Decimal Expansion of Rational Numbers

The decimal expansion of a rational number p/q is the decimal representation obtained by dividing p by q using long division.

There are exactly two types of decimal expansions for rational numbers:

1. Terminating Decimal Expansion:
The decimal has a finite number of digits after the decimal point. The division process eventually gives a remainder of 0.

Examples: 1/4 = 0.25, 7/8 = 0.875, 3/5 = 0.6, 13/20 = 0.65.

2. Non-Terminating Repeating (Recurring) Decimal Expansion:
The decimal goes on forever, but a block of one or more digits repeats indefinitely. The division process never gives a remainder of 0, but the remainders start repeating, causing the quotient digits to repeat.

Examples: 1/3 = 0.333... (3 repeats), 1/7 = 0.142857142857... (142857 repeats), 2/11 = 0.181818... (18 repeats), 1/6 = 0.1666... (6 repeats after the initial 1).

The repeating block is sometimes shown with a bar over it: 1/3 = 0.3 with a bar over 3, or 1/7 = 0.142857 with a bar over 142857.

Theorem (NCERT): Every rational number has either a terminating or a non-terminating repeating decimal expansion. Conversely, every terminating or repeating decimal represents a rational number.

Condition for terminating decimal:
A rational number p/q (in lowest terms, i.e., HCF(p, q) = 1) has a terminating decimal expansion if and only if the prime factorisation of q contains no prime factors other than 2 and 5. In other words, q must be of the form 2^m x 5^n, where m and n are non-negative integers.

If q has any prime factor other than 2 or 5, the decimal expansion will be non-terminating repeating.

Why does a rational number always give a terminating or repeating decimal? (Proof using the Pigeonhole Principle)

This is an elegant argument that every Class 9 student should understand.

Step 1: Consider the rational number p/q where q > 0. Perform long division of p by q.

Step 2: At each step of the division, you get a remainder. The remainder must be one of the values: 0, 1, 2, 3, ..., (q - 1). So there are exactly q possible remainders.

Step 3 (Case 1 — Terminating): If at any step the remainder is 0, the division is complete and the decimal terminates. For example, 3/8: the remainders are 6, 4, 0 — and the division stops. Result: 0.375.

Step 4 (Case 2 — Repeating): If the remainder never becomes 0, then we are cycling through remainders from the set {1, 2, ..., q - 1}, which has (q - 1) elements. By the Pigeonhole Principle, after at most (q - 1) steps, some remainder must appear for the second time.

Step 5: Once a remainder repeats, the entire sequence of quotient digits and remainders from that point onward will be identical to the sequence that followed the first occurrence of that remainder. This creates a repeating cycle in the decimal.

Step 6: Therefore, the decimal expansion of p/q either terminates (remainder becomes 0) or repeats (a remainder recurs). There is no third possibility.

Derivation of the terminating condition:

Why does q = 2^m x 5^n guarantee termination?

Consider p/q where q = 2^m x 5^n. We can multiply numerator and denominator to make the denominator a power of 10:
If m > n: p/q = p x 5^(m-n) / (2^m x 5^m) = p x 5^(m-n) / 10^m.
If n > m: p/q = p x 2^(n-m) / (2^n x 5^n) = p x 2^(n-m) / 10^n.
If m = n: p/q = p / 10^m.

In every case, the denominator becomes a power of 10, so the decimal terminates with at most max(m, n) decimal places.

Conversely, if q has a prime factor other than 2 or 5, we cannot convert the denominator to a power of 10, so the decimal cannot terminate.

Decimal expansions of rational numbers fall into exactly two categories:

1. Terminating Decimals:

These are decimals that have a finite number of digits after the decimal point.

Examples and their fraction forms:

1/2 = 0.5 (denominator 2 = 2^1)

3/4 = 0.75 (denominator 4 = 2^2)

7/8 = 0.875 (denominator 8 = 2^3)

3/5 = 0.6 (denominator 5 = 5^1)

9/25 = 0.36 (denominator 25 = 5^2)

7/20 = 0.35 (denominator 20 = 2^2 x 5)

In each case, the denominator (when the fraction is in lowest terms) has only 2 and/or 5 as prime factors.

2. Non-Terminating Repeating Decimals:

These are decimals that go on forever with a repeating block.

(a) Pure repeating decimals: The repeating block starts immediately after the decimal point.

1/3 = 0.333... (block: 3, length 1)

1/7 = 0.142857142857... (block: 142857, length 6)

1/9 = 0.111... (block: 1, length 1)

5/11 = 0.454545... (block: 45, length 2)

(b) Mixed repeating decimals: There are some non-repeating digits before the repeating block begins.

1/6 = 0.1666... (non-repeating: 1, repeating: 6)

7/12 = 0.58333... (non-repeating: 58, repeating: 3)

1/30 = 0.0333... (non-repeating: 0, repeating: 3)

Mixed repeating decimals occur when the denominator (in lowest terms) has factors of 2 or 5 in addition to other prime factors.

Key distinction from irrational numbers:

Irrational numbers have non-terminating, non-repeating decimals — they go on forever with NO repeating block. Examples: sqrt(2) = 1.41421356..., pi = 3.14159265..., sqrt(3) = 1.73205080... The digits appear random with no pattern that ever repeats.

Understanding decimal expansions of rational numbers has practical applications across many fields:

Currency and Finance: Money is naturally expressed in terminating decimals. Rs 3.50, Rs 129.75, etc. When interest rates produce repeating decimals (like 1/3 percent), they are rounded to a suitable number of decimal places. Understanding why some fractions terminate helps in quick mental calculations.

Measurement and Engineering: Measurements often involve fractions that need decimal conversion. Knowing whether 7/16 of an inch will terminate (it does: 0.4375) helps machinists set precise measurements without rounding errors.

Computer Science: Computers store numbers in binary (base 2). Interestingly, some decimals that terminate in base 10 (like 0.1) become non-terminating in binary, leading to floating-point errors. Understanding the relationship between bases and termination is critical for programmers.

Fractions in Recipes and Daily Life: Converting recipe measurements (like 1/3 cup, 2/7 of a litre) to decimal helps when using digital measuring tools. Knowing that 1/3 = 0.333... helps you understand why 3 x 0.333 gives 0.999 (approximately 1, not exactly 1 in finite decimal form).

Cryptography: Properties of decimal expansions of 1/p (where p is prime) are used in certain cryptographic algorithms. The length of the repeating period relates to the concept of multiplicative order in number theory.