2. Classification of Numbers

Figure: How the major number sets nest inside one another.

Natural Numbers (N): Counting numbers starting from 1 — {1, 2, 3, 4, 5, …}. There is no upper limit; there is a lower limit of 1.

Whole Numbers (W): Natural numbers plus zero — {0, 1, 2, 3, …}. Every natural number is a whole number, but 0 is whole, not natural.

Integers (Z): All whole numbers and their negatives — {…, -3, -2, -1, 0, 1, 2, 3, …}. Integers have no fractional or decimal part.

Rational Numbers (Q): Any number expressible as p/q where p and q are integers and q ≠ 0. This includes all integers (5 = 5/1), all terminating decimals (0.75 = 3/4), and all recurring decimals (0.333… = 1/3).

Irrational Numbers: Numbers that cannot be written as a ratio of two integers. Their decimal expansion is non-terminating and non-repeating. Examples: √2, √3, √5, π, e. Note carefully: √4 = 2 is NOT irrational (it simplifies to an integer); only surds of non-perfect-squares/cubes are irrational.

Real Numbers (R): The union of all rational and irrational numbers — essentially every number you can place on a number line.

Even Numbers: Divisible by 2 — {…, -4, -2, 0, 2, 4, …}.

Odd Numbers: Not divisible by 2 — {…, -3, -1, 1, 3, …}.

Prime Numbers: Natural numbers greater than 1 that have exactly two distinct factors — 1 and themselves. Smallest prime is 2 (the only even prime). Sequence: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 (25 primes below 100 — worth memorising for exam speed).

Composite Numbers: Natural numbers greater than 1 with more than two factors. Example: 4 (factors 1, 2, 4), 9 (factors 1, 3, 9).

Important exception: The number 1 is neither prime nor composite — it has only one factor (itself).

Co-prime (Relatively Prime) Numbers: A pair of numbers whose HCF is 1. They need not individually be prime — e.g., 8 and 9 are co-prime (HCF = 1) though neither is prime. Any two consecutive integers are always co-prime.

Why this matters: Almost every “which of the following is/is not…” question in Set A of any SSC paper directly tests this classification. Misclassifying 1 as prime, or 0 as a natural number, is the single most common silly mistake aspirants make (see Section 5).