A repeating decimal is a decimal number in which, after a certain point, a particular digit or sequence of digits repeats itself indefinitely. Another name for a repeating decimal is a “recurring” decimal. The number of digits that repeat is called the period of the repeating decimal.
Key Facts
- Definition: A repeating decimal is a decimal number in which, after a certain point, a particular digit or sequence of digits repeats itself indefinitely.
- Period: The number of digits that repeat in a repeating decimal is called the period. It is the smallest number of digits that repeat in the decimal representation.
- Representation: Repeating decimals are often represented by placing dots (…) over the first and last digits of the repeating pattern or by a line over the pattern.
- Examples: Some examples of repeating decimals include:
- 1/3 = 0.333… (the digit 3 repeats forever).
- 1/7 = 0.142857142857… (the sequence “142857” repeats forever).
- 77/600 = 0.128333… (the digit 3 repeats forever).
Period of a Repeating Decimal
The period of a repeating decimal is the smallest number of digits that repeat in the decimal representation. For example, the period of the repeating decimal 0.333… is 1, since the digit 3 repeats forever. The period of the repeating decimal 0.142857142857… is 6, since the sequence “142857” repeats forever.
Representation of Repeating Decimals
Repeating decimals are often represented by placing dots (…) over the first and last digits of the repeating pattern or by a line over the pattern. For example, the repeating decimal 0.333… can be represented as 0.3̅ or 0.3. The repeating decimal 0.142857142857… can be represented as 0.142857̅ or 0.142857.
Examples of Repeating Decimals
Some examples of repeating decimals include:
- 1/3 = 0.333… (the digit 3 repeats forever)
- 1/7 = 0.142857142857… (the sequence “142857” repeats forever)
- 77/600 = 0.128333… (the digit 3 repeats forever)
Conclusion
Repeating decimals are a common occurrence in mathematics. They can be used to represent rational numbers that cannot be expressed as a terminating decimal.
Sources:
- https://www.learnalberta.ca/content/memg/division03/repeating%20decimal/index.html
- https://www.merriam-webster.com/dictionary/repeating%20decimal
- https://www.mathsisfun.com/definitions/recurring-decimal.html
FAQs
What is a repeating decimal?
A repeating decimal is a decimal number in which, after a certain point, a particular digit or sequence of digits repeats itself indefinitely.
What is the period of a repeating decimal?
The period of a repeating decimal is the smallest number of digits that repeat in the decimal representation.
How do you represent a repeating decimal?
Repeating decimals are often represented by placing dots (…) over the first and last digits of the repeating pattern or by a line over the pattern.
What are some examples of repeating decimals?
Some examples of repeating decimals include:
- 1/3 = 0.333… (the digit 3 repeats forever)
- 1/7 = 0.142857142857… (the sequence “142857” repeats forever)
- 77/600 = 0.128333… (the digit 3 repeats forever)
Why do repeating decimals occur?
Repeating decimals occur because some rational numbers cannot be expressed as a terminating decimal. This is because the denominator of the fraction that represents the rational number has prime factors other than 2 or 5.
Are all repeating decimals rational numbers?
Yes, all repeating decimals are rational numbers. This is because they can be expressed as a fraction of two integers.
Can repeating decimals be converted to fractions?
Yes, repeating decimals can be converted to fractions. There are a few different methods for doing this, but one common method is to use long division.
What are some applications of repeating decimals?
Repeating decimals are used in a variety of applications, including:
- Representing rational numbers that cannot be expressed as a terminating decimal
- Performing calculations with rational numbers
- Approximating irrational numbers