Rational Numbers: Understanding 0.1 Repeating
In mathematics, rational numbers are a fundamental concept that represents a ratio of two integers. One example of a rational number that often raises questions is 0.1 repeating, also known as 0.111... . In this article, we will explore why 0.1 repeating is indeed a rational number.
What is a Rational Number?
A rational number is a number that can be expressed as the ratio of two integers, i.e., a fraction. It can be written in the form:
a/b
where a and b are integers, and b is non-zero. Examples of rational numbers include 3/4, 22/7, and -1/2.
Why is 0.1 Repeating a Rational Number?
At first glance, 0.1 repeating may not seem like a rational number because it appears to be an infinite, non-terminating decimal. However, we can prove that it is indeed a rational number by writing it as a fraction.
0.1 repeating = 1/9
How did we arrive at this fraction? Let's multiply 0.1 repeating by 9:
0.1 repeating × 9 = 0.9 + 0.09 + 0.009 + ...
= 0.99... (an infinite geometric series)
Since the decimal expansion of 0.9 is also an infinite, non-terminating sequence of 9s, we can rewrite the equation as:
0.1 repeating × 9 = 1
Dividing both sides by 9, we get:
0.1 repeating = 1/9
Conclusion
In conclusion, 0.1 repeating is a rational number because it can be written as a fraction, specifically 1/9. This example illustrates that rational numbers can have infinite, non-terminating decimal expansions, which can sometimes be counterintuitive. Understanding rational numbers is essential in mathematics, and recognizing 0.1 repeating as a rational number helps solidify our grasp of this fundamental concept.
