Lecture 1

Introduction to Whole Numbers

  • Whole numbers consist of:

    • Basic counting numbers: 0, 1, 2, 3, 4, 5, 6, ...

    • Include natural numbers starting from 1.

Difference Between Whole Numbers and Natural Numbers

  • Definitions:

    • Whole Numbers (W): {0, 1, 2, 3, ...}

    • Natural Numbers (N): {1, 2, 3, ...}

  • Key Points:

    • The smallest whole number is 0.

    • The smallest natural number is 1.

    • Every natural number is a whole number.

    • All whole numbers are natural numbers except for 0.

Properties of Whole Numbers

  • Basic Operations: Addition, Subtraction, Multiplication, and Division lead to the following properties:

Closure Property

  • The sum and product of two whole numbers is always a whole number.

  • Example:

    • 7 + 3 = 10 (whole number)

    • 7 × 2 = 14 (whole number)

Associative Property

  • Changing the grouping of numbers does not change the sum or product.

  • Example:

    • 10 + (7 + 12) = (10 + 7) + 12 = (10 + 12) + 7 = 29

Commutative Property

  • The order of sum and product does not affect the result.

  • Example:

    • 10 + 19 = 29 = 19 + 10

Distributive Property

  • Multiplication distributes over addition or subtraction.

  • Example:

    • a × (b + c) = (a × b) + (a × c)

Types of Numbers

Integers (Z)

  • Definition: Whole numbers and their negatives.

  • Example: {..., -2, -1, 0, 1, 2, ...}

Rational Numbers (Q)

  • Definition: Fractions where the numerator and denominator are integers (denominator ≠ 0).

  • Examples:

    • 1/2, 3/4, 7/2, 4/3, 4/1

Irrational Numbers

  • Definition: Numbers that cannot be expressed in fraction form (no repeating decimal).

  • Examples:

    • √2, √5, π

Real Numbers (R)

  • Definition: All numbers that can be represented as decimals.

  • Examples:

    • 0.5, 0.75, 2.35

Imaginary Numbers

  • Definition: Numbers that square to a negative result (square roots of negative numbers).

  • Example:

    • √-2 represented by i (i = √-1).

Operations with Whole Numbers

Addition and Subtraction

  • Focus on basic addition and subtraction of whole numbers.

Multiplication

  • Multiplication table for small whole numbers.

  • Conventions for multiplication:

    • Product of 86 × 7 = 602

    • Product of 764 × 38 = 6112

Multiplication with Different Signs

  • Rule: Multiply the absolute values, result is negative if one is negative.

  • Example:

    • 178 × -46 = -8188 (as it essentially means negative multiplication)

Division

  • Example: Determine 1834 ÷ 7 using short division method.

    • Steps:

      1. 7 into 18 goes 2, remainder 4.

      2. 7 into 43 goes 6, remainder 1.

      3. 7 into 14 goes 2, remainder 0.

    • Result: 1834 ÷ 7 = 262.

Practice Problems

  • Test understanding through calculations without calculators.

    1. (a) 78 x 6

    2. (a) £261 x 7

    3. (a) 783 kg x 11

    4. (a) 27mm x 13

    5. (a) 448 x 23

    • Continue through 10 with expressions using multiplication and division.

Conclusion

  • Summary of key concepts surrounding whole numbers, operations, and types of numbers for a strong foundational understanding.