Negative Numbers and Integers

Understanding Negative Numbers

Negative numbers are numbers less than zero, written with a minus sign (-) in front. Together with positive numbers and zero, they form the set of integers.

The Integer Family

Integers include:

• Positive integers: 1, 2, 3, 4, 5… (numbers greater than zero)
• Zero: 0 (neither positive nor negative)
• Negative integers: -1, -2, -3, -4, -5… (numbers less than zero)

We can write this as: {..., -3, -2, -1, 0, 1, 2, 3, ...}

The Number Line: Your Best Friend

Imagine a horizontal line with zero in the middle. Positive numbers extend to the right, and negative numbers extend to the left:

← -5  -4  -3  -2  -1  0  1  2  3  4  5 →
     negative      zero    positive

Key insight: The further left you go on the number line, the smaller the value. So -5 is smaller than -2, even though 5 is larger than 2 in the positive world.

Real-World Examples of Negative Numbers

Negative numbers appear everywhere in daily life:

1. Temperature: -10°C means 10 degrees below freezing
2. Banking: -$50 means you owe$50 (a debt or overdraft)
3. Elevation: -50m means 50 metres below sea level
4. Floors: B2 means 2 floors below ground level
5. Time: Year -500 represents 500 BC (before year 0)
6. Golf: A score of -3 means 3 under par
7. Profit/Loss: -$200 means a loss of$200

Think of negative numbers as representing opposites: debt vs. savings, below vs. above, loss vs. gain, backward vs. forward.

Comparing and Ordering Integers

The Rules for Comparing

1. Any positive number is greater than any negative number

   • 1 > -100 (even though 100 is a larger distance from zero)

2. Zero is greater than any negative number

   • 0 > -1, 0 > -50, 0 > -1000

3. For negative numbers, the one closer to zero is greater

   • -2 > -5 (because -2 is further right on the number line)
   • -1 > -99

Visual Strategy

Think of a thermometer or number line. The higher up (or further right) a number is, the greater its value.

Example 1: Comparing Two Integers

Compare: -7 and -3

Solution: -7 < -3

Explanation: On a number line, -7 is further left than -3. Think of temperature: -7°C is colder (smaller) than -3°C.

Example 2: Ordering Multiple Integers

Order from smallest to largest: 5, -8, 0, -2, 3, -10

Solution: -10, -8, -2, 0, 3, 5

Explanation: Start with the negative numbers from smallest (furthest left) to largest, then zero, then positive numbers.

Absolute Value: Distance from Zero

The absolute value of a number is its distance from zero on the number line, regardless of direction. We write it with vertical bars: |n|

• |5| = 5 (five steps from zero)
• |-5| = 5 (also five steps from zero)
• |0| = 0 (zero steps from zero)

Absolute value is always positive or zero, never negative.

Example 3: Understanding Absolute Value

Find: |-12|

Solution: 12

Explanation: -12 is 12 units away from zero on the number line, so |-12| = 12.

Adding Integers

Same Signs: Add and Keep the Sign

When adding integers with the same sign, add their absolute values and keep the sign.

Positive + Positive = Positive

• 8 + 5 = 13

Negative + Negative = More Negative

• (-8) + (-5) = -13
• Think: If you owe $8 and borrow another$5, you owe $13

Example 4: Adding Negative Numbers

Calculate: -15 + (-9)

Solution: -24

Explanation: Both numbers are negative. Add their absolute values (15 + 9 = 24) and keep the negative sign.

Different Signs: Subtract and Use the Larger Sign

When adding integers with different signs, subtract the smaller absolute value from the larger absolute value, and use the sign of the number with the larger absolute value.

Think of it as:

• Positive + Negative = Find the difference
• Which is bigger? That determines the sign

Example 5: Adding Different Signs (Positive Result)

Calculate: 12 + (-5)

Solution: 7

Explanation: Different signs, so subtract: 12 - 5 = 7. Since 12 has the larger absolute value and is positive, the answer is positive.

Visual: Start at 12 on the number line, move 5 steps left, land on 7.

Example 6: Adding Different Signs (Negative Result)

Calculate: 8 + (-15)

Solution: -7

Explanation: Different signs, so subtract: 15 - 8 = 7. Since -15 has the larger absolute value and is negative, the answer is -7.

Visual: Start at 8, move 15 steps left, land on -7.

Example 7: Adding to Get Zero

Calculate: -20 + 20

Solution: 0

Explanation: They have equal absolute values but opposite signs. They cancel each other out, giving zero. These are called opposites or additive inverses.

Subtracting Integers

The Golden Rule of Subtraction

Subtracting a number is the same as adding its opposite.

Change subtraction to addition:

• a - b = a + (-b)
• a - (-b) = a + b

Two Key Patterns:

1. Subtracting a positive = Adding a negative

   • 5 - 3 = 5 + (-3) = 2

2. Subtracting a negative = Adding a positive

   • 5 - (-3) = 5 + 3 = 8
   • “Two negatives make a positive”

Why Does This Work?

Think about real-life situations:

• Removing a debt is like gaining money: If you owe $20 and the debt is cancelled, it's like receiving$20
• 7 - (-3): You have $7, and someone removes your$3 debt = $7 +$3 = $10

Example 8: Subtracting a Positive

Calculate: 10 - 15

Solution: -5

Explanation:

10 - 15
= 10 + (-15)    (Change to addition)
= -5            (Different signs: subtract and use the larger sign)

Real-world: You have $10 and spend$15. You’re $5 in debt.

Example 9: Subtracting a Negative

Calculate: 6 - (-4)

Solution: 10

Explanation:

6 - (-4)
= 6 + 4         (Subtracting a negative = adding a positive)
= 10

Real-world: You have $6, and someone cancels your$4 debt. You now effectively have $10 in value.

Example 10: Negative Minus Positive

Calculate: -8 - 5

Solution: -13

Explanation:

-8 - 5
= -8 + (-5)     (Change to addition)
= -13           (Same signs: add and keep negative)

Real-world: You owe $8 and borrow another$5. You owe $13 total.

Example 11: Negative Minus Negative

Calculate: -3 - (-7)

Solution: 4

Explanation:

-3 - (-7)
= -3 + 7        (Subtracting a negative = adding a positive)
= 4             (Different signs: 7 - 3 = 4, positive because 7 is larger)

Multiplying Integers

Sign Rules for Multiplication

Same Signs → Positive Product

• Positive × Positive = Positive: 6 × 4 = 24
• Negative × Negative = Positive: (-6) × (-4) = 24

Different Signs → Negative Product

• Positive × Negative = Negative: 6 × (-4) = -24
• Negative × Positive = Negative: (-6) × 4 = -24

Memory trick: “Same signs give a plus, different signs give a minus.”

Example 12: Multiplying Same Signs

Calculate: (-7) × (-3)

Solution: 21

Explanation: Both negative (same signs), so the product is positive. Multiply the absolute values: 7 × 3 = 21.

Example 13: Multiplying Different Signs

Calculate: 8 × (-5)

Solution: -40

Explanation: Different signs (positive × negative), so the product is negative. Multiply absolute values: 8 × 5 = 40, then apply the negative sign.

Dividing Integers

Division follows the same sign rules as multiplication.

Same Signs → Positive Quotient

• Positive ÷ Positive = Positive: 24 ÷ 6 = 4
• Negative ÷ Negative = Positive: (-24) ÷ (-6) = 4

Different Signs → Negative Quotient

• Positive ÷ Negative = Negative: 24 ÷ (-6) = -4
• Negative ÷ Positive = Negative: (-24) ÷ 6 = -4

Example 14: Dividing Same Signs

Calculate: (-36) ÷ (-4)

Solution: 9

Explanation: Both negative (same signs), so the quotient is positive. 36 ÷ 4 = 9.

Example 15: Dividing Different Signs

Calculate: (-48) ÷ 6

Solution: -8

Explanation: Different signs, so the quotient is negative. 48 ÷ 6 = 8, with a negative sign.

Order of Operations with Integers

When working with integers in complex expressions, remember PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (left to right)
• Addition and Subtraction (left to right)

Example 16: Complex Expression

Calculate: -3 + 5 × (-2)

Solution: -13

Explanation:

-3 + 5 × (-2)
= -3 + (-10)       (Multiply first: 5 × (-2) = -10)
= -13              (Then add: -3 + (-10) = -13)

Common Errors and How to Avoid Them

| Error | Incorrect Thinking | Correction | Explanation | | ----------------------------- | ------------------------------------ | ---------------- | ----------------------------------------- | --- | --- | --- | --------------------------------- | | Thinking -10 > -5 | -10 is bigger than -5 because 10 > 5 | -10 < -5 | Further left on number line means smaller | | Adding instead of subtracting | 5 + (-3) = 8 | 5 + (-3) = 2 | Adding negative means subtract | | Subtracting incorrectly | 5 - (-3) = 2 | 5 - (-3) = 8 | Subtracting negative means add | | Wrong multiplication sign | (-3) × (-4) = -12 | (-3) × (-4) = 12 | Same signs give positive product | | Ignoring order of operations | -5 + 3 × 2 = -4 | -5 + 6 = 1 | Multiply first, then add | | Confusing absolute value | | -8 | = -8 | | -8 | = 8 | Absolute value is always positive |

Practice Problems

Comparing and Ordering

1. Which is greater: -15 or -8? Answer: -8

2. Order from smallest to largest: 3, -7, 0, -2, 5 Answer: -7, -2, 0, 3, 5

3. What is |-23|? Answer: 23

Addition

4. Calculate: -18 + (-7) Answer: -25

5. Calculate: 14 + (-9) Answer: 5

6. Calculate: -25 + 40 Answer: 15

7. Calculate: -30 + 30 Answer: 0

Subtraction

8. Calculate: 12 - 20 Answer: -8

9. Calculate: 8 - (-5) Answer: 13

10. Calculate: -6 - 9 Answer: -15

11. Calculate: -4 - (-10) Answer: 6

Multiplication

12. Calculate: (-8) × 6 Answer: -48

13. Calculate: (-7) × (-9) Answer: 63

14. Calculate: 12 × (-3) Answer: -36

Division

15. Calculate: (-56) ÷ 8 Answer: -7

16. Calculate: (-45) ÷ (-5) Answer: 9

17. Calculate: 72 ÷ (-8) Answer: -9

Mixed Operations

18. Calculate: -7 + 3 × (-2) Answer: -13

19. Calculate: (-4) × 5 - 8 Answer: -28

20. Calculate: 15 - 3 × (-4) Answer: 27

Real-World Applications

Application 1: Temperature Changes

Problem: At 6 AM, the temperature is -8°C. By noon, it rises by 15°C. What’s the temperature at noon?

Solution:

-8 + 15 = 7

The temperature at noon is 7°C.

Application 2: Bank Account

Problem: Your bank account has -$45 (overdrawn). You deposit$120. What’s your new balance?

Solution:

-45 + 120 = 75

Your new balance is $75.

Application 3: Elevation Change

Problem: A submarine is at -250 metres (250m below sea level). It rises 180 metres. What’s its new depth?

Solution:

-250 + 180 = -70

The submarine is now at -70 metres (70m below sea level).

Application 4: Profit and Loss

Problem: A business had a loss of $3,500 (can be written as -$3,500) in January, a profit of $4,200 in February, and a loss of$800 in March. What was the overall result for the three months?

Solution:

-3500 + 4200 + (-800)
= 700 + (-800)
= -100

The business had an overall loss of $100 for the three months.

Application 5: Golf Scoring

Problem: In golf, par is the standard score. Scores are often shown relative to par. If a golfer scores -3 on the first nine holes and -5 on the second nine holes, what’s their total score relative to par?

Solution:

-3 + (-5) = -8

The golfer is 8 under par (excellent!).

Why This Matters

Understanding integers and negative numbers is essential because they:

• Represent real situations like temperature, debt, altitude, and time
• Are foundational for algebra, where variables can be any integer
• Appear in science (charges in physics, vectors in motion)
• Help in finance (debts, losses, negative balances)
• Enable problem-solving in countless real-world scenarios

From checking your bank balance to understanding weather forecasts, from playing video games (health points going negative) to analyzing business profits and losses, integers are everywhere in the real world.

Connection to Other Topics

Integers connect to many mathematical concepts:

• Coordinate Plane: Negative numbers allow us to plot points in all four quadrants
• Algebra: Variables can represent any integer, including negatives
• Rational Numbers: Integers are a subset of rational numbers (fractions)
• Number Lines: Visual representation of all integers
• Absolute Value: Understanding distance regardless of direction
• Inequalities: Comparing integers is fundamental to solving inequalities

Mental Math Strategies

Quick Addition Tips

1. Look for opposites: -7 + 12 + 7 = 12 (the -7 and 7 cancel)
2. Group same signs: -5 + (-8) + 3 + (-2) = (-5 - 8 - 2) + 3 = -15 + 3 = -12
3. Use benchmarks: -28 + 30 = -28 + 28 + 2 = 0 + 2 = 2

Quick Subtraction Tips

1. Change to addition: 10 - (-5) = 10 + 5 = 15
2. Think of it as distance: “How far from -3 to 5?” → 8 units
3. Use number line visualization: Picture yourself moving left or right

Quick Multiplication Tips

1. Signs first, then multiply: (-6) × (-7) → same signs = positive → 6 × 7 = 42
2. Count negatives: Even number of negatives = positive, odd number = negative
3. Zero wipes out everything: Any number × 0 = 0

Historical Note

Negative numbers weren’t always accepted in mathematics! Ancient Greek mathematicians rejected them as “absurd.” Chinese and Indian mathematicians were among the first to use negatives (around 200 BC), representing debts with red rods and assets with black rods. European mathematicians didn’t fully accept negative numbers until the 1600s. Today, we can’t imagine mathematics without them.

Challenge Yourself

Try these mental math challenges:

1. Pattern Recognition: What’s the next number? 8, 4, 0, -4, -8, ___

   • Answer: -12 (subtracting 4 each time)

2. Missing Number: -7 + ___ = -15

   • Answer: -8

3. Temperature Puzzle: The temperature dropped from 5°C to -8°C. How many degrees did it drop?

   • Answer: 13°C (distance from 5 to -8)

Mastering integers opens the door to advanced mathematics and helps you understand the world with greater precision and clarity!