Painting Half a Room Each Day

Date: 03/20/2002 at 21:32:03
From: Jen crowley
Subject: Painting half a room

If Bob paints half of a room one day and decides to keep painting half 
each day, why is the answer infinite?

Date: 03/20/2002 at 22:24:57
From: Doctor Twe
Subject: Re: Painting half a room

Hi - thanks for writing to Dr. Math.

Suppose the room's walls are a total of 64 square feet.

The 1st day, Bob paints 32 sq.ft., leaving 32 sq.ft. unpainted.
The 2nd day, Bob paints 16 sq.ft., leaving 16 sq.ft. unpainted.
The 3rd day, Bob paints 8 sq.ft., leaving 8 sq.ft. unpainted.
The 4th day, Bob paints 4 sq.ft., leaving 4 sq.ft. unpainted.
The 5th day, Bob paints 2 sq.ft., leaving 2 sq.ft. unpainted.
The 6th day, Bob paints 1 sq.ft., leaving 1 sq.ft. unpainted.
The 7th day, Bob paints 1/2 sq.ft., leaving 1/2 sq.ft. unpainted.
The 8th day, Bob paints 1/4 sq.ft., leaving 1/4 sq.ft. unpainted.
The 9th day, Bob paints 1/8 sq.ft., leaving 1/8 sq.ft. unpainted.
 :

And so on. Every day, he paints half of what's left, but leaves the 
other half unpainted - and half of any non-zero number is smaller, but 
not zero. So *in theory*, he never gets done.

In practical terms, however, there would come a time when either: 
(a) the area left unpainted would be so small we would not be able to 
detect it, or (b) the area left unpainted would be smaller than the 
tip of the bristles of the brush, and it would be impossible to cover 
only half of that area.

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum
  http://mathforum.org/dr.math/