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