Repeating decimal

Definition

A repeating decimal is a decimal fraction in which a digit or group of digits repeats infinitely. These decimals can be expressed as fractions, highlighting their relationship with rational numbers. Understanding repeating decimals is crucial for converting between fractions and decimals, as well as for performing arithmetic operations involving these forms.

5 Must Know Facts For Your Next Test

1. Repeating decimals can be represented using a bar notation over the repeating part, such as 0.333... being written as 0. extoverline{3}.
2. Any repeating decimal can be converted into a fraction through algebraic manipulation, making it a key concept in understanding rational numbers.
3. The length of the repeating sequence determines the complexity of converting a repeating decimal into its fractional form.
4. Some decimals are non-repeating and non-terminating, known as irrational numbers, which cannot be expressed as fractions.
5. Not all fractions will yield repeating decimals; fractions with denominators that are products of only the primes 2 and/or 5 will convert to terminating decimals instead.

Review Questions

• How can you convert a repeating decimal into a fraction, and what steps are involved in this process?
  • To convert a repeating decimal into a fraction, you can set the decimal equal to a variable. For instance, if you let x = 0.666..., then multiply both sides by 10 to shift the decimal point: 10x = 6.666.... Subtracting the original equation from this new equation gives you 9x = 6. Solving for x results in x = 6/9, which simplifies to 2/3. This process showcases how repeating decimals are rational numbers that can be expressed as fractions.
• Discuss how repeating decimals relate to rational numbers and why understanding them is important for performing mathematical operations.
  • Repeating decimals are inherently linked to rational numbers because they can be expressed as fractions. Recognizing this connection is essential for performing mathematical operations accurately. When adding or subtracting numbers that include repeating decimals, it's crucial to convert them to their fractional forms to maintain precision. Additionally, understanding repeating decimals aids in grasping concepts of limits and convergence in more advanced math topics.
• Evaluate the implications of recognizing different types of decimals (repeating vs. terminating) when conducting conversions and calculations.
  • Recognizing whether a decimal is repeating or terminating significantly impacts how we approach conversions and calculations. Repeating decimals will often require more complex fraction representation and careful handling during arithmetic operations to avoid errors due to their infinite nature. In contrast, terminating decimals simplify calculations since they have fixed precision. Understanding these differences can enhance accuracy in computations and deepen comprehension of number theory concepts, especially regarding rational versus irrational numbers.

Related terms

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

terminating decimal: A terminating decimal is a decimal that has a finite number of digits after the decimal point, as opposed to repeating indefinitely.

fraction: A fraction represents a part of a whole and is expressed as the quotient of two integers, typically written in the form 'a/b'.