Introduction to Real Numbers

The Number System — Building Up from Natural Numbers

Chapter 1 of Class 9 Mathematics opens by revisiting the number system you have been building since primary school and extending it in an important new direction. The journey begins with the most familiar sets and shows how each one contains the previous as a subset.

• Natural Numbers (N) — the counting numbers: 1, 2, 3, 4, … They do not include zero.
• Whole Numbers (W) — natural numbers plus zero: 0, 1, 2, 3, … Every natural number is a whole number, but 0 is not a natural number.
• Integers (Z) — all whole numbers and their negatives: …, −3, −2, −1, 0, 1, 2, 3, … Every whole number is an integer.
• Rational Numbers (Q) — any number expressible as p/q where p and q are integers and q ≠ 0. This includes fractions, terminating decimals, and repeating decimals. Every integer is a rational number.

The relationship is always one-way: every natural number is rational, but not every rational number is natural. The converses of these inclusion statements are false, which is a common exam question in CBSE, Telangana, and Andhra Pradesh boards.

Representing Rational Numbers on the Number Line

Any rational number can be located precisely on the number line. The method is to first identify which two consecutive integers the fraction lies between, then divide that unit interval into equal parts matching the denominator.

For example, 7/4 = 1¾, so it lies between 1 and 2. Dividing the segment from 1 to 2 into four equal parts, 7/4 falls at the third mark after 1. For a negative fraction like −7/4 = −1¾, the same logic applies on the negative side: it sits between −1 and −2, three quarters of the way from −1 toward −2.

Finding Rational Numbers Between Two Given Numbers

A key insight in this introduction is that between any two rational numbers there are infinitely many other rational numbers. Two methods make this practical.

Method 1 — Mean method: The arithmetic mean of two rational numbers a and b, which is (a + b)/2, always lies between them. To find further numbers, take the mean again between the new number and one of the original endpoints. For instance, the mean of 4 and 5 is 9/2, and the mean of 4 and 9/2 is 17/4, giving 4 < 17/4 < 9/2 < 5.

Method 2 — Denominator method: To find n rational numbers between two integers in one step, rewrite both integers as fractions with denominator (n + 1) and read off the fractions in between. To find 7 rationals between 4 and 5, use denominator 8: 4 = 32/8 and 5 = 40/8, so 33/8, 34/8, 35/8, 36/8, 37/8, 38/8, and 39/8 are the seven required numbers.

Decimal Form of Rational Numbers

Every rational number, when divided out, produces a decimal that either terminates or repeats. There is no third possibility — this is a defining property of rational numbers.

• Terminating decimal — the division ends with a remainder of zero. Example: 3/8 = 0.375.
• Non-terminating recurring decimal — the digits repeat in a fixed pattern forever. Example: 2/11 = 0.181818… = 0.̄18, and 7/6 = 1.1666… = 1.1̄6.

What Determines Whether a Decimal Terminates?

There is a reliable rule based on the prime factorisation of the denominator (in its simplest form). This rule is one of the most frequently tested facts in Class 9 board exams.

• If the denominator's prime factors are only 2s and 5s — that is, of the form 2ᵐ × 5ⁿ — the decimal terminates. Examples: 3/8 (denominator = 2³) gives 0.375; 9/25 (denominator = 5²) gives 0.36.
• If the denominator has any prime factor other than 2 or 5, the decimal is non-terminating and recurring. Examples: 5/9 (denominator = 3²) gives 0.5̄; 8/15 (denominator = 3 × 5) gives 0.53̄.

Converting Decimals Back to Fractions

The reverse process — turning a decimal back into a p/q form — is also important. For a terminating decimal, write the digits over the appropriate power of 10 and simplify: 0.375 = 375/1000 = 3/8. For a purely recurring decimal like 0.̄18, place the repeating block over the same number of 9s: 0.̄18 = 18/99 = 2/11. For a mixed case like 1.1̄6 (where only part of the decimal repeats), the rule involves subtracting the non-repeating part, giving 1.1̄6 = 7/6.

What This Lesson Prepares You For

Understanding rational numbers and their decimal behaviour is essential before tackling irrational numbers and surds, which are introduced later in this chapter. The terminating/non-terminating distinction directly sets up the concept of irrational numbers — numbers whose decimals neither terminate nor repeat. These ideas continue into Exercise 1.1 and Exercise 1.2, and resurface in Class 10 in Real Numbers where the Fundamental Theorem of Arithmetic is used to prove irrationality formally.