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Topic: Mantissa and Characteristic
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Ron Ferguson

Posts: 169
Registered: 12/6/04
Mantissa and Characteristic
Posted: Mar 13, 2003 2:17 PM
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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
non-zero 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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