−4) = 12 |
Different signs | Negative | 12 ÷ (−3) = −4 |
Multiplication by zero gives zero. Division by zero is undefined: there is no number that, multiplied by zero, produces a nonzero dividend. These sign rules describe multiplication and division, not addition. For instance, adding a positive number to a negative number can have either sign.
4. Follow the Structure of an Expression
Evaluate grouping symbols first, then powers, then multiplication and division from left to right, then addition and subtraction from left to right. Multiplication does not automatically come before division; they share a level. Addition and subtraction also share a level.
Worked Example: Several Operations
Evaluate 18 ÷ 3 × 2 − (5 − 8)².
First, 5 − 8 = −3 and (−3)² = 9. The expression becomes 18 ÷ 3 × 2 − 9.
Work left to right through division and multiplication: 18 ÷ 3 = 6, then 6 × 2 = 12. Finally, 12 − 9 = 3.
Common Mistakes
Do not apply multiplication sign rules to addition. For example, −7 + 2 = −5.
Keep parentheses when squaring a negative base: (−3)² = 9, but −3² = −9.
Do not multiply before an earlier division at the same operation level.
5. Practice
1. Put −4, 2, −7, and 0 in increasing order.
2. Calculate −9 + 14 and 3 − (−8).
3. Evaluate 24 ÷ (−6) × 2 + 5.
4. Evaluate (−2)³ + 3(4 − 1).
Worked Solutions
1. The order is −7, −4, 0, 2, from left to right on the number line.
2. −9 + 14 = 5. Also, 3 − (−8) = 3 + 8 = 11.
3. 24 ÷ (−6) = −4, then −4 × 2 = −8, and −8 + 5 = −3.
4. (−2)³ = −8 and 4 − 1 = 3. Thus −8 + 3(3) = −8 + 9 = 1.
6. Summary
- A number's position and its distance from zero are different ideas.
- Subtracting a number means adding its opposite.
- Use sign rules for multiplication and division, and respect grouping and left-to-right order.