Definition of a Repeating Decimal
A repeating decimal, also known as a recurring decimal, is a decimal representation of a number where one or more digits endlessly repeat after a certain point. This repeating sequence of digits is called the repetend. For example, in the decimal 0.333..., the digit '3' is the repetend, repeating infinitely.
How to Identify and Denote Them
You can identify a repeating decimal by observing a pattern of digits that reoccurs indefinitely. To denote a repeating decimal concisely, a bar (vinculum) is placed over the repeating digits. For instance, 0.333... is written as 0.3̄, and 0.121212... is written as 0.12̄. If the repetition doesn't start immediately after the decimal point, the bar only covers the repeating block, like 0.1666... being 0.16̄.
Relationship to Rational Numbers
Every repeating decimal represents a rational number, which is any number that can be expressed as a fraction p/q, where p and q are integers and q is not zero. Conversely, every rational number, when expressed as a decimal, will either be a terminating decimal (like 1/2 = 0.5) or a repeating decimal (like 1/3 = 0.333...). For example, the fraction 2/7 results in the repeating decimal 0.285714285714..., which can be written as 0.285714̄.
Importance in Number Systems
Repeating decimals are crucial for understanding the distinction between rational and irrational numbers. They demonstrate that division of integers can result in infinite, yet predictable, decimal expansions. This concept is fundamental in number theory and practical applications where precise fractional values need to be represented in decimal form, providing a complete picture of the rational number system.