A problem that makes my head spin

[piano guy]

     I was thinking the other day, and I came up with the craziest musical problem... It deals largely with math, so I decided to post it here instead of the Music Theory board...

     Here's the problem: How many dots would one have to add to a quarter note to make it equal a half note? Seems like a simple question at first, but then you think about it... A dotted quarter note = a quarter note + an eighth note. A quarter note with two dots = a quarter note + an eighth note + a sixteenth note, and if you keep adding dots, the value added to the previous note continues to half, so, mathematically, the question is essentially this:

     If we continue this pattern: 1+1/2+1/4+1/8, etc., How many additions are we going to have to make to the original 1 to make the answer equal to 2? I guess the question is really "WILL it ever equal 2?"

     Math people help me out here! This one makes my head spin!

[steveie986]

1+1/2+1/4+1/8... = 2

The sum of the series 1/(2^n) approaches 1 as n approaches infinity.

[piano guy]

     ...so what you're basically saying is that, musically, it would take an infinite number of dots to make a quarter note equal a half note, and mathematically, it would take an infinite number of additions to the original 1 in the pattern for it to equal 2?

[steveie986]

You got it.

[invictious]

Damn, screw it, steveie beat me to it

You will never ever get there, just like 0.9999999999999999999 =/= 1

*shoots steveie*

[stevie]

*shoots steveie*

shoot him twice, on behalf of me.

[pianolearner]

Shoot him 0.999...  times

[pianistimo]

xenakis?  i'd just make it simple and write the half note.

[prometheus]

Shoot him 0.999...  times