.9 Repeating as a Fraction
The decimal .9 repeating, also known as .9̄, is a non-terminating repeating decimal that has a fascinating property - it can be expressed as a simple fraction.
What is .9 Repeating?
.9 repeating, in its simplest form, is a decimal that goes on indefinitely in a repeating pattern of 9s: .9, .99, .999, .9999, and so on. This type of decimal is known as a repeating decimal or a recurrent decimal.
Converting .9 Repeating to a Fraction
So, how do we convert .9 repeating to a fraction? To do this, we can use a simple trick:
Let x = .9̄ (the repeating decimal)
Multiply both sides by 10:
10x = 9.9̄
Now, subtract x from both sides:
9x = 9
Divide both sides by 9:
x = 1
So, .9 repeating is equal to 1! This may seem surprising at first, but it's true - .9 repeating is equivalent to the fraction 1/1, which is simply 1.
Proof and Explanation
To understand why this works, let's dive deeper into the structure of repeating decimals.
A repeating decimal like .9̄ can be thought of as an infinite geometric series:
.9̄ = .9 + .09 + .009 + .0009 + ...
Using the formula for an infinite geometric series, we can rewrite this as:
.9̄ = .9 / (1 - .1)
Simplifying this expression, we get:
.9̄ = 1
Thus, we have shown that .9 repeating is indeed equal to 1.
Conclusion
In conclusion, .9 repeating is a fascinating mathematical object that can be expressed as a simple fraction - 1/1, or simply 1. This result may seem counterintuitive at first, but it's a beautiful example of the elegance and beauty of mathematics.
