1.1 Definition of Numbers

A number is a mathematical entity used to count, measure, and label. Numbers form the foundation of mathematics and are used in calculations, financial transactions, measurements, and problem-solving.

1.2 Classification of Numbers

Numbers can be classified into different categories based on their properties:

1.     Natural Numbers (NNN)

    • The set of counting numbers: 1, 2, 3, 4, 5, …
    • Zero is not included in natural numbers.
    • Example: If a child counts apples, they use 1, 2, 3, ….

2.     Whole Numbers (WWW)

    • Includes all natural numbers plus zero.
    • W = {0, 1, 2, 3, 4, …}
    • Example: The number of pencils in a box can be 0, 1, 2, ….

3.     Integers (ZZZ)

    • Includes positive numbers, negative numbers, and zero.
    • Z = {..., -3, -2, -1, 0, 1, 2, 3, …}
    • Example: A bank account balance can be negative (-500) if overdrawn, zero (0) if empty, or positive (500) if money is deposited.

4.     Even Numbers

    • Numbers divisible by 2 (without a remainder).
    • Example: 2, 4, 6, 8, 10, …

5.     Odd Numbers

    • Numbers not divisible by 2.
    • Example: 1, 3, 5, 7, 9, …

6.     Prime Numbers

    • Numbers greater than 1 with only two factors: 1 and itself.
    • Example: 2, 3, 5, 7, 11, 13, …
    • Note: 2 is the only even prime number.

7.     Composite Numbers

    • Numbers with more than two factors.
    • Example: 4 (1, 2, 4), 6 (1, 2, 3, 6), 9 (1, 3, 9).
    • Note: 1 is neither prime nor composite.

8.     Rational Numbers (QQQ)

    • Numbers that can be written as a fraction (pq\frac{p}{q}qp​, where ppp and qqq are integers, q≠0q \neq 0q=0).
    • Example: 12\frac{1}{2}21​, −3-3−3 (which can be written as −31\frac{-3}{1}1−3​), 0.750.750.75 (which is 34\frac{3}{4}43​).

9.     Irrational Numbers

    • Cannot be expressed as a fraction and have non-repeating, non-terminating decimals.
    • Example: π (3.14159...), √2 (1.414...), e (2.718...).

10.  Real Numbers (RRR)

    • All rational and irrational numbers combined.
    • Example: 5, -3.2, 0, 73\frac{7}{3}37​, π, √2.

11.  Imaginary Numbers

    • Numbers involving the square root of negative numbers.
    • Example: i=−1i = \sqrt{-1}i=−1​, 2i,5i,−3i2i, 5i, -3i2i,5i,−3i.

12.  Complex Numbers

    • A combination of real and imaginary numbers in the form a + bi.
    • Example: 2+3i2 + 3i2+3i, −4−5i-4 - 5i−4−5i.

1.3 Examples and Exercises

✅ Example 1: Identify the type of number:

  • 7 → Prime, Natural, Whole, Integer, Rational, Real
  • 0 → Whole, Integer, Rational, Real
  • -2 → Integer, Rational, Real
  • 3.14 → Irrational (if π), Rational (if terminating decimal)
  • 52\frac{5}{2}25​ → Rational, Real
  • √-4 → Imaginary

💡 Exercise 1: Classify the following numbers into Natural, Whole, Integer, Rational, or Irrational.

  • 1, -5, 0, 2.5, √3, 8, ⅔, -10, 0.3333…

💡 Exercise 2: Identify the prime and composite numbers:

  • 3, 4, 7, 9, 11, 14, 15, 17