Multiple Solutions

Date: 02/08/2002 at 09:27:43
From: Ryan Morris
Subject: Why?

Why is there more than one way of doing one type of math problem?

Date: 02/08/2002 at 10:53:40
From: Doctor Ian
Subject: Re: Why?

Hi Ryan,

That's an interesting question!

In one sense, math is the search for easier ways to solve certain 
kinds of problems. For example, suppose you're in a library, and you 
want to know how many books it contains.  

There is a simple solution: Count them, one by one. But that takes 
time, and if you lose track in the middle, you have to start all over 
again. Is there a better way?

There is. You can count the number of books on a few shelves, and find 
the average number of books per shelf. Then you can multiply that by 
the number of shelves. You lose a little accuracy (your answer won't 
be _exactly_ right, but the more shelves you use in finding an 
average, the closer your answer will be to the truth); but you gain 
quite a lot of speed. But if there are lots of shelves, this can still 
be tedious, especially if you need only an approximate answer. Is 
there a _better_ way? 

There is. You can estimate the volume of a single book, compute the 
volume of the space taken up by the books, and divide the latter by 
the former. And you can improve that solution by correcting for the 
fact that some of the volume of any bookcase is left unfilled; and so 
forth, and so on. 

Now, when you come up with a _new_ way to solve the problem, what 
happens to all the _old_ ways? Do they just disappear?  

I hope this helps. Write back if you'd like to talk about this some 
more, or if you have any other questions. 

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