Lecture Notes 5 - Operation on Integers

 

Operations on Integers

Integers are whole numbers that can be positive, negative, or zero. They are a fundamental concept in mathematics and are used in various fields such as science, finance, engineering, and everyday life. Understanding operations on integers is crucial for problem-solving and mathematical reasoning. 

Definition of Integers

Integers consist of all whole numbers, both positive and negative, including zero. The set of integers can be represented as:

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

Integers do not include fractions or decimals, making them distinct from rational and real numbers.

Operations on Integers

2.1 Addition of Integers

Addition is one of the fundamental operations performed on integers. When adding two integers, the result can be either positive, negative, or zero.

2.1.1 Rules for Addition

1. Same Signs: If both integers have the same sign, add their absolute values and keep the common sign.

   • Example:
     • 3 + 5 = 8 (both positive)
     • -4 + -6 = -10 (both negative)

2. Different Signs: If the integers have different signs, subtract the smaller absolute value from the larger absolute value and keep the sign of the integer with the larger absolute value.

   • Example:
     • 7 + (-2) = 5 (7 is larger)
     • -3 + 5 = 2 (5 is larger)

2.1.2 Properties of Addition

• Commutative Property: The order of addition does not affect the sum.

  • Example: 4 + 6 = 6 + 4

• Associative Property: The grouping of numbers does not affect the sum.

  • Example: (2 + 3) + 5 = 2 + (3 + 5)

• Identity Property: Adding zero to any integer does not change the integer.

  • Example: 9 + 0 = 9

2.1.3 Examples of Addition

1. 8 + 3 = 11
2. -5 + 4 = -1
3. 0 + 7 = 7
4. -2 + (-6) = -8
5. -9 + 5 = -4

2.2 Subtraction of Integers

Subtraction is the process of taking one integer away from another. Subtraction can be viewed as adding a negative integer.

2.2.1 Rules for Subtraction

1. Subtracting a Positive Integer: To subtract a positive integer, add its negative.

   • Example:
     • 5 - 3 = 5 + (-3) = 2

2. Subtracting a Negative Integer: To subtract a negative integer, add its positive.

   • Example:
     • 6 - (-2) = 6 + 2 = 8

2.2.2 Properties of Subtraction

• Not Commutative: The order of subtraction affects the result.

  • Example: 5 - 3 ≠ 3 - 5

• Not Associative: The grouping of integers affects the result.

  • Example: (6 - 2) - 1 ≠ 6 - (2 - 1)

2.2.3 Examples of Subtraction

1. 10 - 4 = 6
2. -2 - 5 = -7
3. 3 - (-2) = 5
4. -7 - 4 = -11
5. 0 - 9 = -9

2.3 Multiplication of Integers

Multiplication is another basic operation involving integers. It can be thought of as repeated addition.

2.3.1 Rules for Multiplication

1. Same Signs: If both integers have the same sign, the product is positive.

   • Example:
     • 3 × 4 = 12 (both positive)
     • -2 × -3 = 6 (both negative)

2. Different Signs: If the integers have different signs, the product is negative.

   • Example:
     • 5 × (-3) = -15
     • -4 × 2 = -8

2.3.2 Properties of Multiplication

• Commutative Property: The order of multiplication does not affect the product.

  • Example: 4 × 5 = 5 × 4

• Associative Property: The grouping of numbers does not affect the product.

  • Example: (2 × 3) × 4 = 2 × (3 × 4)

• Distributive Property: Multiplication distributes over addition.

  • Example: 2 × (3 + 4) = 2 × 3 + 2 × 4

• Identity Property: Multiplying any integer by one gives the integer.

  • Example: 7 × 1 = 7

• Zero Property: Multiplying any integer by zero results in zero.

  • Example: 8 × 0 = 0

2.3.3 Examples of Multiplication

1. 6 × 3 = 18
2. -4 × 5 = -20
3. -7 × -2 = 14
4. 0 × 9 = 0
5. 8 × (-1) = -8

2.4 Division of Integers

Division is the process of distributing a number into equal parts. It is the inverse operation of multiplication.

2.4.1 Rules for Division

1. Dividing by a Positive Integer: When dividing by a positive integer, the quotient is positive or negative, depending on the sign of the dividend.

   • Example:
     • 10 ÷ 2 = 5 (both positive)
     • -10 ÷ 2 = -5

2. Dividing by a Negative Integer: When dividing by a negative integer, the quotient is negative or positive, depending on the sign of the dividend.

   • Example:
     • 10 ÷ (-2) = -5
     • -10 ÷ (-2) = 5

3. Division by Zero: Division by zero is undefined.

   • Example: 5 ÷ 0 is undefined.

2.4.2 Properties of Division

• Not Commutative: The order of division affects the result.

  • Example: 10 ÷ 5 ≠ 5 ÷ 10

• Not Associative: The grouping of integers affects the result.

  • Example: (20 ÷ 4) ÷ 2 ≠ 20 ÷ (4 ÷ 2)

2.4.3 Examples of Division

1. 20 ÷ 4 = 5
2. -15 ÷ 3 = -5
3. 8 ÷ (-2) = -4
4. -12 ÷ -3 = 4
5. 0 ÷ 5 = 0

Absolute Value of Integers

The absolute value of an integer is its distance from zero on the number line, regardless of direction. It is always a non-negative value.

• Notation: The absolute value of an integer x is denoted as |x|.

3.1 Examples of Absolute Value

1. |5| = 5
2. |-5| = 5
3. |0| = 0
4. |-12| = 12
5. |7| = 7

Applications of Integer Operations

Understanding operations on integers is crucial for various real-world applications:

4.1 Financial Calculations

In finance, integers are used to represent profits and losses. Positive integers represent profits, while negative integers represent losses.

• Example: A company earns $500 (positive integer) but incurs a loss of $200 (negative integer). The net gain is calculated as:

Net Gain = Profit + Loss
Net Gain = 500 + (-200) = 300

4.2 Temperature Changes

In weather reporting, temperatures can be represented as integers. Positive integers indicate above-freezing temperatures, while negative integers indicate below-freezing temperatures.

• Example: If the temperature is -5°C and it rises by 10°C, the new temperature can be calculated as:

New Temperature = Current Temperature + Change
New Temperature = -5 + 10 = 5°C

4.3 Sports Scoring

In many sports, scores can be represented as integers. A team can have a positive score, while penalties can be represented as negative scores.

• Example: A basketball team scores 80 points but incurs 5 penalties worth -2 points each. The total score is calculated as:

Total Score = Points + Penalties
Total Score = 80 + (-10) = 70

4.4 Distance and Direction

In navigation, positive and negative integers can represent distances in different directions. Positive integers may represent north or east, while negative integers represent south or west.

• Example: If a person moves 5 units north and then moves 3 units south, the net distance can be calculated as:

Net Distance = Distance North + Distance South
Net Distance = 5 + (-3) = 2 units north

Practice Problems

5.1 Addition Problems

1. Calculate: 7 + (-3) = ?
2. Calculate: -8 + 4 + (-2) = ?
3. Calculate: 0 + (-9) + 5 = ?

5.2 Subtraction Problems

1. Calculate: 6 - 10 = ?
2. Calculate: -5 - (-3) = ?
3. Calculate: 8 - 3 - 2 = ?

5.3 Multiplication Problems

1. Calculate: -7 × 3 = ?
2. Calculate: 4 × (-2) × (-1) = ?
3. Calculate: 0 × (-5) = ?

5.4 Division Problems

1. Calculate: -12 ÷ 4 = ?
2. Calculate: 15 ÷ (-3) = ?
3. Calculate: 0 ÷ 7 = ?

5.5 Mixed Problems

1. Calculate: 5 + (-2) - 3 × 2 = ?
2. Calculate: -8 ÷ (-4) + 6 = ?
3. Calculate: (3 × 4) + (-5) - 2 = ?

Operations on integers are foundational skills in mathematics that facilitate understanding of more complex concepts. By mastering addition, subtraction, multiplication, and division of integers, students develop essential problem-solving skills applicable in various real-world scenarios. Practice with a variety of problems enhances proficiency and prepares learners for advanced mathematical challenges.

By applying the rules and properties of integer operations, individuals can tackle mathematical tasks with confidence and accuracy. As integers are integral to many disciplines, a solid understanding of their operations is vital for success in both academic and practical contexts. Continue practicing these operations to build a strong mathematical foundation.