Problem of the week #4

October 24th, 2008 | Categories: general math, Problem of the week | Tags:

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

  1. jOkAmE
    October 25th, 2008 at 15:58
    Reply | Quote | #1

    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.

  2. HP
    October 26th, 2008 at 00:58
    Reply | Quote | #2

    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

  3. hp
    October 26th, 2008 at 06:50
    Reply | Quote | #3

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

  4. Mike Croucher
    October 26th, 2008 at 12:14
    Reply | Quote | #4

    Hi HP

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

  5. October 27th, 2008 at 22:16
    Reply | Quote | #5

    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

  6. October 27th, 2008 at 23:14
    Reply | Quote | #6

    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.

  7. wayne
    November 2nd, 2008 at 13:29
    Reply | Quote | #7

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

  8. David P
    November 7th, 2008 at 13:15
    Reply | Quote | #8

    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.