Give a description of the set of rational numbers whose decimal expansions terminate. (Alternati

step 1 So in this problem we have been asked that write down the decimal expansions of those rational n

Give a description of the set of rational numbers whose...

Key Concepts

Reduced Form of Fractions

Expressing fractions in their simplest or reduced form (where the numerator and denominator have no common factors other than 1) is an essential step in identifying whether a rational number will have a terminating decimal. The process ensures that extraneous factors are removed, leaving only the prime factors that dictate the behavior of the decimal expansion.

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient of two integers, typically in the form p/q, where p and q are integers and q is non-zero. This concept is fundamental in mathematics as it covers a wide range of numbers and their properties, including how they are represented in decimal form.

Terminating Decimal Expansion

A terminating decimal expansion is a decimal representation that ends after a finite number of digits. For rational numbers, this property is closely studied because it distinguishes between numbers with finite versus infinite repeating decimals, and it has implications in understanding number precision and representation.

Prime Factorization of the Denominator

The prime factorization of the denominator in a reduced fraction is crucial in determining if the decimal expansion terminates. Specifically, a rational number will have a terminating decimal expansion if and only if, when the fraction is in lowest terms, the denominator's prime factors are only 2, 5, or both. This connection arises because the base of the decimal system is 10, which factors into 2 and 5.

Transcript

Please give Ace some feedback