Repeating Decimals

Definition

Repeating decimals, also known as infinite decimals or recurring decimals, are decimal numbers where one or more digits in the decimal part repeat infinitely. They occur when a fraction cannot be expressed as a terminating decimal.

5 Must Know Facts For Your Next Test

1. Repeating decimals are denoted by placing a bar over the repeating digits, such as $\overline{0.3}$ for the repeating decimal 0.333333...
2. The decimal expansion of a fraction with a denominator that has prime factors other than 2 or 5 will result in a repeating decimal.
3. Repeating decimals can be converted to fractions by using the formula: $\frac{a}{10^n - 1}$, where $a$ is the repeating digit(s) and $n$ is the number of repeating digits.
4. Operations with repeating decimals, such as addition, subtraction, multiplication, and division, can be performed by converting them to fractions first.
5. Repeating decimals are a subset of rational numbers, and every rational number can be expressed as a repeating decimal or a terminating decimal.

Review Questions

• Explain how repeating decimals are related to fractions and rational numbers.
  • Repeating decimals are closely related to fractions and rational numbers. The decimal expansion of a fraction with a denominator that has prime factors other than 2 or 5 will result in a repeating decimal. This is because the denominator cannot be fully factored into powers of 10, which is the base of the decimal system. Repeating decimals are a subset of rational numbers, and every rational number can be expressed as either a repeating decimal or a terminating decimal.
• Describe the process of converting a repeating decimal to a fraction.
  • To convert a repeating decimal to a fraction, you can use the formula: $\frac{a}{10^n - 1}$, where $a$ is the repeating digit(s) and $n$ is the number of repeating digits. For example, to convert $\overline{0.3}$ to a fraction, we have $a = 3$ and $n = 1$, so the fraction would be $\frac{3}{10^1 - 1} = \frac{3}{9} = \frac{1}{3}$. This method works because the repeating decimal can be expressed as a geometric series, which can be summed to a finite value.
• Analyze the significance of repeating decimals in the context of real numbers and algebra essentials.
  • Repeating decimals are an important concept in the study of real numbers and algebra essentials. They highlight the fact that not all rational numbers can be expressed as terminating decimals, and that the decimal representation of some fractions is an infinite, repeating pattern. This understanding is crucial for working with rational numbers, performing operations with repeating decimals, and recognizing the limitations of the decimal system. Mastering the properties and conversion of repeating decimals is essential for solving algebraic problems involving rational expressions and equations.

Related terms

Terminating Decimal: A decimal number that has a finite number of digits and eventually stops repeating.

Fraction: A numerical quantity that is expressed as a ratio of two integers, the numerator and denominator.

Rational Number: A number that can be expressed as a ratio of two integers, including both terminating and repeating decimals.