.99999 Repeating as a Fraction
The infinite repeating decimal .99999 is a mathematical curiosity that has puzzled many people for centuries. While it may seem like a strange and unusual number, .99999 repeating has a fascinating property: it can be expressed as a simple fraction.
What is .99999 Repeating?
.99999 repeating is a decimal that goes on forever in a repeating pattern of 9s. It can be written as:
.99999 = 0.9 + 0.09 + 0.009 + 0.0009 + ...
This sequence of 9s continues indefinitely, with each term getting smaller and smaller.
Converting .99999 to a Fraction
So, how can we express .99999 as a fraction? The answer lies in the concept of geometric series.
A geometric series is a type of infinite series that has a constant ratio between each term. In this case, the ratio is 0.1 (or 1/10). We can write the series as:
.99999 = 0.9 + 0.09 + 0.009 + 0.0009 + ...
= 0.9(1 + 0.1 + 0.01 + 0.001 + ...)
Using the formula for an infinite geometric series, we can simplify this expression to:
.99999 = 0.9 / (1 - 0.1)
= 0.9 / 0.9
= 1
So, .99999 repeating is equal to 1!
Proof and Implications
This result may seem counterintuitive, but it can be proven mathematically. In fact, this result is a fundamental property of real numbers and has far-reaching implications in mathematics and physics.
For example, this result shows that the rational number 1 can be expressed as an infinite, non-terminating decimal. This has implications for our understanding of rational and irrational numbers, as well as the nature of infinity itself.
Conclusion
.99999 repeating is a fascinating mathematical curiosity that has a simple and elegant solution. By using the concept of geometric series, we can show that .99999 is equal to 1, a result that has profound implications for our understanding of mathematics and the world around us.
