.9999 Equals 1: A Mathematical Proof
One of the most debated topics in mathematics is whether .9999 (a non-terminating, repeating decimal) is equal to 1. While some argue that it is not, others claim that it is indeed equal to 1. In this article, we will provide a mathematical proof to settle this dispute once and for all.
The Problem with .9999
At first glance, it seems logical to assume that .9999 is not equal to 1. After all, .9999 has an infinite number of 9s, whereas 1 is a finite number. However, this intuition is misleading. The key to understanding why .9999 equals 1 lies in the concept of limits.
The Limit of .9999
Consider the sequence .9, .99, .999, .9999, ... . As we add more 9s to the end of the sequence, the value of the sequence approaches 1. In mathematical notation, we can write this as:
$\lim_{n\to\infty} \sum_{i=1}^n \frac{9}{10^i} = 1$
This equation states that as the number of terms in the sequence approaches infinity, the value of the sequence approaches 1.
Proof by Algebraic Manipulation
Another way to prove that .9999 equals 1 is through algebraic manipulation. Let's start with the equation:
$x = .9999...$
We can multiply both sides of the equation by 10 to get:
$10x = 9.9999...$
Subtracting the original equation from the new equation, we get:
$9x = 9$
Dividing both sides of the equation by 9, we arrive at:
$x = 1$
This proves that .9999 is indeed equal to 1.
Geometric Series
The .9999 sequence can also be viewed as a geometric series with first term .9 and common ratio .1. The formula for the sum of an infinite geometric series is:
$S = \frac{a}{1-r}$
where a is the first term and r is the common ratio. Plugging in the values, we get:
$S = \frac{.9}{1-.1} = 1$
Conclusion
Through the use of limits, algebraic manipulation, and geometric series, we have provided irrefutable proof that .9999 equals 1. This mathematical truth may challenge our initial intuition, but it is an essential concept to grasp in order to fully understand mathematics. So, the next time someone asks you whether .9999 equals 1, you can confidently say, "Yes, it does!"
