Unit 7, Lesson 10# Converting Repeating Decimals to Fractions

EngageNY 45 min(s)

Students develop a convincing argument establishing that every real number with a repeating decimal is a rational number. In Lesson 8, we say that every rational number, that is, every fraction, has a decimal expansion that falls into a repeating pattern. A natural question now is the converse: If a real number has an infinitely long decimal expansion with a repeating pattern, must it be a rational number? We begin by observing the effect of multiplying decimals by powers of 10.