Exploring the Rationality of Repeating Decimals- Unraveling the Mystery Behind Infinite Sequences

Is a repeating decimal a rational number? This question often arises in the study of mathematics, particularly in the realm of number theory. To understand the answer, we must delve into the definitions of rational numbers and repeating decimals, and explore the properties that link them together.

Rational numbers are numbers that can be expressed as a fraction of two integers, where the denominator is not zero. This means that every rational number can be written in the form p/q, where p and q are integers. On the other hand, a repeating decimal is a decimal representation of a number that has a repeating pattern of digits after the decimal point. For instance, 0.3333… (where the 3s repeat indefinitely) is a repeating decimal.

The key to determining whether a repeating decimal is a rational number lies in the fact that it can be expressed as a fraction. To illustrate this, let’s consider the repeating decimal 0.3333… (where the 3s repeat indefinitely). We can represent this decimal as an infinite geometric series:

0.3 + 0.03 + 0.003 + 0.0003 + …

This series can be expressed as a sum of an infinite geometric series with a common ratio of 0.1:

S = 0.3 / (1 – 0.1)

Simplifying the expression, we get:

S = 0.3 / 0.9

S = 1/3

Therefore, the repeating decimal 0.3333… can be expressed as the fraction 1/3, which is a rational number.

In general, any repeating decimal can be expressed as a fraction by using the same method. For a repeating decimal with a repeating block of n digits, we can represent it as an infinite geometric series with a common ratio of 1/10^n. By finding the sum of this series, we can obtain the corresponding fraction, proving that the repeating decimal is indeed a rational number.

In conclusion, the answer to the question “Is a repeating decimal a rational number?” is a resounding yes. The ability to express repeating decimals as fractions demonstrates the fascinating connections between different mathematical concepts and their underlying properties. Understanding these connections can deepen our appreciation for the beauty and consistency of mathematics.