Integers

Detailed Notes on Integers

1. Properties of Addition and Subtraction of Integers

In this section, we explore various properties of integers, focusing on addition and subtraction.

1.1 Closure Property

• Closure under Addition:

  • The sum of any two integers is always an integer. For example, adding positive, negative integers, or combinations of both will yield an integer.
  • Statements to observe:
    1. 17 + 24 = 41 (Integer)
    2. (-10) + 3 = -7 (Integer)
    3. (-75) + 18 = -57 (Integer)
    4. 19 + (-25) = -6 (Integer)
    5. 27 + (-27) = 0 (Integer)
    6. (-20) + 0 = -20 (Integer)
    7. (-35) + (-10) = -45 (Integer)
  • General statement: For any two integers a and b, a + b is an integer.

• Closure under Subtraction:

  • Similar to addition, the difference of two integers is also an integer.
  • Examples:
    1. 7 - 9 = -2 (Integer)
    2. 17 - (-21) = 38 (Integer)
    3. (-8) - (-14) = 6 (Integer)
    4. (-21) - (-10)= -11 (Integer)
    5. 32 - (-17) = 49 (Integer)
  • Conclusion: Integers are closed under subtraction as well.

1.2 Commutative Property

• Addition:

  • Addition of integers is commutative.
  • For example:
    • 5 + (-6) = (-6) + 5 = -1
    • This holds for any integers, i.e., a + b = b + a.

• Subtraction:

  • Subtraction is not commutative.
  • For instance:
  • 5 - (-3) = 8, but (-3) - 5 = -8.

1.3 Associative Property

• Addition:
  • Addition of integers is associative.
  • That is, (a + b) + c = a + (b + c).
• Practice confirming this property using different combinations of integers.

1.4 Additive Identity

• The number 0 is called the additive identity because for any integer a:
  • a + 0 = a and 0 + a = a.
• Examples:
  • (-8) + 0 = -8, 0 + (-8) = -8

2. Multiplication of Integers

In this section, the focus shifts to multiplication and its properties.

2.1 Multiplication Basics

• Products involving integers can result in positive or negative integers.
• Multiplying a positive integer with a negative integer gives a negative integer, and the product of two negative integers is positive.
• General rules:
  • a × (-b) = - (a × b)
  • (-a) × (-b) = a × b.

3. Properties of Multiplication of Integers

3.1 Closure, Commutativity, and Associativity

• Integers are closed under multiplication, meaning a × b remains an integer.
• Multiplication is commutative (a × b = b × a) and associative [(a × b) × c = a × (b × c)].
• The multiplicative identity is 1, i.e., a × 1 = a.

3.2 Distributive Property

• This property connects multiplication with addition:
  • a × (b + c) = (a × b) + (a × c).

4. Division of Integers

4.1 Basic Concepts

• Division is not commutative; a ÷ b ≠ b ÷ a generally.
• Division by zero is undefined.
• When dividing integers, the result is:
  • Positive, when positives are divided, or when negatives are divided by negatives.
  • Negative, when a positive integer is divided by a negative integer.

4.2 Properties of Division of Integers

• Integers are not closed under division.
• Division of integers does not satisfy commutativity or associativity.
• For any integer a, a ÷ 1 = a remains true.

5. Key Equations and Examples

• Important operations illustrated through examples enable practical understanding of these properties.
• Solving practical problems, understanding temperature changes, test scoring, or sales calculations involves applying these properties of integers effectively.

1. Closure: Integers are closed under both addition and subtraction.
2. Commutative Property: Addition is commutative, but subtraction is not.
3. Associative Property: Addition is associative; grouping does not affect the sum.
4. Identity Elements: Zero is the additive identity; it does not change the value of integers.
5. Multiplication Rules: Positive times negative equals negative; negative times negative equals positive.
6. Multiplication Properties: Integers are closed under multiplication, and multiplication is commutative and associative.
7. Distributive Property: a × (b + c) = a × b + a × c for any integers a, b, and c.
8. Division of Integers: Results in negative if one is negative; division by zero is undefined.
9. Groups: Integer operations (addition, multiplication) follow distinct patterns and properties for efficient calculation.
10. Non-compliance: Division is neither commutative nor associative for integers.