Proof That .99999… Is Equal To 1

Interesting Mathematics

I will start this article off with a statement. Point 9 repeating (aka .999…) is equal to the number one. No, it’s not just really close to one so we just round up, it is another way of writing the number one and in this article, I will show some proofs of this.

Contrary to what many people believe, an infinite number can be represented as a finite number and vice versa.

Different Ways to Represent the Number 1

(a) 2/2 = 1
(b) 1³ = 1
(c) 4⁰ = 1
(d) 3/3 = 1
(e) .999... = 1

The number ‘1’ can be written as 2/2 or 1³² or 4⁰ or 10/10 or .999… (point 9 repeating). All of those are different ways to write the number 1, they are all equivalent to the number 1. Another way you might see the number represented is 3/3, which brings us to our first proof to show that .9999…. = 1 is true.

Proof #1: Proof 1 = 0.999… (Using Fractions)

Let's start off with the number 1. 
Step (1): 1 = 3/3 ← This is true
Step (2): 3/3 = 3(1/3) ← This is true
Step (3): 3(1/3) = (1/3) + (1/3) + (1/3) ← This is true
Step (4): (1/3) = .33333… ← This is true 
Step (5): (1/3) + (1/3) + (1/3) = .333… + .333… + .333…
Step (6): .333… + .333… + .333… = .999…

From the above proof, you can see that we started with the number 1 and ended with the number .999….

Note: 0.333… is point 3 repeating the number never ends and .999… is point 9 repeating, these numbers don’t have an end, and so I can’t write an infinite number in finite space in that form. But you can understand the representation of both numbers.

Proof #2: Proof 1 = 0.999… (Using Algebra)

Step A: Let X = .99999…..
Step B: If X = .99999….. Then 10X = 9.99999…..
Step C: 10X - X = 9X and this implies 9.99999….. - .99999….. = 9
Step D: Therefore 9X = 9 and this implies X = 9/9 = 1
Step E: So, X = .99999….. at the beginning and X = 1 at the end of the proof. This implies .99999…… = 1

Looking at the above proof, X = .99999….. at the beginning and X = 1 at the end of the proof. This implies .99999…… = 1. Let’s do another proof to show this is true.