Is Full Equal to Empty?

Date: 06/15/99 at 13:03:06
From: Joseph
Subject: A weird question

Hi, Joseph again! I have a question I could not understand. Could you 
help me? thanks!

1/2 full = 1/2 empty

So when we multiply both sides of the equation by 1/2, won't we get 
full = empty?

Date: 06/15/99 at 17:39:53
From: Doctor Rick
Subject: Re: A weird question

Hi, Joseph.

Cute! This is just one of many examples of why we don't use English as 
the language of mathematics. "Half empty" does not mean "1/2 times 
empty." 

If we let full mean V (volume of a full glass) and empty mean 0 
(nothing in the glass), then you would have

   1/2 * V = 1/2 * 0

which is, of course, false (unless the glass can't hold anything - 
then V = 0).

What we really mean by "half empty" is halfway from full to empty, 
while "half full" means halfway from empty to full. If I were to 
translate the phrases into math, they would be

   half full:  0 + 1/2 * V
   half empty: V - 1/2 * V

These, of course, are equal.

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