## Problem of the week #4

I haven’t done a ‘problem of the week’ for a while so I thought I would throw a fun one out there to see what happens. Prove (or otherwise) that 0.9 recurring (that is 0.999999999999…… etc) is equal to one.

**Update:** Several solutions have been posted in the comments section

Let x = 0.9999999999…, so 10*x = 9.99999999999999. Then, 10x – x = 9x = 9.9999999…-0.99999999… = 9.

Thus, if 9x=9, x=1. Q.E.D.

Hello,

I don’t know how to write this in latex, so bear with me.

First, Notice that .9999999999999999999 = 9*10^0 – 9 + 9*10^-1 + 9*10^-2 + 9*10^-3 ……………………

thus, .99999999999999999…………. = sum(9/10^i, for i =0 to infinity) – 9.

So we have an geometric series, so we get 9/(1-1/10) – 9 = 10 – 9 = 1

so .9999999999999999999999999999999 is 1

Weird, i posted my proof on here, but it disappeared.

Hi HP

Sorry about that – I just didn’t get around to approving all the posts. Fixed now :)

This is a specific case of a more general one —

period of the repeating decimal representation —I posted here http://problemasteoremas.wordpress.com/2008/09/20/numeros-racionais-exercicio-sobre-dizimas-periodicas-e-serie-geometrica/ on Sept, 20th.0.99999… = 9 / (10^1-1) = 9 /9 = 1

I used to teach a math 101 course and I’d show them two proofs for this. The first was the first one presented here, by jOkAmE.

The other way is to consider the distance between 0.999… and 1; say d = |1 – 0.999…|. How small is d? Well, it has to be smaller than 1/10 because 0.999… is closer to 1 than 0.9 is, and 0.9 is distance 1/10 from 1. It has to be smaller than 1/100 as well, since 0.999… is closer to 0.99 which is 1/100 from 1. Continuing like this, you have that the distance between 0.999… and 1 is smaller than any positive number you pick, so the distance between them must be zero, so they must be equal.

This is a little bit hand-wavy, I guess, but you get the idea.

i remembered when i was a student in a elementary school , i used this tricky question to disturb the whole classroom , :)

I have also seen people say: 1/9 = 0.1111111… and 2/9 = 0.22222222…. and 3/9 = 0.33333…. and so forth. Thus 9/9 = 0.9999999…. and there’s your proof.