Quote:
Originally Posted by antoniseb
I had the impression that dividing zero by zero is undefined, but dividing a non-zero number by zero puts you in the realm of the transfinite.
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No.
Division by zero is just plain old undefined. It doesn't matter what is in the numerator.
"Transfinite" generally refers to either a means of proof based on well-ordering of arbitrary sets (transfinite induction) or else to the cardinal or ordinal numbers. Even in the setting of cardinal and ordinal numbers you cannot divide by zero.
About the closes that you can come is to look at something like Y/X and consider the limit as X tends to zero. That may have meaning. It may grow without bound, in which case it is sometimes said to
tend to infinity. But the actualy expression Y/0 is meaningless.
The problem is that 1/x has the property that x * 1/x = 1. That is what division actually means. And there is no number that is a multiplicative inverse for 0. That is because 0*x = (0 + 0)*x = 0*x + 0*x and subtracting 0*x from both sides you get 0*x=0. So there is nothing that you can multiply times 0 and get 1.