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Floating-point representation is similar in concept to scientific notation. Logically, a floating-point number consists of: A signed (meaning positive or negative) digit string of a given length in a given base (or radix). This digit string is referred to as the significand, mantissa, or coefficient.
For example, while a fixed-point representation that allocates 8 decimal digits and 2 decimal places can represent the numbers 123456.78, 8765.43, 123.00, and so on, a floating-point representation with 8 decimal digits could also represent 1.2345678, 1234567.8, 0.000012345678, 12345678000000000, and so on.
The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point arithmetic originally established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard addressed many problems found in the diverse floating-point implementations that made them difficult to use reliably and ...
A floating-point variable can represent a wider range of numbers than a fixed-point variable of the same bit width at the cost of precision. A signed 32-bit integer variable has a maximum value of 2 31 − 1 = 2,147,483,647, whereas an IEEE 754 32-bit base-2 floating-point variable has a maximum value of (2 − 2 −23) × 2 127 ≈ 3.4028235 ...
The significand [1] (also coefficient, [1] sometimes argument, [2] or more ambiguously mantissa, [3] fraction, [4] [5] [nb 1] or characteristic [6] [3]) is the first (left) part of a number in scientific notation or related concepts in floating-point representation, consisting of its significant digits.
By contrast, in computer science, addition and multiplication of floating point numbers are not associative, as different rounding errors may be introduced when dissimilar-sized values are joined in a different order. [7] To illustrate this, consider a floating point representation with a 4-bit significand:
Floating-point representation is similar to scientific notation, except computers use base two (with rare exceptions), rather than base ten. The encoding scheme stores the sign, the exponent (in base two for Cray and VAX , base two or ten for IEEE floating point formats, and base 16 for IBM Floating Point Architecture ) and the significand ...
In simple settings, the resulting value is usually one of various primitive types, such as string, Boolean, or numerical (such as integer, floating-point, or complex). In computer algebra , formulas are viewed as expressions that can be evaluated as a Boolean, depending on the values that are given to the variables occurring in the expressions.