Repeating decimal

Definition

A repeating decimal is a decimal representation of a number that has a digit or group of digits that repeat infinitely. This type of decimal occurs when a fraction is expressed in decimal form and can be traced back to rational numbers, where the numerator and denominator have no common factors other than one. Repeating decimals highlight the relationship between fractions and their decimal equivalents, showcasing the nature of rational numbers as they can be expressed in both forms.

5 Must Know Facts For Your Next Test

1. Repeating decimals can be represented using bar notation, where the repeating part is indicated by a line or dots above the digits.
2. Not all fractions result in repeating decimals; some result in terminating decimals depending on their denominators.
3. The length of the repeating portion in a repeating decimal can vary, ranging from one digit to several digits.
4. Every repeating decimal can be converted back into a fraction, proving its status as a rational number.
5. Common examples of repeating decimals include 1/3, which equals 0.333..., and 2/9, which equals 0.222...

Review Questions

• How do you convert a repeating decimal into a fraction?
  • To convert a repeating decimal into a fraction, you can set the decimal equal to a variable (like x), then multiply both sides by a power of ten that shifts the decimal point to align with the start of the repeat. After this, subtract the original equation from this new equation to eliminate the repeating part, allowing you to solve for x as a fraction. This process effectively demonstrates how every repeating decimal corresponds to a rational number.
• Compare and contrast repeating decimals with terminating decimals in terms of their properties and origins from rational numbers.
  • Repeating decimals and terminating decimals both stem from rational numbers, but they differ significantly in their characteristics. A terminating decimal ends after a finite number of digits (like 0.75), while a repeating decimal continues indefinitely with one or more digits repeating (like 0.333...). The key difference lies in their denominator's prime factors; terminating decimals have denominators that are products of only 2s and/or 5s when simplified, while repeating decimals have denominators that include other prime factors.
• Evaluate how understanding repeating decimals enhances your comprehension of rational numbers and their behaviors in mathematical operations.
  • Understanding repeating decimals deepens your grasp of rational numbers by illustrating how they exist in multiple forms—both as fractions and as decimals. This knowledge helps clarify concepts like addition and subtraction of decimals, especially when dealing with non-terminating values. Furthermore, recognizing that all repeating decimals can be converted back into fractions reinforces the idea that rational numbers are closed under basic arithmetic operations, allowing for more confidence in handling various mathematical problems involving fractions and their decimal equivalents.

Related terms

Rational Number: A rational number is any number that can be expressed as the quotient or fraction of two integers, where the denominator is not zero.

Terminating Decimal: A terminating decimal is a decimal number that has a finite number of digits after the decimal point, representing fractions with denominators that are powers of 10.

Fraction:

A fraction represents a part of a whole and consists of a numerator and a denominator, indicating division.