Decimal Expansion of Rationals (Revisited)

A rational number is any number that can be expressed in the form p/q, where p and q are integers and q is not equal to zero. Every rational number, when divided, produces a decimal expansion. This decimal expansion takes one of two forms:

1. Terminating Decimal Expansion: The decimal representation ends after a finite number of digits. For example, 7/8 = 0.875. The division process eventually leaves a remainder of zero, so the decimal terminates.

2. Non-Terminating Repeating Decimal Expansion: The decimal representation goes on forever, but a block of digits repeats indefinitely. For example, 1/3 = 0.333... (the digit 3 repeats), and 1/7 = 0.142857142857... (the block 142857 repeats). This repeating block is called the period of the decimal, and it is often written with a bar over the repeating digits.

The key theorem in Class 10 states: Let x = p/q be a rational number, where p and q are co-prime (that is, HCF of p and q is 1). Then:

(i) x has a terminating decimal expansion if and only if the prime factorisation of q is of the form 2^n * 5^m, where n and m are non-negative integers.

(ii) x has a non-terminating repeating decimal expansion if the prime factorisation of q contains a prime factor other than 2 or 5.

This theorem connects the algebraic representation of a fraction to its decimal behaviour. The denominator's prime factorisation is the sole deciding factor. If the denominator (after reducing the fraction to lowest terms) has only 2s and 5s in its prime factorisation, the decimal will terminate. If any other prime (such as 3, 7, 11, 13, etc.) appears, the decimal will be non-terminating repeating.

The proof of the theorem on terminating decimal expansions relies on the Fundamental Theorem of Arithmetic and the structure of our base-10 number system.

Why base 10 matters: Our decimal system is based on powers of 10. When we write a terminating decimal like 0.375, we mean 375/1000 = 375/(10^3). Since 10 = 2 * 5, any power of 10 has the form 10^k = 2^k * 5^k. Therefore, a fraction p/q can be expressed as a terminating decimal only if q can divide some power of 10.

Forward direction — If q = 2^n * 5^m, then the decimal terminates:

Suppose q = 2^n * 5^m. We want to express p/q as a fraction with a power of 10 in the denominator. If n is greater than or equal to m, multiply numerator and denominator by 5^(n - m): p/q = p * 5^(n - m) / (2^n * 5^m * 5^(n - m)) = p * 5^(n - m) / (2^n * 5^n) = p * 5^(n - m) / 10^n. This gives a terminating decimal with at most n decimal places. Similarly, if m is greater than n, multiply by 2^(m - n) to get a denominator of 10^m.

Example: Consider 7/8 = 7/2^3. Here q = 2^3, so n = 3 and m = 0. Multiply numerator and denominator by 5^3 = 125: 7 * 125 / (2^3 * 5^3) = 875/1000 = 0.875. The decimal terminates.

Reverse direction — If the decimal terminates, then q = 2^n * 5^m:

Suppose p/q has a terminating decimal expansion. Then p/q = d/10^k for some integers d and k. So p/q = d/(2^k * 5^k). Since p and q are co-prime, q must divide 2^k * 5^k. By the Fundamental Theorem of Arithmetic, the only prime factors of 2^k * 5^k are 2 and 5. Therefore, q can only have 2 and 5 as prime factors, meaning q = 2^n * 5^m for some non-negative integers n and m.

Why non-terminating repeating occurs:

When we perform long division of p by q, at each step we get a remainder. The possible remainders are 0, 1, 2, ..., q - 1. If the remainder is ever 0, the division terminates. If the remainder is never 0, then by the Pigeonhole Principle, within at most q steps, some remainder must repeat. Once a remainder repeats, the sequence of digits in the quotient begins to repeat, producing a non-terminating repeating decimal. This is guaranteed when q has a prime factor other than 2 or 5, because then q cannot divide any power of 10, so the remainder can never become 0.

Length of the repeating block: The length of the repeating block (period) divides the value of Euler's totient function applied to q. For simple cases like 1/7, the period is 6 (since the remainders cycle through 1, 3, 2, 6, 4, 5 before repeating). For 1/3, the period is 1.

The decimal expansion of rational numbers falls into exactly two categories:

1. Terminating Decimals: These are decimals that have a finite number of digits after the decimal point. They occur when the denominator (in lowest terms) has no prime factors other than 2 and 5.

• 1/2 = 0.5 (denominator = 2 = 2^1)
• 3/4 = 0.75 (denominator = 4 = 2^2)
• 7/8 = 0.875 (denominator = 8 = 2^3)
• 13/20 = 0.65 (denominator = 20 = 2^2 * 5)
• 9/125 = 0.072 (denominator = 125 = 5^3)

2. Non-Terminating Repeating Decimals: These are decimals that go on forever with a repeating block of digits. They occur when the denominator (in lowest terms) has at least one prime factor other than 2 or 5.

• 1/3 = 0.333... (denominator = 3, repeating block: 3)
• 2/7 = 0.285714285714... (denominator = 7, repeating block: 285714)
• 5/6 = 0.8333... (denominator = 6 = 2 * 3, repeating block: 3)
• 1/11 = 0.090909... (denominator = 11, repeating block: 09)
• 7/12 = 0.58333... (denominator = 12 = 2^2 * 3, repeating block: 3)

Important note: Every terminating decimal can also be written as a non-terminating repeating decimal by appending repeating zeroes. For example, 0.5 = 0.5000... But conventionally, we consider it terminating.

Mixed type: Some decimals like 7/12 = 0.58333... have a non-repeating part (58) followed by a repeating part (3). This still counts as non-terminating repeating.

Method 1: Prime Factorisation of the Denominator

To determine whether a rational number p/q has a terminating or non-terminating repeating decimal expansion:

Step 1: Reduce p/q to its lowest terms by dividing both p and q by their HCF.

Step 2: Find the prime factorisation of q.

Step 3: Check if q = 2^n * 5^m. If yes, the decimal terminates. If q has any other prime factor, the decimal is non-terminating repeating.

Method 2: Long Division

Perform long division of p by q. If the remainder eventually becomes 0, the decimal terminates. If a remainder repeats, the decimal is non-terminating repeating. This method directly produces the decimal expansion and is useful for finding the actual digits.

Method 3: Converting Denominator to a Power of 10

If the denominator is of the form 2^n * 5^m, multiply both numerator and denominator by 2^(m-n) or 5^(n-m) (whichever is needed) to make the denominator a power of 10. This directly gives the terminating decimal.

Method 4: Converting Repeating Decimals to Fractions

To convert a repeating decimal back to a fraction: (a) Let x = the repeating decimal. (b) Multiply x by 10^k, where k is the number of digits in the repeating block. (c) Subtract the original equation from the new one to eliminate the repeating part. (d) Solve for x.

Understanding the decimal expansion of rational numbers has several practical applications across mathematics and everyday life.

In Banking and Finance: Interest rates, currency conversions, and tax calculations often produce decimal results. Knowing whether a division will yield a terminating or repeating decimal helps in rounding correctly and avoiding errors in financial computations.

In Engineering and Measurement: When converting between units (for instance, fractions of an inch to millimetres), the nature of the decimal expansion determines precision. Terminating decimals give exact values, while repeating decimals require rounding, introducing small errors.

In Computer Science: Computers store numbers in binary (base 2). A fraction that terminates in base 10 may not terminate in base 2 (for example, 1/5 = 0.2 in base 10 but 0.001100110011... in base 2). Understanding decimal expansions helps programmers anticipate floating-point representation errors.

In Cryptography: Properties of repeating decimals and modular arithmetic are used in number theory, which forms the basis of encryption algorithms.

In Examinations: CBSE Class 10 board exams frequently test the ability to determine whether a fraction has a terminating decimal expansion, making this a high-scoring and essential topic.