Transfinite Numbers – Yes

The reason being that division by zero is not allowable for reasons very simply stated. Let a and b be any two numbers, and then consider the equation:

a x 0 = b x 0

If division by zero is allowable, you can then simply divide both sides of that equation by zero to get a=b for all a and b. Since a and b can be any two numbers, it would then follow that every number is equal to every other number! Which, of course, is nonsensical. For that reason alone, division by zero cannot be allowed.

You will sometimes hear it said that the result of dividing by zero is infinity, but for a mathematician the result of dividing by zero is simply undefined. Apart from anything else, the question would arise, “Which infinity?” Since the end of the nineteenth century, following a discovery by the German mathematician Georg Cantor, it has been known that there is more than one infinite number and that, in fact, there are an infinite number of infinite numbers! These infinite numbers are strange things, and they obey very different rules compared to ordinary finite numbers.

For example, if A an B are two infinite numbers, and B is the larger of the two, then A+B=B. On the face of it, if you subtracted B from both sides of that equation you could obtain the result that A (an infinite number) is equal to zero. As was the case with division by zero, that result is clearly nonsensical. For that reason, not only does division by an infinite number have to be ruled out, but so does subtraction of an infinite number!

The infinite number corresponding to the number of natural numbers is commonly denoted as N by mathematicians. Now you might think that if you took some subset of the natural numbers, say all the even numbers, 2, 4 , 6 , 8, and so on, then there would be fewer of even numbers than there are natural numbers – after all you are missing out every other number. But contrary to expectations and common sense, the number of even numbers is also N. So if you have a set of N numbers, you can take N numbers out of that set, and still be left with N numbers!

If all this is making your head spin, then it should be said that these transfinite numbers (as they are called) are really just the play thing of pure mathematicians, and that they have no real practical application.