.99999 Repeating Equals One: A Mathematical Proof
One of the most counterintuitive and debated topics in mathematics is the concept of .99999 repeating equals one. Many people, including some mathematicians, have argued that .99999 repeating is less than one, or that it is a distinct number that is not equal to one. However, we will explore the mathematical proof that shows why .99999 repeating is indeed equal to one.
What is .99999 Repeating?
.99999 repeating is a non-terminating decimal that goes on indefinitely in a repeating pattern. It can be represented as:
.9 + .09 + .009 + .0009 + ...
The Mathematical Proof
One way to prove that .99999 repeating equals one is to use a simple algebraic trick. Let's assume that .99999 repeating is equal to a variable, x.
x = .9 + .09 + .009 + .0009 + ...
Now, let's multiply both sides of the equation by 10.
10x = 9 + .9 + .09 + .009 + ...
Subtracting the original equation from the new equation, we get:
9x = 9
Dividing both sides by 9, we get:
x = 1
Therefore, .99999 repeating is equal to one.
Another Proof Using Geometry
Another way to prove that .99999 repeating equals one is to use geometric concepts. Imagine a circle with a circumference of 1 unit.
Divide the circle into 10 equal parts, and shade one part. The shaded area represents .1 of the circle.
Now, divide each of the unshaded parts into 10 equal sub-parts, and shade one of the sub-parts. The newly shaded area represents .01 of the circle.
Continuing this process indefinitely, we can create an infinite series of shaded areas that represent .001, .0001, .00001, and so on.
The key observation is that the shaded areas will eventually cover the entire circle, implying that the sum of the infinite series is equal to one.
Conclusion
The concept of .99999 repeating equals one may seem counterintuitive at first, but the mathematical proof and geometric illustration demonstrate that it is indeed true. This concept has important implications in calculus, algebra, and other areas of mathematics.
In conclusion, .99999 repeating is not just a close approximation of one, but rather an equal representation of one.
