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Mantissa and Characteristic
Posted:
Mar 13, 2003 2:17 PM


From a historical perspective these concepts allow you to review the power of logarithms for computations.
"The miraculous power of modern calculations are due to three inventions:the Arabic Notation, Decimal fractions and Logarithms." F. Cajori
Although John Napier coined the word logarithm in 1616 (Kepler abbreviated it to log in 1624), Briggs suggested the usage of the common base 10. Base 10 by use of Mantissa and Characteristic was quite amicable to a table format. This made approximation of complex calculations even easier. (One might haul out an antique log 10 table or even a slide rule for show and tell).
However, one also might argue that the advent of modern calculating machines should excuse us from such antiquities. After all base e is far more suitable for the construction of "real world" models beyond the needs of simple calculation. And for many people base e is likely satisfactory although its introduction is not so intuitive as bases such as 2 or 10.
But there may be some who interested in how modern calculating machines are able to do their calculations. Indeed, the internal representation of finite precision decimals in most computers employs a representation similar to the mantissa and characteristic. For example IEEE standard 754 is likely the most common method for machine representation of floating point numbers. In the IEEE standard their are three parts to the internal representation: the sign,the characteristic and the mantissa.
For example, in the IEEE single precision, floating point decimal, one bit is the sign bit. The next 8 bits represent the exponent (or characteristic), followed by 23 bits representing the mantissa (or fraction or significand) for a total of 32 bits (4 bytes). Usually the mantissa has an implied radix point which in a normalized form occurs after the first nonzero digit. The exponent field may include a bias, often 127 which is added to true exponent so that negative exponents can be represented as a kind of complement. So that an exponent of 10 is represented by 137 and an exponent of 3 is represented by 124. You can likely compare this to some of the shenanigans associated with negative characteristics in base 10 logarithms. The mantissa or significand is usually the precision bits of the number. This would ease their addition, however a true logarithmic mantissa would improve the multiplication.
As an aside, note that if base e is the more useful base in practice, that base 10 can be approached for logarithms based on the change of base formula: log x = (ln x)/(ln 10). And the function y = 10^x can be defined so that 10^x = e^(x ln 10), so that the graphs of these base 10 functions are simply transformations of the corresponding base e functions. One remarkable property of logarithms is that if y = log( ax) is regarded as a horizontal stretch of y = log x and if y = log x + c is regarded as a vertical shift of y = log x, then because log(ax) = log a + log x so that if c = log a then y = log(ax) = log x + c: We cannot distinguish a horizontal stretch of a logarithmic function from a vertical shift.
cordially,  Ron
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