Considering floating-point numbers as a subset of the real numbers

• The real numbers are dense; between any two real numbers lies another. A floating-point number system is discrete; the gap between neighboring elements is determined by the precision and the elements' values.
• The real numbers are unbounded. Floating-point numbers are bounded according to the maximum exponent of a normal number.
• Real numbers may be found arbitrarily close to zero. The smallest nonzero floating-point value is the smallest subnormal number.
• The real numbers are fully precise. Floating-point numbers have a limited precision, so that results of calculations are usually just approximate. In the real number system, 1/3 has a well-defined, fully precise meaning. In binary floating-point arithmetic, the value 1/3 is represented by rounding it to a nearby representable number.
• Most real numbers require an infinite binary (or decimal) expansion. For example, , where the bar indicates a repeating decimal digit, and , where the ellipsis indicates an infinite, nonrepeating fraction. By definition, floating-point values can be expressed with a finite number of bits (or digits); generally, it takes as many decimal digits as binary bits to represent a binary fraction exactly in decimal.