.9 Repeating as a Fraction
Introduction
The decimal .9 repeating, also known as .9~, is a non-terminating decimal that has been a subject of interest in mathematics. Many people have wondered how to convert this decimal into a fraction. In this article, we will explore how to convert .9 repeating as a fraction.
The Pattern of .9 Repeating
Before we dive into converting .9 repeating as a fraction, let's take a closer look at the pattern of this decimal. The decimal .9 repeating can be written as:
.9, .99, .999, .9999, ...
As you can see, the pattern consists of an infinite number of 9s. This pattern will be useful in helping us convert .9 repeating as a fraction.
Converting .9 Repeating as a Fraction
To convert .9 repeating as a fraction, we can use the following method:
Let x = .9~
Since x is equal to .9~, we can multiply both sides of the equation by 10 to get:
10x = 9.~
Subtracting x from both sides of the equation gives us:
9x = 9
Dividing both sides of the equation by 9 gives us:
x = 1
So, .9 repeating is equal to 1 in fractional form!
Proof
To prove that .9 repeating is equal to 1, we can use a geometric series. The sum of an infinite geometric series can be calculated using the formula:
a / (1 - r)
where a is the first term and r is the common ratio. In the case of .9 repeating, the first term is .9 and the common ratio is .1 (since each term is obtained by multiplying the previous term by .1).
Applying the formula, we get:
.9 / (1 - .1) = .9 / .9 = 1
This proves that .9 repeating is indeed equal to 1.
Conclusion
In conclusion, .9 repeating can be converted as a fraction, and its value is equal to 1. This may seem counterintuitive at first, but the mathematical proof is sound. Understanding the pattern of .9 repeating and using the correct method to convert it as a fraction can help us appreciate the beauty of mathematics.
