# floating-point notation

**floating-point notation** A representation of real numbers that enables both very small and very large numbers to be conveniently expressed. A floating-point number has the general form ±*m* × *R ^{e}* where

*m*is called the

*mantissa*,

*R*is the radix (or base) of the number system, and

*e*is the

*exponent*.

In the context of computing, a more common name for the mantissa is the

*significand*.

IEEE Standard 754 defines the most commonly used representations for real numbers on computers. It defines 32-bit (single precision) and 64-bit (double precision) as follows.

The first bit is a

*sign bit*, denoting the sign of the significand. This is followed by a fixed number of bits representing the exponent, which is in turn followed by another fixed number of bits representing the magnitude of the significand.

The exponent is often represented using

*excess-n notation*. This means that a number, called the

*characteristic*(or

*biased exponent*), is stored instead of the exponent itself. To derive the characteristic for a floating-point number from its exponent, the

*bias*(or

*excess factor*)

*n*is added to the exponent. For example, for an 8-bit characteristic, exponents in the range −128 to +127 are represented in excess-128 notation by characteristics in the range 0 to 255.

IEEE 754 specifies an 8-bit single-precision exponent, with a bias of 127, and an 11-bit double-precision exponent, with a bias of 1023. A nonzero floating-point number is

*normalized*if the leading digit in its significand is nonzero. Since the only possible nonzero digit in base 2 is 1, the leading nonzero digit in the significand need not be explicitly represented. This means that the 23-bit significand in the IEEE 754 single-precision floating-point representation effectively provides 24 bits of resolution, and the 52-bit double-precision significand provides 53 bits of resolution.

Although normalized floating-point numbers are most frequently used, unnormalized representations are also needed to represent numbers close to zero.

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**floating-point notation**