Editing Division by zero (section)

=== Floating-point arithmetic ===

In computing, most numerical calculations are done with [[floating-point arithmetic]], which since the 1980s has been standardized by the [[IEEE 754]] specification. In IEEE floating-point arithmetic, numbers are represented using a sign (positive or negative), a fixed-precision [[significand]] and an integer [[exponent]]. Numbers whose exponent is too large to represent instead "overflow" to positive or negative [[infinity]] (+∞ or −∞), while numbers whose exponent is too small to represent instead "[[Arithmetic underflow|underflow]]" to [[signed zero|positive or negative zero]] (+0 or −0). A [[NaN]] (not a number) value represents undefined results.

In IEEE arithmetic, division of 0/0 or ∞/∞ results in NaN, but otherwise division always produces a well-defined result. Dividing any non-zero number by positive zero (+0) results in an infinity of the same sign as the dividend. Dividing any non-zero number by [[negative zero]] (−0) results in an infinity of the opposite sign as the dividend. This definition preserves the sign of the result in case of [[arithmetic underflow]].<ref>{{citation|last=Cody|first=W. J.|title=Analysis of Proposals for the Floating-Point Standard|journal=Computer|date=March 1981 |volume=14|issue=3|pages=65|doi=10.1109/C-M.1981.220379|s2cid=9923085|quote=With appropriate care to be certain that the algebraic signs are not determined by rounding error, the affine mode preserves order relations while fixing up overflow. Thus, for example, the reciprocal of a negative number which underflows is still negative.}}</ref>

For example, using single-precision IEEE arithmetic, if {{nowrap|1=''x'' = −2<sup>−149</sup>}}, then ''x''/2 underflows to −0, and dividing 1 by this result produces 1/(''x''/2) = −∞. The exact result −2<sup>150</sup> is too large to represent as a single-precision number, so an infinity of the same sign is used instead to indicate overflow.