Division by Zero

Division by zero is the problematic special case in mathematics where the divisor in a fraction is zero. Under the ordinary definition of a quotient, in which a divided by b equals c only if c times b equals a, the expression a divided by zero has no sensible value, because no number times zero can produce a nonzero dividend, so the ordinary arithmetic of the real numbers and other structures called fields leaves division by zero undefined, and zero divided by zero undefined as well. In calculus, when a function can be written as a fraction whose denominator tends to zero, the function value tends to infinity, a type of mathematical singularity, as with the reciprocal function one over x as x tends to zero; when both numerator and denominator tend to zero at once the limit is called an indeterminate form because its value depends on the specific functions involved. Some alternative number systems instead define a result for division by zero, for instance by introducing an explicit point at infinity or a signed infinity, and in computing an attempt to divide by zero may return an infinite value, a special not-a-number value, or cause the program to crash depending on context.