The above explanation may be too abstract and technical for many purposes, but if one assumes the existence and properties of the rational numbers, as is commonly done in elementary mathematics, the "reason" that division by zero is not allowed is hidden from view. Nevertheless, a (non-rigorous) justification can be given in this setting.
A compelling reason for not allowing division by zero is that, if it were allowed, many absurd results (i.e., fallacies) would arise. When working with numerical quantities it is easy to determine when an illegal attempt to divide by zero is being made. For example, consider the following computation.
In abstract algebra, the integers, the rational numbers, the real numbers, and the complex numbers can be abstracted to more general algebraic structures, such as a commutative ring, which is a mathematical structure where addition, subtraction, and multiplication behave as they do in the more familiar number systems, but division may not be defined. Adjoining a multiplicative inverses to a commutative ring is called localization. However, the localization of every commutative ring at zero is the trivial ring, where 0 = 1 \displaystyle 0=1 , so nontrivial commutative rings do not have inverses at zero, and thus division by zero is undefined for nontrivial commutative rings.
Nevertheless, any number system that forms a commutative ring can be extended to a seldom used structure called a wheel in which division by zero is always possible. However, the resulting mathematical structure is no longer a commutative ring, as multiplication no longer distributes over addition. Furthermore, in a wheel, division of an element by itself no longer results in the multiplicative identity element 1 \displaystyle 1 , and if the original system was an integral domain, the multiplication in the wheel no longer results in a cancellative semigroup.
The concepts applied to standard arithmetic are similar to those in more general algebraic structures, such as rings and fields. In a field, every nonzero element is invertible under multiplication; as above, division poses problems only when attempting to divide by zero. This is likewise true in a skew field (which for this reason is called a division ring). However, in other rings, division by nonzero elements may also pose problems. For example, the ring Z/6Z of integers mod 6. The meaning of the expression 2 2 \textstyle \frac 22 should be the solution x of the equation 2 x = 2 \displaystyle 2x=2 . But in the ring Z/6Z, 2 is a zero divisor. This equation has two distinct solutions, x = 1 and x = 4, so the expression 2 2 \textstyle \frac 22 is undefined.
Integer division by zero is usually handled differently from floating point since there is no integer representation for the result. Some processors generate an exception when an attempt is made to divide an integer by zero, although others will simply continue and generate an incorrect result for the division. The result depends on how division is implemented, and can either be zero, or sometimes the largest possible integer.
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