📐 Concept diagram

00-05 — Integers and Directed Numbers

Phase: 0 — Arithmetic & Number Foundations Subject: 00-05 Prerequisites: 00-01 — Whole Number Arithmetic, 00-02 — Fractions, 00-03 — Decimals, 00-04 — Percentages Next subject: 00-06 — Powers and Roots

Learning Objectives

By the end of this subject, you will be able to:

1. Represent integers on a number line and understand negative numbers as directed quantities
2. Compute absolute values of integers and interpret them geometrically as distance from zero
3. Add, subtract, multiply, and divide any two integers — including negative numbers — confidently
4. Apply the order of operations correctly to expressions containing negative numbers
5. Solve real-world problems involving temperature changes, bank balances, elevation, and other directed quantities

Core Content

1. What Are Integers?

The integers extend the whole numbers by including negative values. Symbolically:

ℤ = {..., −3, −2, −1, 0, 1, 2, 3, ...}

Where: - Negative integers are numbers less than zero: −1, −2, −3, ... - Zero is neither positive nor negative — it is the boundary - Positive integers are the whole numbers greater than zero: 1, 2, 3, ...

The set of integers is denoted by ℤ (from the German "Zahlen" meaning "numbers").

⚠️ THIS IS CRITICAL — Negative numbers are used constantly in algebra, coordinate geometry, physics, finance, and calculus. You cannot do algebra without mastering integer operations.

2. The Number Line

The number line is a straight line where every point corresponds to a number:

$    ← more negative                                    more positive →
...  -5   -4   -3   -2   -1    0    1    2    3    4    5  ...
    |    |    |    |    |    |    |    |    |    |    |
$

Key observations: - Zero sits at the center - Numbers increase as you move right, decrease as you move left - The number −3 is 3 units left of zero - The number 3 is 3 units right of zero - −3 and 3 are opposites — they are the same distance from zero but on opposite sides

Thinking of integers as positions on a thermometer:

A temperature of −5°C is 5 degrees below freezing (0°C). A temperature of +5°C is 5 degrees above freezing. The number line is just a thermometer turned sideways.

3. Comparing Integers

On the number line, larger means farther to the right:

$... -5 < -4 < -3 < -2 < -1 < 0 < 1 < 2 < 3 < 4 < 5 ...
$

Therefore: - −3 < 2 (negative three is less than two — it's to the left) - −10 < −5 (negative ten is more negative, so it's smaller) - 0 > −1 (zero is greater than negative one)

Common pitfall: Many beginners think −10 > −5 because "10 is bigger than 5." But on the number line, −10 is farther left (more negative) than −5, so −10 < −5. The more negative a number, the SMALLER it is.

Memory aid: Imagine owing money. Owing £10 (−10) is worse (less money) than owing £5 (−5).

4. Absolute Value

The absolute value of a number is its distance from zero on the number line, ignoring direction.

|x| = the distance of x from 0

Definition:

|x| = x if x ≥ 0 |x| = −x if x < 0

For example: - |5| = 5 (distance from 0 is 5) - |−5| = 5 (distance from 0 is 5) - |0| = 0 (distance from 0 is 0)

Why does |−x| = |x|? Because distance is never negative. The absolute value strips away the sign.

Geometric interpretation: |a − b| is the distance between a and b on the number line.

|−3 − 2| = |−5| = 5 — the distance between −3 and 2 is 5 units ✓ |2 − (−3)| = |5| = 5 — same distance ✓

Common pitfall: |−x| is NOT the same as −|x|. - |−3| = 3 - −|3| = −3 These are different. The absolute value makes things non-negative; putting a minus sign outside makes it negative.

5. Adding Integers

Adding integers is combining directed movement on the number line.

Rule for adding two numbers with the same sign:

Add their absolute values. Keep the common sign.

5 + 3 = 8 (both positive → positive sum) (−5) + (−3) = −8 (both negative → negative sum)

Rule for adding two numbers with opposite signs:

Subtract the smaller absolute value from the larger. Keep the sign of the number with the larger absolute value.

5 + (−3) = 2 (|5| > |3|, so positive) (−5) + 3 = −2 (|−5| > |3|, so negative)

Number line visualization for 5 + (−3):

Start at 5. "Add −3" means move 3 units LEFT:

$    ...  2    3    4    5    6 ...
         ↑    |    |    |
         |←---3 units---→|
$

Start at 5 → move left 3 → land at 2.

Number line visualization for (−5) + 3:

Start at −5. "Add 3" means move 3 units RIGHT:

$    ... -5   -4   -3   -2   -1    0 ...
         |----→|----→|----→|
                     ↑
$

Start at −5 → move right 3 → land at −2.

Why these rules work: Addition is commutative and associative for integers, just as for whole numbers. The rules above follow directly from the definition of integers on the number line.

6. Subtracting Integers

⚠️ THIS IS CRITICAL — The subtraction rule for integers is the most important single rule in this subject.

a − b = a + (−b)

Subtracting any number is the SAME as adding its opposite.

This single rule converts every subtraction problem into an addition problem.

Examples: - 7 − 3 = 7 + (−3) = 4 - 7 − (−3) = 7 + 3 = 10 ← subtracting a negative = adding a positive - (−7) − 3 = (−7) + (−3) = −10 - (−7) − (−3) = (−7) + 3 = −4

The "double negative" rule:

Subtracting a negative number is the same as adding a positive number. a − (−b) = a + b

Why? Because −(−b) = b. The opposite of the opposite of b is b itself. Think of "−" as "opposite of": the opposite of the opposite of east is east.

Number line intuition for 7 − (−3):

Start at 7. "Subtract −3" → add the opposite → add 3 → move right 3 → land at 10.

Common pitfall — the "two negatives make a positive" confusion:

This applies ONLY when you are multiplying/dividing or when subtracting a negative: - (−2) × (−3) = 6 ✓ (multiplication of two negatives → positive) - 5 − (−3) = 5 + 3 = 8 ✓ (subtracting a negative → add) - But: (−2) + (−3) = −5 (adding two negatives → negative, NOT positive!)

Don't over-apply the "two negatives make a positive" rule. Context matters.

7. Multiplying Integers

Rule for signs in multiplication:

Same signs → positive product. Different signs → negative product.

Examples: - 4 × 3 = 12 (both positive) - 4 × (−3) = −12 (positive × negative) - (−4) × 3 = −12 (negative × positive) - (−4) × (−3) = 12 (both negative → positive)

Why does negative × negative = positive?

Proof by pattern:

$3 × 3 = 9
2 × 3 = 6      (decrease by 3 each time)
1 × 3 = 3
0 × 3 = 0
(−1) × 3 = −3   (pattern continues: decrease by 3)
(−2) × 3 = −6
(−3) × 3 = −9
$

Now consider (−3) × 3 = −9. Now follow a different pattern:

$(−3) × 3 = −9
(−3) × 2 = −6    (increase by 3 each time!)
(−3) × 1 = −3
(−3) × 0 = 0
(−3) × (−1) = 3  (pattern continues: increase by 3)
(−3) × (−2) = 6
(−3) × (−3) = 9
$

This shows (−3) × (−3) = 9, confirming negative × negative = positive.

Another way: Multiplication by −1 is a 180° rotation on the number line. Doing it twice (multiplying by −1 twice) = 360° rotation = back to original = positive.

8. Dividing Integers

The division sign rules are identical to multiplication:

Same signs → positive quotient. Different signs → negative quotient.

• 12 ÷ 3 = 4
• 12 ÷ (−3) = −4
• (−12) ÷ 3 = −4
• (−12) ÷ (−3) = 4

Why? Division is the inverse of multiplication. Since (−4) × 3 = −12, it must be that (−12) ÷ 3 = −4.

Common pitfall — dividing by zero:

Just as with whole numbers, division by zero is undefined for integers too: - 5 ÷ 0 = undefined - (−5) ÷ 0 = undefined - 0 ÷ (−5) = 0 (zero divided by anything non-zero is zero)

9. Order of Operations with Negative Numbers

The order of operations (PEMDAS/BODMAS) still applies. The trick is handling negative signs correctly.

⚠️ THIS IS CRITICAL: When you see −3², this means −(3²) = −9, NOT (−3)² = 9.

The exponent applies ONLY to the number it's directly attached to. The minus sign is treated as multiplication by −1, and exponents come before multiplication in PEMDAS.

Examples:

1. Evaluate 10 − 3 × (−2): $10 − 3 × (−2) = 10 − (−6) ← Multiplication: 3 × (−2) = −6 = 10 + 6 ← Subtracting a negative = adding positive = 16$

2. Evaluate (−2)³ + 4 × (−3): $(−2)³ + 4 × (−3) = (−8) + (−12) ← Exponent: (−2)³ = −8; Multiplication: 4 × (−3) = −12 = −20$

3. Evaluate 6 − 2 × (3 − 8)²: $6 − 2 × (3 − 8)² = 6 − 2 × (−5)² ← Parentheses: 3 − 8 = −5 = 6 − 2 × 25 ← Exponent: (−5)² = 25 = 6 − 50 ← Multiplication = −44 ← Subtraction$

4. Evaluate (−8 + 3) × (−1) − (−6) ÷ 2: $(−8 + 3) × (−1) − (−6) ÷ 2 = (−5) × (−1) − (−3) ← Parentheses: −8+3 = −5; Division: (−6)÷2 = −3 = 5 − (−3) ← Multiplication: (−5)×(−1) = 5 = 5 + 3 ← Subtract negative = add positive = 8$

Common pitfalls with negative numbers and PEMDAS: - −3² ≠ (−3)² — the first equals −9, the second equals 9 - 4 − (−3) = 4 + 3 = 7, but 4 − −3 is messy notation — use parentheses - (−2 + 5)² = 3² = 9, NOT (−2)² + 5² = 4 + 25 = 29 — the addition happens inside parentheses FIRST

10. Applications of Integers

Integers model any situation with two opposite directions:

Example: A submarine at −200 m (200 m below sea level) rises 45 m, then descends 67 m. What is its final position?

$(−200) + 45 + (−67)
= (−200) + 45 − 67
= −155 − 67
= −222
$

The submarine is at −222 m (222 m below sea level).

Key Terms

• Absolute value
• Integers
• Negative integers
• Order of operations
• Positive integers
• Zero

Worked Examples

Example 1: Integer Addition and Subtraction

Calculate: (−15) + 8 − (−6) − 4 + (−9)

Solution:

$(−15) + 8 − (−6) − 4 + (−9)
= −15 + 8 − (−6) − 4 − 9          ← Remove parentheses around −9
$

Now handle the subtraction of negative:

$= −15 + 8 + 6 − 4 − 9              ← Subtract (−6) → add 6
$

Group and compute step by step:

$= (−15 + 8) + 6 − 4 − 9           ← Add left to right where possible
= −7 + 6 − 4 − 9
= (−7 + 6) − 4 − 9
= −1 − 4 − 9
= −5 − 9
= −14
$

Answer: −14

Alternative method — combine all positives and negatives:

Separate into positive terms and negative terms: Positives: +8, +6 → +14 Negatives: −15, −4, −9 → −28

Result: 14 + (−28) = −14 ✓

Example 2: Multiplication and Division with Signs

Calculate: (−12) × (−3) ÷ (−6) × (−2)

Solution:

Work left to right (multiplication and division have equal precedence):

$(−12) × (−3) ÷ (−6) × (−2)
$

Step 1: (−12) × (−3) = +36

Step 2: 36 ÷ (−6) = −6

Step 3: (−6) × (−2) = +12

Answer: 12

Sign shortcut: Count the number of negative signs! 4 negative signs in the expression. Even number → positive result ✓

Example 3: Order of Operations with Negatives

Calculate: −2³ − 3 × (−4 + 1)² + (−15) ÷ (−5)

Solution:

$−2³ − 3 × (−4 + 1)² + (−15) ÷ (−5)
$

Step 1 — Parentheses: (−4 + 1) = −3

$= −2³ − 3 × (−3)² + (−15) ÷ (−5)
$

Step 2 — Exponents. ⚠️ −2³ = −(2³) = −8, and (−3)² = 9

$= −8 − 3 × 9 + (−15) ÷ (−5)
$

Step 3 — Multiplication and Division (L→R): 3 × 9 = 27, (−15) ÷ (−5) = 3

$= −8 − 27 + 3
$

Step 4 — Addition/Subtraction (L→R):

$= −35 + 3
= −32
$

Answer: −32

Common mistake check: If you read −2³ as (−2)³ = −8, you'd get the same answer here (by coincidence). But −2³ = −8 and (−2)³ = −8 are both −8! This is because the exponent is odd. For even exponents: −2² = −4 but (−2)² = 4 — very different.

Practice Problems

(Answers are below. Try each problem before checking.)

Problem 1: Evaluate: (−18) + 25 − (−7) − 11 + (−3)

Problem 2: Evaluate: (−6) × (−5) × (−2)

Problem 3: Evaluate: (−48) ÷ 8 − (−3) × (−2)

Problem 4: Evaluate: −3² + (−4)² − 5 × (−2)

Problem 5: Evaluate: (7 − 12) × (−3 + 1)² − (−24) ÷ (−6)

Problem 6: A mountain climber starts at sea level. She ascends 450 m, descends 120 m into a valley, then ascends 280 m. What is her final elevation?

Problem 7: The temperature at midnight was −8°C. By 6 AM it had fallen 5°C. By noon it had risen 11°C. What was the temperature at noon?

Summary

1. Integers include zero, positive whole numbers, and negative whole numbers, extending the whole numbers to directed quantities
2. Absolute value |x| is the distance from zero — always non-negative — and |a − b| measures the distance between a and b
3. Adding integers: same signs → add absolute values, keep sign; opposite signs → subtract absolute values, keep sign of larger absolute value
4. Subtracting integers: a − b = a + (−b) — ALWAYS convert subtraction to addition of the opposite number; subtracting a negative becomes adding a positive
5. Multiplying/dividing integers: same signs → positive; different signs → negative; negative × negative = positive
6. Order of operations with negatives: ⚠️ −3² = −(3²) = −9, NOT (−3)² = 9 — the exponent attaches only to what it's directly next to

Pitfalls

• Misreading −3² as (−3)². The expression −3² means −(3²) = −9 because exponents bind tighter than the minus sign (which is multiplication by −1). With parentheses: (−3)² = 9. The distinction matters enormously for even exponents.
• Thinking −10 > −5 because \"10 is bigger than 5.\" On the number line, −10 is farther left (more negative) than −5, so −10 < −5. More negative means smaller. Think of owing £10 vs owing £5.
• Over-applying \"two negatives make a positive\" to addition. Two negatives multiply to a positive, and subtracting a negative IS adding a positive. But adding two negatives stays negative: (−2) + (−3) = −5, NOT +5. Context matters.
• Forgetting to treat subtraction as adding the opposite. a − b = a + (−b) is the universal rule. Skipping this step leads to sign errors in chains like (−15) + 8 − (−6) − 4 + (−9).
• Losing track of signs across multiple operations. When an expression has several negatives, count signs systematically for multiplication/division (even number → positive, odd → negative) and convert all subtractions to additions for addition/subtraction.

Quiz

Answer each question, then read the explanation for your choice.

Q1: Evaluate: (−7) + (−3) − (−5)

A) −15 B) −5 C) 5 D) −1

Q2: What is |−7| − |3| + |−2|?

A) −8 B) 6 C) 12 D) 8

Q3: (−4) × (−6) ÷ (−3) = ?

A) −8 B) 8 C) −12 D) 2

Q4: Evaluate: −6 + 4 × (−2)² − (−10) ÷ 2

A) −10 B) 5 C) 15 D) −5

Q5: What is the distance between −12 and 7 on the number line?

A) −19 B) 19 C) 5 D) −5

Q6: Which of the following is the LARGEST number?

A) −15 B) −3 C) −8 D) −1

Q7: What is (−3)³ − 2 × (−5)?

A) −37 B) −17 C) 17 D) −23

Q8: A bank account has a balance of −£45 (overdraft). The account holder deposits £120, then writes a cheque for £90. What is the final balance?

A) £75 B) −£15 C) £15 D) −£75

Next Steps

Move on to 00-06 — Powers and Roots to learn about squares, square roots, cubes, cube roots, exponent laws, and working with negative and fractional exponents.

Q5: A submarine is at −200 m. It rises 75 m, then descends 130 m. What is its new depth?

A) −255 m B) −145 m C) −55 m D) −405 m

Answer: A) −255 m

−200 + 75 + (−130) = −200 + 75 − 130 = −125 − 130 = −255 m.