Operations with Integers

CHAPTER 2: OPERATIONS WITH INTEGERS - COMPREHENSIVE NOTES

2.1 A QUICK RECAP OF INTEGERS

Understanding Integers Through Problem-Solving

**Integers** are numbers that include positive numbers, negative numbers, and zero: ..., -3, -2, -1, 0, 1, 2, 3, ...

#### Rakesh's Puzzle Game

The puzzle teaches us to find pairs of numbers when we know their sum and difference. This is a foundational skill for understanding how numbers relate to each other.

**How to solve**: Try different pairs and check:

1. Do the two numbers add up to the given sum?

2. Is the difference between them (first number - second number) equal to the given difference?

**Example**: Sum = 25, Difference = 11

Try First Number = 18, Second Number = 7

Check: 18 + 7 = 25 ✓

Check: 18 - 7 = 11 ✓

These numbers are correct!

**Important observation**: If you change "difference = 11" to "difference = -11", the answer simply swaps: First Number = 7, Second Number = 18. This shows how negative differences reverse the order.

#### Finding Pairs from Sums and Differences

For **Sum = 27, Difference = 9**:

Try pairs systematically

Let's say first number is 18, second is 9

Check: 18 + 9 = 27 ✓

Check: 18 - 9 = 9 ✓

This can become a game where two players take turns giving sums and differences, and the other must find the pairs.

The Carrom Coin Model

The carrom coin model is a powerful way to represent integer operations using movement on a number line.

#### Moving in One Direction Only (All Positive)

On a number line:

```

0 1 2 3 4 5 6 7 8 9 10 11 12

```

If a coin starts at 0 and is struck:

First strike: 4 units right → position = 4

Second strike: 3 units right → position = 4 + 3 = 7

**Formula**: P = a + b, where P is final position, a is first movement, b is second movement

#### Moving in Both Directions (Positive and Negative)

We need to represent movement as both **magnitude** (how far) and **direction** (which way).

**Convention**:

**Positive integers** (+): Movement to the right

**Negative integers** (-): Movement to the left

Number line representation:

```

... -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 ...

←── negative direction ← ← → positive direction ───→

```

#### Worked Example: Multiple Movements

**Example**: Coin starts at 0. Movements are: 5 units right, then 7 units left.

First movement = +5 (right)

Second movement = -7 (left)

Final position = 5 + (-7) = -2

The coin is 2 units to the left of 0, at position -2.

**Another Example**: Movements of 1, -2, 3, -4, ..., -10

Position = 1 + (-2) + 3 + (-4) + 5 + (-6) + 7 + (-8) + 9 + (-10)

Pairing: (1 - 2) + (3 - 4) + (5 - 6) + (7 - 8) + (9 - 10)

= (-1) + (-1) + (-1) + (-1) + (-1) = -5

Token Model for Addition and Subtraction

**Token Representation**:

**Green token** (●) = represents +1

**Red token** (●) = represents -1

One green and one red token together = **zero pair** (they cancel out)

#### Subtraction Using Tokens

**Problem**: Find 7 - 18 using tokens

Step 1: Start with 7 green tokens (representing +7)

```

● ● ● ● ● ● ● (7 positives)

```

Step 2: We need to remove 18 positives, but we only have 7. So we add zero pairs.

We need 11 more positives, so we add 11 zero pairs:

```

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

●●●●●●●●●●● ●●●●●●●●●●●

(7 positives) (11 zero pairs)

```

Step 3: Now remove 18 positives (green tokens)

```

Remaining: ●●●●●●●●●●● (11 red tokens = -11)

```

**Answer**: 7 - 18 = -11

**Connection to Addition**: Subtracting a number is the same as adding its **additive inverse**.

7 - 18 = 7 + (-18) [Because the additive inverse of 18 is -18]

4 - (-12) = 4 + 12 [Because the additive inverse of -12 is +12]

Additive Inverse

**Definition**: The **additive inverse** of an integer a is represented as -a.

**Examples**:

Additive inverse of 18 = -(18) = -18

Additive inverse of -18 = -(-18) = 18

Additive inverse of -5 = -(-5) = 5

**Key Property**: When you add a number and its additive inverse, you get zero.

18 + (-18) = 0

(-25) + 25 = 0

---

2.2 MULTIPLICATION OF INTEGERS

Token Model for Multiplication

The token model helps us understand what multiplication means and why certain rules apply.

#### Case 1: Positive × Positive

**Interpretation**: Place tokens into a bag multiple times.

**Example**: 4 × 2

Place 2 positive (green) tokens into an empty bag 4 times

After each placement, we have: 2, then 4, then 6, then 8 positive tokens

**Result**: 4 × 2 = 8

Visual representation:

```

Bag starts empty ∅

After 1st placement: ● ●

After 2nd placement: ● ● ● ●

After 3rd placement: ● ● ● ● ● ●

After 4th placement: ● ● ● ● ● ● ● ● (8 tokens)

4 × 2 = 8

```

#### Case 2: Positive × Negative

**Interpretation**: Place negative (red) tokens into a bag multiple times.

**Example**: 4 × (-2)

Place 2 negative (red) tokens into an empty bag 4 times

After each placement: -2, then -4, then -6, then -8 negative tokens

**Result**: 4 × (-2) = -8

**Rule**: When multiplying a positive number by a negative number, the result is negative.

#### Case 3: Negative × Positive

**Interpretation**: Remove tokens from the bag multiple times.

**Example**: (-4) × 2

We need to remove 2 positive tokens 4 times

But the bag is empty! We need to add zero pairs first

Add 2 zero pairs 4 times (8 zero pairs total)

Then remove 2 green tokens 4 times, leaving 8 red tokens

```

Start: ∅ (empty bag)

Add 2 zero pairs, 4 times:

● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ●

Remove 2 green tokens, 4 times:

Remaining: ● ● ● ● ● ● ● ● (8 red tokens = -8)

(-4) × 2 = -8

```

**Rule**: When multiplying a negative number by a positive number, the result is negative.

#### Case 4: Negative × Negative

**Interpretation**: Remove negative tokens from the bag multiple times.

**Example**: (-4) × (-2)

We need to remove 2 negative tokens 4 times

Start with empty bag, add 2 zero pairs 4 times

Remove 2 red tokens 4 times, leaving 8 green tokens

```

Start: ∅ (empty bag)

Add 2 zero pairs, 4 times:

● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ●

Remove 2 red tokens, 4 times:

Remaining: ● ● ● ● ● ● ● ● (8 green tokens = +8)

(-4) × (-2) = 8

```

**Rule**: When multiplying a negative number by another negative number, the result is positive.

Complete Multiplication Rules for Integers

**Summary of all cases**:

| Case | Calculation | Rule | Example |

|------|------------|------|---------|

| Positive × Positive | 4 × 2 | = Positive | = 8 |

| Positive × Negative | 4 × (-2) | = Negative | = -8 |

| Negative × Positive | (-4) × 2 | = Negative | = -8 |

| Negative × Negative | (-4) × (-2) | = Positive | = 8 |

**Memory trick**:

Same signs (++ or --) → Result is **positive** (+)

Different signs (+- or -+) → Result is **negative** (-)

Working with Related Products

**If you know one product, you can find others**:

Given: 123 × 456 = 56,088

(-123) × 456 = -56,088 (one negative, one positive → negative)

(-123) × (-456) = 56,088 (both negative → positive)

123 × (-456) = -56,088 (one positive, one negative → negative)

Special Cases in Multiplication

#### 1 × a = a (for any integer a)

Using tokens: Place 'a' tokens into the bag once. The bag contains 'a' tokens.

**Examples**:

1 × 5 = 5

1 × (-8) = -8

1 × 0 = 0

#### -1 × a = -a (for any integer a)

Using tokens: Remove 'a' tokens from the bag once.

**Examples**:

(-1) × 5 = -5

(-1) × (-8) = 8 (remove 8 negatives, get 8 positives)

(-1) × 10 = -10

**Key idea**: Multiplying by -1 gives you the additive inverse (opposite) of the number.

Patterns in Integer Multiplication

#### Pattern 1: Positive Multiplicand, Changing Multiplier

Look at the times 3 table:

```

4 × 3 = 12

3 × 3 = 9 (decreased by 3)

2 × 3 = 6 (decreased by 3)

1 × 3 = 3 (decreased by 3)

0 × 3 = 0 (decreased by 3)

-1 × 3 = -3 (decreased by 3)

-2 × 3 = -6 (decreased by 3)

-3 × 3 = -9 (decreased by 3)

```

**Pattern**: When the multiplicand is positive (3), each time the multiplier decreases by 1, the product decreases by 3 (the value of the multiplicand).

#### Pattern 2: Negative Multiplicand, Changing Multiplier

Look at the times -3 table:

```

4 × (-3) = -12

3 × (-3) = -9 (increased by 3)

2 × (-3) = -6 (increased by 3)

1 × (-3) = -3 (increased by 3)

0 × (-3) = 0 (increased by 3)

-1 × (-3) = 3 (increased by 3)

-2 × (-3) = 6 (increased by 3)

-3 × (-3) = 9 (increased by 3)

```

**Pattern**: When the multiplicand is negative (-3), each time the multiplier decreases by 1, the product increases by 3 (the magnitude of the multiplicand).

#### Complete Times 3 Tables

**Times +3**:

```

1 × 3 = 3 -1 × 3 = -3

2 × 3 = 6 -2 × 3 = -6

3 × 3 = 9 -3 × 3 = -9

4 × 3 = 12 -4 × 3 = -12

5 × 3 = 15 -5 × 3 = -15

```

**Times -3**:

```

1 × (-3) = -3 -1 × (-3) = 3

2 × (-3) = -6 -2 × (-3) = 6

3 × (-3) = -9 -3 × (-3) = 9

4 × (-3) = -12 -4 × (-3) = 12

5 × (-3) = -15 -5 × (-3) = 15

```

Commutative Property of Multiplication

**Definition**: The **commutative property** means the order of multiplication doesn't matter.

For integers: **a × b = b × a** for any integers a and b

**Proof idea**:

The magnitude of the product depends only on the magnitudes of the numbers

The magnitudes don't change when we swap the numbers

The sign depends on whether the numbers have the same or different signs

The signs don't change when we swap the numbers

**Examples**:

3 × (-4) = -12 and (-4) × 3 = -12 ✓

(-15) × 8 = -120 and 8 × (-15) = -120 ✓

(-6) × (-7) = 42 and (-7) × (-6) = 42 ✓

Brahmagupta's Historical Contribution

**Brahmagupta** (628 CE) was an Indian mathematician who wrote the **Brāhmasphuṭasiddhānta**. He was the first to clearly articulate rules for multiplication and division of positive and negative numbers.

**His language**:

**Fortune** (dhana) = positive numbers

**Debt** (ṛṇa) = negative numbers

**Brahmagupta's Rules** (Verse 18.30-32):

1. The product or quotient of two fortunes is a fortune (positive × positive = positive)

2. The product or quotient of two debts is a fortune (negative × negative = positive)

3. The product or quotient of a debt and a fortune is a debt (negative × positive = negative)

4. The product or quotient of a fortune and a debt is a debt (positive × negative = negative)

This was a revolutionary contribution to mathematics because it was the first formal articulation of these rules!

Worked Examples: Multiplication of Integers

#### Example 1: Exam Marks Calculation

**Problem**: An exam has 50 multiple choice questions. 5 marks are given for each correct answer and 2 negative marks for each wrong answer. Mala had 30 correct answers and 20 wrong answers. What are her total marks?

**Solution**:

Marks for correct answers = 30 × 5 = 150

Marks for wrong answers = 20 × (-2) = -40

Total marks = 150 + (-40) = 110

**Answer**: Mala got 110 marks.

**Additional questions**:

Maximum possible marks = 50 × 5 = 250 (all correct)

Minimum possible marks = 50 × (-2) = -100 (all wrong)

#### Example 2: Elevator in Mining Shaft

**Problem**: An elevator in a mining shaft moves 3 metres per minute. Positions above ground are positive, below ground are negative.

**(a) If it descends from ground level (0), what will be its position after one hour?**

**Solution - Method 1 (Using Subtraction)**:

Distance moved in 1 hour = 60 minutes × 3 metres/minute = 180 metres

Starting position = 0

Since it's descending: 0 - 180 = -180

**Answer**: The elevator will be at -180 metres (180 metres below ground)

**Solution - Method 2 (Using Multiplication)**:

Speed while descending = -3 metres per minute (negative because going down)

Time = 60 minutes

Position change = 60 × (-3) = -180

Final position = 0 + (-180) = -180

**Answer**: The elevator will be at -180 metres

**(b) If it begins to descend from 15 m above the ground, what will be its position after 45 minutes?**

**Solution - Method 1 (Using Subtraction)**:

Starting position = 15 metres above ground

Distance moved down = 45 × 3 = 135 metres

Final position = 15 - 135 = -120

**Answer**: The elevator will be 120 metres below ground (-120 metres)

**Solution - Method 2 (Using Multiplication)**:

Starting position = +15

Velocity while descending = -3 metres per minute

Distance moved = 45 × (-3) = -135

Final position = 15 + (-135) = -120

**Answer**: The elevator will be at -120 metres

Finding Products Using Known Products

**Worked Example**:

If 123 × 456 = 56,088, find:

(-123) × 456 = ?

(-123) × (-456) = ?

123 × (-456) = ?

**Solution**:

For (-123) × 456:

One number is negative, one is positive

Result is negative

Magnitude stays the same: 123 × 456 = 56,088

**Answer**: (-123) × 456 = -56,088

For (-123) × (-456):

Both numbers are negative

Result is positive

Magnitude stays the same: 123 × 456 = 56,088

**Answer**: (-123) × (-456) = 56,088

For 123 × (-456):

One number is positive, one is negative

Result is negative

Magnitude stays the same: 123 × 456 = 56,088

**Answer**: 123 × (-456) = -56,088

---

2.3 DIVISION OF INTEGERS

Connection Between Division and Multiplication

**Key idea**: Division is the inverse operation of multiplication.

If a ÷ b = c, then c × b = a

**Example**: (-100) ÷ 25 = ?

Think: What number times 25 gives -100?

We need: ? × 25 = -100

Since 25 is positive and the result is negative, the unknown must be negative

Check: (-4) × 25 = -100 ✓

Therefore: (-100) ÷ 25 = -4

Division Rules for Integers

Just like multiplication, division follows the same sign rules:

**Division Rules**:

| Case | Signs | Result Sign | Example |

|------|-------|-------------|---------|

| Positive ÷ Positive | (+) ÷ (+) | Positive | 100 ÷ 25 = 4 |

| Positive ÷ Negative | (+) ÷ (-) | Negative | 100 ÷ (-25) = -4 |

| Negative ÷ Positive | (-) ÷ (+) | Negative | (-100) ÷ 25 = -4 |

| Negative ÷ Negative | (-) ÷ (-) | Positive | (-100) ÷ (-25) = 4 |

**Memory trick**: The same rule as multiplication!

Same signs → Positive result

Different signs → Negative result

Worked Examples: Division of Integers

#### Example 1: Simple Division

**Find**: (-48) ÷ 6

**Method**: Using multiplication fact

We need: ? × 6 = -48

One negative, one positive → result is negative

8 × 6 = 48, so the unknown is -8

Check: (-8) × 6 = -48 ✓

**Answer**: (-48) ÷ 6 = -8

#### Example 2: Finding Unknown in Division

**Find**: 84 ÷ (-7)

**Method**: Using multiplication fact

We need: ? × (-7) = 84

Both are the same sign in the answer (positive), so signs must match

The multiplier is negative, so the unknown must be negative

12 × 7 = 84, so the unknown is -12

Check: (-12) × (-7) = 84 ✓

**Answer**: 84 ÷ (-7) = -12

Verification Using Multiplication

**Always verify division answers using multiplication**:

If you claim that a ÷ b = c, then check: c × b = a

**Example**: Verify that (-36) ÷ (-4) = 9

Check: 9 × (-4) = -36 ✓

Both are negative, result is negative: ✓

So the answer is correct!

---

BRAHMAGUPTA'S RULES SUMMARY

**Historical Context**: In 628 CE, Brahmagupta documented these rules for the first time ever in his mathematical text **Brāhmasphuṭasiddhānta**. Using ancient Indian terminology of "fortune" (positive) and "debt" (negative), he established rules that form the foundation of modern arithmetic with negative numbers.

**Complete Rules** (for both multiplication and division):

1. **Fortune × Fortune = Fortune** (Positive × Positive = Positive)

Example: 5 × 3 = 15 ✓

Example: 5 ÷ 3 = 5/3 (positive)

2. **Debt × Debt = Fortune** (Negative × Negative = Positive)

Example: (-5) × (-3) = 15 ✓

Example: (-12) ÷ (-4) = 3 ✓

3. **Debt × Fortune = Debt** (Negative × Positive = Negative)

Example: (-5) × 3 = -15 ✓

Example: (-12) ÷ 4 = -3 ✓

4. **Fortune × Debt = Debt** (Positive × Negative = Negative)

Example: 5 × (-3) = -15 ✓

Example: 12 ÷ (-4) = -3 ✓

---

MAGIC GRIDS OF INTEGERS

Understanding the Magic Grid

A magic grid is a special arrangement of numbers where no matter which numbers you choose (following specific rules), you always get the same product!

**Sample Grid**:

```

8 -4 12 -6

-28 14 -42 21

12 -6 18 -9

20 -10 30 -15

```

How to Play the Magic Grid Game

**Rules**:

1. Circle any number in the grid

2. Strike out (cross) the entire row and entire column of that number

3. Circle any other number that is not struck out

4. Strike out its row and column

5. Continue until all numbers are either circled or struck out

6. Multiply all circled numbers

**Key observation**: No matter which path you take, you always get the same final product!

Why Does the Magic Occur?

**The Magic is in Multiplication!**

The grid is constructed using multiplication patterns. Each number is a product of specific factors arranged in rows and columns.

**Example Analysis**:

```

8 -4 12 -6

-28 14 -42 21

12 -6 18 -9

20 -10 30 -15

```

Notice:

Row 1: 8, -4, 12, -6 (multiples related to 4 and 2 with different signs)

Row 2: -28, 14, -42, 21 (multiples involving 7 and 14)

Row 3: 12, -6, 18, -9 (multiples related to 3 and 6)

Row 4: 20, -10, 30, -15 (multiples involving 5 and 10)

**Why the product is constant**:

When you select one number from each row and each column (which is what the game forces you to do), you're essentially multiplying factors that are related by the grid's construction. The combination of multiplication properties (commutative, associative) ensures the product remains constant.

Creating Your Own Magic Grid

**Method**:

1. Choose a target product (e.g., 360)

2. Choose factors that multiply to give this target

3. Arrange them so each cell is a product of its row factor and column factor

4. The magic will work!

**Example**: To make a 2×2 magic grid with product 24:

Use row factors: 2, -3

Use column factors: 4, -2

Grid:

```

2×4 = 8 2×(-2) = -4

(-3)×4 = -12 (-3)×(-2) = 6

```

Any path gives: 8 × (-2) × (-3) × (-4) or (-4) × (-12) × (-3) × 4, etc., all equal to 24 (or ±24 depending on selection)

---

KEY FORMULAS AND RULES SUMMARY

Integer Arithmetic Formulas

**Addition/Subtraction with Number Line**:

Final position = Starting position + Movement 1 + Movement 2 + ... + Movement n

**Multiplication Signs**:

(Positive) × (Positive) = Positive

(Positive) × (Negative) = Negative

(Negative) × (Positive) = Negative

(Negative) × (Negative) = Positive

**Division Signs** (same as multiplication):

(Positive) ÷ (Positive) = Positive

(Positive) ÷ (Negative) = Negative

(Negative) ÷ (Positive) = Negative

(Negative) ÷ (Negative) = Positive

**Identity Properties**:

1 × a = a (for any integer a)

(-1) × a = -a (for any integer a)

**Commutative Property**:

a × b = b × a

a ÷ b ≠ b ÷ a (division is NOT commutative)

**Inverse Relationship**:

a ÷ b = c means c × b = a

Common Mistakes to Avoid

1. **Mistake**: Thinking (-) ÷ (-) = (-)

**Correct**: (-) ÷ (-) = (+) because same signs make positive

2. **Mistake**: Forgetting that subtraction changes signs

**Correct**: a - b = a + (-b), so be careful with the sign of b

3. **Mistake**: Using wrong direction on number line

**Correct**: Positive movements go right, negative movements go left

4. **Mistake**: Not changing the sign when using additive inverse

**Correct**: To subtract b, add its additive inverse -b

5. **Mistake**: Assuming division is commutative

**Correct**: 12 ÷ 4 = 3 but 4 ÷ 12 = 1/3 (very different!)

6. **Mistake**: Making errors with multiple negatives

**Correct**: (-1) × (-1) × (-1) = -1 (odd number of negatives = negative result)

**Correct**: (-1) × (-1) × (-1) × (-1) = 1 (even number of negatives = positive result)

---

WORKED PRACTICE PROBLEMS

Problem Set 1: Finding Pairs from Sums and Differences

**Problem 1**: Find two numbers whose sum is 27 and difference is 9

**Solution**:

Let first number = x, second number = y

x + y = 27

x - y = 9

From the second equation: x = y + 9

Substitute: (y + 9) + y = 27

2y + 9 = 27

2y = 18

y = 9, so x = 18

Check: 18 + 9 = 27 ✓ and 18 - 9 = 9 ✓

**Answer**: The numbers are 18 and 9

**Problem 2**: Find two numbers whose sum is -7 and difference is -1

**Solution**:

Let first number = x, second number = y

x + y = -7

x - y = -1

From second equation: x = y - 1

Substitute: (y - 1) + y = -7

2y - 1 = -7

2y = -6

y = -3, so x = -4

Check: (-4) + (-3) = -7 ✓ and (-4) - (-3) = -4 + 3 = -1 ✓

**Answer**: The numbers are -4 and -3

Problem Set 2: Token Model and Subtraction

**Problem**: Using the token model, find 5 - 9

**Solution**:

1. Start with 5 green tokens (positive 5)

2. Need to remove 9 green tokens, but only have 5

3. Need 4 more, so add 4 zero pairs

4. Now we have: 5 green tokens + 4 zero pairs = 5 green + 4 red + 4 green = 9 green + 4 red

5. Remove 9 green tokens

6. Remaining: 4 red tokens = -4

**Answer**: 5 - 9 = -4

**Verification**: 5 - 9 = 5 + (-9) = -4 ✓

Problem Set 3: Integer Multiplication

**Problem 1**: Calculate (-8) × 7

**Solution**:

One negative, one positive → result is negative

Magnitude: 8 × 7 = 56

Sign: different signs → negative

**Answer**: (-8) × 7 = -56

**Problem 2**: Calculate (-6) × (-9)

**Solution**:

Both negative → result is positive

Magnitude: 6 × 9 = 54

Sign: same signs → positive

**Answer**: (-6) × (-9) = 54

**Problem 3**: If 45 × 32 = 1440, find (-45) × (-32) without multiplying

**Solution**:

Both numbers are negative

Same signs → result is positive

Magnitude doesn't change

(-45) × (-32) = 1440

**Answer**: 1440

Problem Set 4: Division of Integers

**Problem 1**: Find (-56) ÷ 8

**Solution**:

Need: ? × 8 = -56

One negative, one positive → unknown must be negative

7 × 8 = 56, so ? = -7

Check: (-7) × 8 = -56 ✓

**Answer**: (-56) ÷ 8 = -7

**Problem 2**: Find (-72) ÷ (-9)

**Solution**:

Need: ? × (-9) = -72

Both are same sign in result (negative), so signs must be opposite

The divisor is negative, so the unknown must be positive

8 × 9 = 72, so ? = 8

Check: 8 × (-9) = -72 ✓

**Answer**: (-72) ÷ (-9) = 8

Problem Set 5: Real-World Applications

**Problem 1: Temperature Change**

A thermometer shows -5°C at dawn. During the day, the temperature increases by 3°C per hour for 6 hours. What is the temperature in the evening?

**Solution**:

Starting temperature = -5°C

Change per hour = +3°C

Number of hours = 6

Total change = 6 × 3 = 18°C

Final temperature = -5 + 18 = 13°C

**Answer**: 13°C

**Problem 2: Bank Account**

Rohan's bank account starts