Converting Fractions to Decimals

Date: 01/08/97 at 21:36:32
From: Susan Fredrickson
Subject: Question

Dear Dr. Math,

Today in math class we were working on rational and irrational numbers 
and I asked the teacher what the square root of 2.25 is. Well, if 
you punch it into your calculater, it says 1.5. That's not a rational 
number because it's not an integer. You could do 1.5 over one and that 
wouldn't be a rational number, but if you doubled both the top and the 
bottom, it would be 3 over 2 and that would be a rational number. 
Doubling them is kind of like the opposite of reducing and I was 
wondering whether that had a name.  

Also, the equation 3/3 = 1 is correct but I don't see how that can be 
because 1/3 =.3333 and if you multiply 3.3 by three times, it won't be 
one.  It will be 9.999999.....
 
Thank you for your time.

Date: 01/11/97 at 14:44:18
From: Doctor Gerald
Subject: Re: Question

Susan,

Good questions - you are really thinking!   First, remember that a 
rational number is one that CAN be written as one integer over 
another.  So your argument that doubling 1.5 to get 3 over 2 shows 
that it really is rational. 

Both doubling and reducing have the same general name; it's called 
finding EQUIVALENT fractions for rational numbers.  You can multiply 
or divide the top and bottom of any fraction by the same number and 
get an equivalent fraction.  When you divide, you are reducing; when 
you multiply, you are doubling or tripling or whatever. 

Any fraction has countless equivalent fractions. For example:

  .15/.1 = .3/.2 = 1.5/1 = 3/2 = 6/4 = 12/8 = 60/40 = 300/200 = .... 

Can you write others that are equivalent to 1.5?

Your second question gets into something that mathematicians call 
"limits."  You are very insightful to recognize that: 

  3(1/3) = 3(.3333...) = .9999... = 1

.9999... is equal to 1 because no matter how small a difference 
between .9999... and 1 you ask for, I can write enough 9s to get 
within that difference.  So suppose you want it within .00001 of 1.  
I can write .99999.  This can go on and on.

Here is another way of looking at it.  Let's say

n = .9999....

This means that 10 x n = 9.9999....

Now subtract 10n - n = 9.9999... - .9999...
So                9n = 9.0
                   n = 1

This works because that 9s keep repeating and .9999...  - .9999... = 0

You can do this with any repeating decimal to find the fraction that 
it is equivalent to.  For example, what fraction is equivalent to 
2.161616...?

Let    n =   2.161616...
    100n = 216.161616...  (Notice that this time you multiply by 100.          
                           Why?)
100n - n = 216.161616...  - 2.161616... (Subtract)
     99n = 214
       n = 214/99

Plug 214/99 into your calculator and see what you get. Calculators 
can go only one way, from fractions to decimals. You have to figure 
out how to go from decimals to fractions.

-Doctor Gerald,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/