Ratios Containing a Zero

Date: 07/03/2005 at 21:27:57
From: Jackie
Subject: Ratios with 0 in them

What does the ratio 1:0 mean?  What does the ratio 0:1 mean?  What 
does the ratio 0:0 mean?  Are they meaningless like dividing by 0?

I am having trouble understanding whether they exist or are 
meaningless and disallowed?  Please help.

1:2 means that, for example, for every 1 cm on a map, it represents 
2 cm.  Therefore, a ratio of 1:0 means that for every 1cm on a map, it 
represents nothing.  Also, a ratio of 0:1 means that for every 0 cm on 
a map, it represents 1 cm.  Both of those ratios are impossible.  But 
are they meaningless?

Date: 07/03/2005 at 23:28:48
From: Doctor Peterson
Subject: Re: Ratios with 0 in them

Hi, Jackie.

I think you've got the right idea.  I wouldn't actually say that these
ratios are meaningless--they do mean something!--but for many purposes 
such a ratio means that something is useless. The ratios themselves 
are not meaningless, but in a specific application they may be 
invalid.

The ratio 0:1 means that the first item will be zero no matter what
the second is; for example, your map would have no size at all.  So 
you will never find a map with that scale.  Similarly, a ratio of 1:0 
would mean that the map showed nothing.

Ratios often are turned into fractions; 0:1 becomes 0/1, which is the
perfectly respectable value 0, while 1:0 becomes 1/0, which is 
undefined (though in some situations you can think of it as 
"infinite").  One benefit of ratios is that both 0:1 and 1:0 are
meaningful, whereas 1/0 is not defined.

If, for example, they represent the ratio of blue to white paint in a
mixture, the first means pure white, and the second means pure blue.
Both ratios are perfectly fine in that setting!

If you have any further questions, feel free to write back.


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

Date: 07/11/2005 at 01:42:53
From: Jackie
Subject: Thank you (Ratios with 0 in them)

Thanks for making me understand where those ratios may be used.