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Are 0 and 1 Really Rational Numbers?

Date: 11/14/2001 at 11:27:38
From: Richard
Subject: Are 0 and 1 really rational numbers?

Rational number - A rational number is any number that can be 
expressed in the form a(numerator) over b(denominator) when a and 
b are rational numbers, but when b does not equal 0.

Now here's when the laws of rational numbers fall apart...

A) 0/1 = 0           B) 0/0 = 0 and 1?

Example A makes sense, but in example B, we get 0/0 = 0 and 1. But the 
law of rational numbers says that if the denominator is 0, then that 
fractional number IS NOT a rational number. But 0/0 equals 0 and 1.  

So now I am really confused. Are the laws of rational numbers saying 
that 0 and 1 are not rational numbers because x/0 cannot be a rational 
number? Or is the law of rational numbers incomplete because 0/0 can 
equal 1 and 0, which are rational numbers because 0/1 equals 0 and 1/1 
equals 1? Or 0/0 is not equal to 0 or 1 but maybe something else?  

I need your help!  

Thanky you for answering my questions on rational numbers.

Date: 11/14/2001 at 12:08:21
From: Doctor Peterson
Subject: Re: Are 0 and 1 really rational numbers?

Hi, Richard.

Actually, you've understated the "problem": 0/0 is an indeterminate 
expression that can be considered to be equal to EVERY number, not 
just 1 and 0. Consider:

    0 * n = 0  for every n


    0/0 = n  for every n

So are there no rational numbers?

No, you've got things sort of backward. A rational number is any 
number that CAN be expressed as a/b, with b non-zero and a and b 
integers. This doesn't mean that any number that can be expressed as 
a/0 is NOT rational; rather, an irrational number is one that CAN'T be 
expressed as a fraction a/b with b non-zero. And any expression of the 
form a/0 is not merely an irrational number; it is not a number at 

The real problem is simply that you CAN'T work with the expression 
0/0. This is indeterminate, as I said, so it can't really be said to 
be equal to anything. If you say that 0/0 = 0 and 0/0 = 1, then you 
imply that 0 = 1, and that is far worse than just saying that 0 and 1 
are irrational. All we have to do to correct your difficulty is agree 
not to write any equation involving 0/0.

See the Dr. Math FAQ:

   Dividing by 0   

- Doctor Peterson, The Math Forum   
Associated Topics:
High School Number Theory

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