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 Therefore: 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 all. 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 http://mathforum.org/dr.math/faq/faq.divideby0.html - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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