In Exercises 51-58, perform the calculations on a calculator...

In Exercises $51-58,$ perform the calculations on a calculator without rounding. Following Exercise $56,$ show that the decimal equivalents of the following fractions indicate they are rational: (a) $1 / 3$ (b) $5 / 11$ (c) $2 / 5 .$ What is the repeating part of the decimal in (c)?

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction where both the numerator and the denominator are integers and the denominator is not zero. These numbers are characterized by having decimal representations that either terminate or eventually repeat, distinguishing them from irrational numbers.

Decimal Representations

Decimal representations of numbers reveal an important characteristic of rational numbers: they either come to an end (terminate) or they enter a repeating cycle. The process of converting a fraction to a decimal makes it clear whether the expansion is finite or has an infinite repeating segment, both of which confirm the number's rational nature.

Repeating Decimals

Repeating decimals are decimals in which one or more digits repeat infinitely after a certain point. This repeating block is a clear indication that the number is rational. Even if a decimal appears to terminate, it can be understood to have an implicit repeating zero, emphasizing that all decimals from rational numbers either terminate or repeat.

Transcript

Please give Ace some feedback