Is 0.999… really equal to 1?

There are certain concepts in mathematics that are counter-intuitive. In this post, we will discuss one of these concepts — the elementary proof that 0.999… = 1.

The meaning of 0.999… is a tricky concept, and depends on what we allow a number to be.

For most mathematicians, 0.999… represents a series of numbers: 0.9, 0.99, 0.999, 0.9999, and so on

                                 

A common assumption is that numbers cannot be “infinitely close” together — they’re either the same, or they’re not.

If we assume infinitely small numbers don’t exist, we can show 0.999… = 1.

For non-math persons, you will probably disagree with the equality, but there are many elementary proofs that could show it.

Proof 1:      

¹/₉ = 0.111…

                  Multiplying both sides by 9 we have

1 = 0.999...

Proof 2:      

                   ¹/₃ = 0.333 

²/₃ = 0.666         

          ¹/₃ + ²/₃ = 0.333 + 0.666...

          ³/₃ = 0.999...

           But ³/₃ = 1, therefore, 1 = 0.999

Proof 3:  

          This is the most intuitive proof.

                    

Let k = 0.999...

Then 10k = 9.999...

10k - k = 9.999... - 0.999...

9k = 9

k = 1

And k = .999...

Hence, 0.999... = 1