Can you prove 0.bar9 = 1?

2 Answers

Yes, 0.999999…=1

But note that the above holds true only if the 9's are recurring infinitely (which cannot be well-defined with only basic algebra concepts)

Explanation:

Here is one non-rigorous proof.

Consider x such that
x=0.999999…

Multiply both sides by 10:
10x=9.999999…

Subtract the first expression from the second expression:
10x-x=9.999999…-0.999999…
9x=9
x=1

Thus, x=0.999999…=1.

To make it more rigorous, instead of saying 1=0.999999… where 9 is recurring, we should say 0.999999…, where the number of 9's tends to oo, converges to 1. However, this would involve the concepts of limits and convergence, something beyond algebra.

Jan 26, 2018

Yes, 0.bar9=1

Explanation:

Here is one of the proofs:

1/3=0.bar3

1/3*color(blue)3=0.bar3*color(blue)3

3/3=0.bar9

1=0.bar9

Since all of the algebraic manipulations was done correctly, the statement holds true.