.99999 = 1 Proof
The equality .99999 = 1 is a mathematical concept that has been debated by many people. Some argue that .99999 is not equal to 1, while others claim that it is. In this article, we will explore the proof that .99999 is indeed equal to 1.
The Intuitive Argument
At first glance, it may seem that .99999 is not equal to 1. After all, the digits 9 seem to go on forever, and it's hard to imagine that they can be equal to a finite number like 1. However, this intuition is misleading.
The Algebraic Proof
One way to prove that .99999 = 1 is to use algebra. Let's define .99999 as an infinite geometric series:
.99999 = 9/10 + 9/100 + 9/1000 + ...
We can then multiply both sides of the equation by 10 to get:
9.9999 = 9 + 9/10 + 9/100 + ...
Now, subtract the original equation from this new equation to get:
9.9999 - .99999 = 9
which simplifies to:
9 = 9
This shows that .99999 = 1, since the difference between the two numbers is 0.
The Geometric Series Proof
Another way to prove that .99999 = 1 is to use the formula for an infinite geometric series:
a / (1 - r)
where a is the first term, and r is the common ratio. In this case, a = 9/10 and r = 1/10, so we get:
.99999 = 9/10 / (1 - 1/10)
= 9/10 / (9/10)
= 9/9
= 1
The Limit Proof
We can also use the concept of limits to prove that .99999 = 1. Let's define a sequence:
a_n = .9 + .09 + .009 + ... + .000...9 (n times)
We can show that the limit of this sequence as n approaches infinity is 1:
lim (a_n) = 1
This means that as we add more and more terms to the sequence, the sum gets arbitrarily close to 1. Therefore, .99999 = 1.
Conclusion
In conclusion, we have shown that .99999 = 1 using three different methods: algebra, geometric series, and limits. This equality is a fundamental concept in mathematics, and it has many important implications for calculus, analysis, and other areas of mathematics.
