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Non Terminating Repeating Decimal Expansion

A non-terminating repeating decimal expansion is a type of decimal representation for rational numbers where the digits after the decimal point continue infinitely but repeat in a specific pattern. A rational number p/q has this expansion if the prime factorization of its denominator q contains prime factors other than 2 or 5.

Key Concepts

Every rational number expressed in the form p/q, where p and q are co-prime integers and q is not zero, has either a terminating or a non-terminating repeating decimal expansion.

If the prime factorization of the denominator q is not of the form 2^n * 5^m (where n and m are non-negative integers), the decimal expansion will be non-terminating and repeating.

The repeating block of digits in the decimal expansion is called the period, and the number of digits in this repeating block is always less than the denominator q.

Formula & Equation

q != 2^n * 5^m

q is the denominator of the rational number p/q in its simplest form, while n and m are non-negative integers.

Common Misconceptions

Myth: Non-terminating repeating decimals are irrational numbers.

Fact: Non-terminating repeating decimals are always rational numbers because they can be expressed in the fractional form p/q. Only non-terminating non-repeating decimals are irrational.

Real World Applications

Dividing 1 by 3 gives 0.333..., where the digit 3 repeats infinitely, representing a non-terminating repeating decimal.

Fractions like 1/7 result in 0.142857142857..., where the block of digits 142857 repeats continuously.

Frequently Asked Questions

How do you check if a rational number has a non-terminating repeating decimal expansion?

Simplify the fraction to its lowest terms and find the prime factors of the denominator. If there is any prime factor other than 2 or 5, the decimal expansion is non-terminating and repeating.

Can a non-terminating repeating decimal be converted into a fraction?

Yes, every non-terminating repeating decimal can be converted into a fraction of the form p/q using algebraic methods to eliminate the repeating part.

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