Tenth Grade Mathematics: Rational Numbers and Decimal Expansions
Every rational number can be expressed as a quotient of two integers, p/q, but when converted into decimal notation, rational numbers display distinct behavioral characteristics. Some fractions terminate into finite decimal strings, while others produce infinite repeating periodic blocks. In this chapter, we explore the deep algebraic connection between the prime factorization of the denominator and the exact nature of its decimal expansion, enabling us to determine terminating behavior without performing long division.
Classification of Real Decimal Expansions
The decimal representation of any real number falls into one of three comprehensive categories:
Decimal Expansions
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+-----------------------+-----------------------+
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Terminating Decimal Non-Terminating Decimal
(e.g., 0.375, 2.5) |
+-----------------------+-----------------------+
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Non-Terminating Repeating Non-Terminating Non-Repeating
(Rational Numbers) (Irrational Numbers)
(e.g., 0.333..., 0.142857...) (e.g., 1.414213..., 3.141592...)
Terminating decimals and non-terminating repeating decimals together make up the complete set of rational numbers. Non-terminating non-repeating decimals represent irrational numbers.
The Prime Factorization Criterion for Denominators
The core theorem of tenth-grade decimal analysis allows us to classify the decimal expansion of any rational number solely by inspecting its denominator.
Theorem: Let x = p / q be a rational number such that the prime factorization of
the denominator q is of the form:
q = 2^n * 5^m
where n and m are non-negative integers. Then x has a decimal expansion that
terminates.
Conversely, if the prime factorization of q contains any prime factor other than 2 or 5 (that cannot be canceled by the numerator p), then x has a non-terminating repeating decimal expansion.
Condition Check for Rational p/q (in lowest terms, HCF(p, q) = 1):
Denominator q factored into primes
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+----------------------------+----------------------------+
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Only Primes 2 and/or 5 Any other prime factor
q = 2^n * 5^m (e.g., 3, 7, 11, 13...)
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TERMINATING DECIMAL NON-TERMINATING REPEATING
Mathematical Proof of the Terminating Condition
To understand why only powers of 2 and 5 produce terminating decimals, consider the base of our positional number system: base 10.
A decimal expansion terminates after k decimal places if and only if it can be written as an integer divided by 10^k:
x = d / 10^k = d / (2 * 5)^k = d / (2^k * 5^k)
Now let x = p / q be in simplest reduced form (HCF(p, q) = 1). If q = 2^n * 5^m, let k = max(n, m). We can multiply both the numerator and denominator by appropriate powers of 2 or 5 to equalize their exponents to k:
If n >= m: multiply numerator and denominator by 5^(n - m)
q * 5^(n - m) = 2^n * 5^n = 10^n
If m > n: multiply numerator and denominator by 2^(m - n)
q * 2^(m - n) = 2^m * 5^m = 10^m
The denominator becomes an exact power of 10, meaning the fraction converts directly into a terminating decimal with exactly k = max(n, m) decimal places.
Determining the Number of Decimal Places
Without carrying out long division, the number of decimal places before termination is given by the maximum of the exponents n and m in the reduced denominator:
Number of Decimal Places = max(n, m)
Example 1: 3 / 8
Step 1: Simplify fraction: HCF(3, 8) = 1 (already reduced).
Step 2: Factor denominator: 8 = 2^3 = 2^3 * 5^0.
Step 3: Exponents are n = 3, m = 0. Max(3, 0) = 3.
Conclusion: 3 / 8 will terminate after exactly 3 decimal places.
Conversion: 3 / 8 = (3 * 5^3) / (2^3 * 5^3) = (3 * 125) / 10^3 = 375 / 1000 = 0.375.
Example 2: 13 / 125
Step 1: 125 = 5^3 = 2^0 * 5^3.
Step 2: Max(0, 3) = 3 decimal places.
Conversion: 13 / 125 = (13 * 2^3) / (5^3 * 2^3) = (13 * 8) / 1000 = 104 / 1000 = 0.104.
Example 3: 7 / 80
Step 1: 80 = 2^4 * 5^1.
Step 2: Max(4, 1) = 4 decimal places.
Conversion: 7 / 80 = (7 * 5^3) / (2^4 * 5^4) = (7 * 125) / 10^4 = 875 / 10000 = 0.0875.
The Critical Importance of Reducing Fractions First
One of the most common pitfalls in decimal classification is evaluating the denominator before reducing the fraction to lowest terms.
Consider: 6 / 15
Unreduced denominator: 15 = 3 * 5 (Contains 3, falsely suggesting non-terminating!)
Reduced form: Divide numerator and denominator by HCF(6, 15) = 3:
6 / 15 = 2 / 5
Correct denominator: 5 = 2^0 * 5^1
Conclusion: The decimal terminates! Indeed, 2 / 5 = 0.4.
Always ensure that HCF(p, q) = 1 before applying the denominator factorization theorem.
Chapter Exercises
Exercise 1: Without actually performing long division, determine whether each of the following rational numbers has a terminating or a non-terminating repeating decimal expansion: 17/8, 64/455, 15/1600, and 29/343.
Exercise 2: Write down the decimal expansion of 23 / (2^3 * 5^2) by converting the denominator to a power of 10 without using division. State how many places of decimals it has.
Exercise 3: The decimal expansion of a number is given as 0.120120012000120000... State with logical justification whether this number is rational or irrational, and whether it can be expressed in the form p/q.
Solutions and Explanations
Solution 1: For 17/8, the fraction is in lowest terms. The denominator is 8 = 2^3 * 5^0, which is of the form 2^n * 5^m; therefore, it has a terminating decimal expansion. For 64/455, the fraction is in lowest terms. The denominator 455 factors into 5 * 7 * 13. Since it contains the prime factors 7 and 13, it cannot be written in the form 2^n * 5^m; thus, it has a non-terminating repeating decimal expansion. For 15/1600, reduce first by dividing by 5: 15/1600 = 3/320. Factor 320: 320 = 32 * 10 = 2^5 * (2 * 5) = 2^6 * 5^1. Since the denominator contains only primes 2 and 5, it terminates. For 29/343, 343 = 7^3, which contains prime factor 7; therefore, it has a non-terminating repeating decimal expansion.
Solution 2: We have the fraction 23 / (2^3 * 5^2). To equalize the exponents of 2 and 5 in the denominator, multiply both numerator and denominator by 5^1: 23 * 5 / (2^3 * 5^2 * 5^1) = 115 / (2^3 * 5^3) = 115 / (10^3) = 115 / 1000 = 0.115. Because max(3, 2) = 3, the decimal expansion terminates after exactly 3 decimal places.
Solution 3: The given decimal expansion is 0.120120012000120000... Inspecting the sequence of digits reveals that the number of zeros between the blocks of '12' increases successively by one in each group. Consequently, there is no fixed, finite block of repeating digits that repeats periodically. Because the decimal expansion is both non-terminating and non-repeating, it represents an irrational number. By definition, irrational numbers cannot be expressed in the rational quotient form p/q with integer numerator and non-zero integer denominator.