mantissa
A mantissa is the number of digits in a floating point number before the power .
Example: For the number 2.9979 · 10 ^{8} , 2.9979 is the mantissa.
Mantissa for logarithms
When working with decadic logarithms , it is also common to use only the decimal places as the mantissa. The reason for this is that it is they who determine the sequence of digits in the logarithmic number. This property of the decadic logarithms is used when creating number tables to determine decadic logarithms. The decadic logarithms can then be easily determined, even if there is no electronic aid such as B. a calculator or a computer is available.
The decadic logarithm consists of the code number as well as the real decimal fraction, the mantissa, created from an irrational number by rounding. The number always has the same mantissa, regardless of the decimal point. (The code number of a decadic logarithm corresponds to the exponent of the place value of the first non-zero digit of the number.)
Examples:
The decadic logarithm of 299,790,000 (log _{10} 299,790,000 or log 299,790,000) is to be determined. According to the logarithmic laws, this results in:
lg (10 ^{8} * 2.9979) = lg 100,000,000 + lg 2.9979 = lg 10 ^{8} + lg 2.9979 = 8 + 0.4768 = 8.4768
The lg 299.790.000 has the mantissa 4768 and the code number 8, since the first valid number, the 2, has the place value 10 ^{8} .
(299.790.000 = 2 · 10 ^{8} + 9 · 10 ^{7} + 9 · 10 ^{6} + 7 · 10 ^{5} + 9 · 10 ^{4} + 0 · 10 ^{3} + 0 · 10 ^{2} + 0 · 10 ^{1} + 0 · 10 ^{0} )
The decadic logarithm of 0.021544 (log _{10} 0.021544 or log 0.021544) is to be determined. According to the logarithmic laws, this results in:
lg = lg 2.1544 - lg 100 = lg 2.1544 - lg 10 ^{2} = 0.333 33 - 2 = -1.666 67 ^{}
The lg 0.021544 has the mantissa 33333 and the code number -2, since the first valid number, the 2, has the place value 10 ^{-2} .
(0.021544 = 0 · 10 ^{0} + 0 · 10 ^{−1} + 2 · 10 ^{−2} + 1 · 10 ^{−3} + 5 · 10 ^{−4} + 4 · 10 ^{−5} + 4 · 10 ^{−6} )
Mantissa in computer science
In computer science , the mantissas for the representation of floating point numbers are of outstanding importance, which are preferably represented by
- [Sign ( s )] · [mantissa ( m )] · base ( b ) ^{[exponent ( e )]}
However, the definition of the mantissa is apparently not so clear here. Published definitions differ and are sometimes even contradicting. For this reason, the term significand , suggested by some English-speaking authors, is also used in German as a signifier in order to better differentiate it , but it is not generally used.
The x.xxxx form of the mantissa appears most frequently, in which the most significant digit is shifted to the place in front of the decimal point (the exponent carries the information about the thrust and direction ).
- Normalized mantissa (only with base b = 2)
- If the mantissa is in the value range 1 ≤ m <2 (i.e.: the number in front of the decimal point is 1), one speaks of a normalized mantissa.
- Normalized mantissa
- If the mantissa is in the value range 1 / b ≤ m <1 (i.e.: the number in front of the decimal point is 0 and the first decimal place is not equal to 0), one speaks of a standardized mantissa (0.xxxx form).
See also
Web links
Individual evidence
- ↑ Hans Kreul: Mathematics made easy: 781 problems with solutions . 4th edition. Deutsch, Thun 1994, ISBN 3-8171-1356-0 (special edition of the 6th, revised edition of the textbook Moderner Preliminary Course for Elementary Mathematics ).