by Bennett Kalafut

Created on: January 15, 2008   Last Updated: July 31, 2009

Division by zero is impossible in ordinary, everyday number systems such as the rational numbers (whole number fractions), algebraic numbers (solutions to polynomials with whole number coefficients) or real numbers (numbers on the number line). If it were possible, zero would be equal to one, moreover, all numbers would be zero. This contradicts the definition of these number systems, and moreover renders them useless for computation.

It is somewhat astonishing that this question is being asked, and that several hundred voters somehow disagree with trained mathematicians and with logic itself. More astonishing still are the excuses given by well over thirty Helium.com writers for this position. The worst of them are nonsequiturs, bordering on glossolalia, about mirrors or the distribution of wealth. The best make an appeal to "infinity" which I will address at the end of this article. In between are many that indicate severe problems in the way we teach mathematics: they're touchy-feely arguments about physical objects or the words "something" and "nothing" as though division of numbers is a physical process or a matter of semantics. Although manipulation of cubes, strings, and other objects is used to teach math to schoolchildren, numbers are not physical objects or flimsy metaphors. Talk arguments about pieces of pie or pseudophilosophical reflections on the nature of nothing don't help us to understand numbers. When we must address a subtle question-and from Everyman's perspective division by zero is a subtle question even if it is trivial for the mathematically trained-we must turn to the rigorous definition of the numbers and operations in question: a set of axioms.

I prove below that the ability to divide by zero would imply 1=0 for rational, real, and algebraic numbers by proving a more general result for algebraic structures called fields. The proof is elementary in nature and should be understandable even by readers for whom this is the first exposure to formal mathematics. Technical limitations of Helium.com make notation somewhat awkward; this article will be updated if and when Helium upgrades its back-end to allow use of a better character set such as Unicode.

Rational, reals, and algebraic numbers are but familiar examples of a formal algebraic structure known as a "field", a field being a set of numbers F and two operations + and * satisfying what we usually expect from addition and multiplication, defined by the following axioms,

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