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Terminating Decimal Expansion

A terminating decimal expansion is a decimal number that ends or stops after a finite number of decimal places. For a rational number in the form p/q, this occurs strictly when the prime factorization of the denominator q consists only of powers of 2, powers of 5, or both.

Key Concepts

Let x = p/q be a rational number where p and q are co-prime integers. The decimal expansion of x terminates if the prime factorization of q is of the form 2^n * 5^m.

If the denominator q contains any prime factor other than 2 or 5, such as 3 or 7, the rational number will have a non-terminating repeating decimal expansion.

Before checking the prime factors of the denominator, it is mandatory to simplify the fraction to its lowest terms to ensure p and q share no common factors.

Formula & Equation

q = 2^n * 5^m

q represents the denominator of the simplified rational number p/q, while n and m are non-negative integers.

Common Misconceptions

Myth: Students often check the prime factorization of the denominator without simplifying the fraction first.

Fact: Always reduce the rational number p/q to its simplest co-prime form before checking the denominator. For example, 6/15 looks non-terminating because 15 has a factor of 3, but simplified it becomes 2/5, which is terminating.

Real World Applications

Converting fractions like 3/8 into exact decimals like 0.375 for precise measurements in carpentry or engineering.

Financial calculations where currency is divided into finite decimal units, such as calculating exactly 1/4 of a rupee as 0.25 rupees.

Frequently Asked Questions

How do you find if a rational number has a terminating decimal expansion without actual division?

Simplify the fraction to its lowest terms, then find the prime factorization of the denominator. If the factors are only 2, 5, or both, it is a terminating decimal.

Is 7/8 a terminating or non-terminating decimal?

It is a terminating decimal because the denominator 8 can be prime factorized as 2^3, which satisfies the condition of having only 2 as a prime factor.

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