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I know that division by zero is undefined and is also not rational, but I am not sure whether this means it's irrational because it is undefined.

Can anyone clarify?

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Irrational means a number that isn't rational. Division by zero does not give you a number. –  David Mitra Aug 4 '12 at 0:01
(In the mathematical sense,) "irrational" applies only to numbers. Division by zero does not result in a number. Therefore, the term "irrational" does not apply. –  Gerry Myerson Aug 4 '12 at 0:01
$\dfrac{\bullet}{0}$ is not rational, is not irrational, it's innumber (if there's ever such a thing). –  user2468 Aug 4 '12 at 2:08
Your calculator knows, or at least mine does. If I try to divide $45$ by $0$, my calculator shows an E in the display, and refuses to do anything until I reset it. –  André Nicolas Aug 4 '12 at 2:47

2 Answers 2

up vote 6 down vote accepted

An irrational number is a real number that is not rational. Dividing by zero doesn't give you a number at all. Calling irrationals "not rationals" oversimplifies what's going on. Is $i$ rational, for example? Cardinal numbers? You have to be very careful with how you apply definitions.

Irrationals aren't merely "not rational", they're real numbers that aren't rational. Since $\frac{1}{0}$ doesn't evaluate to a real number (or any kind of number at all, if you're working in $\mathbb{R}$), it's neither rational nor irrational. It's non-existent.

See this interfaith description for more information. There are things that aren't in the rationals or irrationals (all reals are one or the other though, so that diagram is somewhat misleading).

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Mathematicians have come to the sad conclusion (well, not that sad), after long years of research and industrial ammounts of coffee, that an expression of the form $\,a/0\,$ cannot be properly, logically and soundly defined within the commonly accepted boundaries of mathematical logic in usual mathematics...and I remark usual because there are some non-standard subjects in mathematics that could possibly deal with, at least, some cases of this problem but I don't know about this.

The reason division by zero is undefined is that if we had some definite real number $\,x\,$ s.t. $\,a/0=x\Longrightarrow a=0\cdot x =0\,$ , rendering $\,a=0\,$...but then any real number $\,x\,$ works! As we want functions to be single-valued (here, within this basic context. In complex variable functions they may not be so and there're lots of fun there with this stuff, but it doesn't belong to basic mathematics) we must keep division by zero out of the game.

You we already explained why isn't it proper to call such an expression "irrational", and instead of that I propose the following name taken from my country's high school lingo: division by zero is a "MLE"=a meaningless expression.

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Mathematical logic in itself has no problem with assigning some meaning to "$a/0$". It is the rules of algebra (or of arithmetic, depending on how you slice the subjects) that will necessarily be violated. –  Henning Makholm Aug 4 '12 at 2:19
The projective line is non-standard mathematics?! –  Hurkyl Aug 4 '12 at 3:26
Well, it definitely is not basic mathematics, but anyway: I can't remember anything equivalent to "division by zero" in basic standard mathematics. There certainly exists an infinite point there, but I can't see how is that parallel even remotely to the above. –  DonAntonio Aug 4 '12 at 10:59

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