A repeating decimal is a decimal number in which one or more digits in the decimal part repeat infinitely. This pattern of repeating digits is a characteristic of certain fractions when expressed as a decimal.
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Repeating decimals are a result of dividing an integer by another integer, where the denominator cannot be expressed as a product of only 2s and 5s.
The repeating pattern in a repeating decimal is called the 'repeating block' or 'repetend'.
Repeating decimals can be converted to fractions using a specific formula: $\frac{a}{10^n - 1}$, where 'a' is the repeating block and 'n' is the number of digits in the repeating block.
Repeating decimals are considered rational numbers because they can be expressed as a ratio of two integers.
Irrational numbers, such as $\pi$ and $\sqrt{2}$, cannot be expressed as a repeating decimal or a ratio of two integers.
Review Questions
Explain how repeating decimals are related to fractions and rational numbers.
Repeating decimals are a result of dividing an integer by another integer, where the denominator cannot be expressed as a product of only 2s and 5s. This means that repeating decimals can be converted to fractions using a specific formula: $\frac{a}{10^n - 1}$, where 'a' is the repeating block and 'n' is the number of digits in the repeating block. Since repeating decimals can be expressed as a ratio of two integers, they are considered rational numbers.
Describe the difference between repeating decimals and terminating decimals, and how they relate to rational and irrational numbers.
Terminating decimals are decimal numbers that have a finite number of digits and do not repeat infinitely, whereas repeating decimals have a pattern of digits that repeat infinitely. Terminating decimals can be expressed as fractions with denominators that are powers of 10, and they are also considered rational numbers. In contrast, irrational numbers, such as $\pi$ and $\sqrt{2}$, cannot be expressed as a repeating decimal or a ratio of two integers, and therefore are not rational numbers.
Analyze the significance of repeating decimals in the context of decimal operations and the relationship between decimals and fractions.
Repeating decimals play a crucial role in understanding decimal operations and the relationship between decimals and fractions. When performing arithmetic operations with repeating decimals, it is important to recognize the repeating pattern and apply appropriate strategies, such as converting the repeating decimal to a fraction. This understanding helps in simplifying calculations, comparing decimal values, and converting between decimal and fractional representations. The connection between repeating decimals and rational numbers also highlights the fundamental link between decimals and fractions, which is a key concept in pre-algebra and algebra.