Mantissa
In mathematics, the mantissa refers to the fractional part of a common (base-10) logarithm, which lies between 0 and 1 and represents the digits after the decimal point, excluding the integer part known as the characteristic.[1] For example, the logarithm of 20 to the base 10 is approximately 1.3010, where 1 is the characteristic and 0.3010 is the mantissa.[2] This term, derived from the Latin mantissa meaning "an addition" or "makeweight," was first used in a mathematical context by English mathematician John Wallis in 1693 and applied to the fractional part of a logarithm by Leonhard Euler in 1748.[3] In scientific notation, the mantissa is equivalently the significand—the leading digits of a number normalized between 1 and 10 (or 1 and the base in other systems)—multiplied by a power of the base to express the full value, as in 5.3266 × 10³ where 5.3266 is the mantissa.[4] The concept extends to computing, where the mantissa (often interchangeably called the significand) forms the fractional component of a floating-point number in binary representation, typically normalized to lie between 1 and 2 (or 0.5 and 1 in some conventions) and combined with an exponent to approximate real numbers with finite precision.[5] In the IEEE 754 standard, widely used in modern processors, the mantissa occupies a significant portion of the bit allocation—23 bits for single precision and 52 bits for double precision—ensuring accurate representation of the number's significant digits while the exponent handles scaling.[6] This usage highlights the mantissa's role in balancing precision and range in numerical computations, though it can introduce rounding errors due to limited bits.[7] Historically, the term's application to floating-point arose in the mid-20th century with the development of electronic computers, adapting the logarithmic notion to binary formats for efficient arithmetic operations.[8]Logarithmic mantissa
Definition and properties
In mathematics, for a positive real number x > 0, the common logarithm is expressed as \log_{10} x = n + m, where n = \lfloor \log_{10} x \rfloor is the integer part, called the characteristic, and m is the fractional part, called the mantissa, satisfying $0 \leq m < 1.[9] The mantissa is thus mathematically defined as m = \log_{10} x - \lfloor \log_{10} x \rfloor.[1] In some texts, the mantissa is denoted using the standard fractional part notation \{ \log_{10} x \}. A key property of the mantissa is that its antilogarithm, $10^m, yields the significand of x (the normalized coefficient between 1 and 10 in scientific notation).[9] For instance, consider x = 23.0, where \log_{10} 23.0 = 1.362; the mantissa is 0.362, and $10^{0.362} \approx 2.30, matching the significand.[9] Another important property is that the mantissa remains unchanged when x is scaled by any integer power of 10, since \log_{10}(10^k x) = k + \log_{10} x for integer k, preserving the fractional part.[10] The following table illustrates mantissa values for numbers between 1 and 10, highlighting their distribution on a logarithmic scale (values approximated to four decimal places):| x | \log_{10} x | Mantissa |
|---|---|---|
| 1 | 0.0000 | 0.0000 |
| 2 | 0.3010 | 0.3010 |
| 3 | 0.4771 | 0.4771 |
| 4 | 0.6021 | 0.6021 |
| 5 | 0.6990 | 0.6990 |
| 6 | 0.7782 | 0.7782 |
| 7 | 0.8451 | 0.8451 |
| 8 | 0.9031 | 0.9031 |
| 9 | 0.9542 | 0.9542 |
| 10 | 1.0000 | 0.0000 |