If the range of numbers is very large, the scientific notation n = f x 10e is used, where f is the fraction or mantissa, and e is a positive or negative exponent. There is only one digit before the decimal point. Some examples are:
3.14
= 0.314 x 101 = 3.14
x 100
0.000001
= 0.1 x 10-5
= 1.0 x 10-6
1941
= 0.1941 x 104 = 1.941 x 103
The range is determined by the number of digits in the exponent and the precision is determined by the number of digits in the fraction. f is in the range 0.1 <= | f | < 1 or zero and a signed two-digit exponent. These numbers range in magnitude from +0.100 x 10-99 to +0.999 x 10+99, a span of nearly 199 orders of magnitude, yet only five digits and two signs are needed to store a number.
Floating-point numbers can be used to model the real-number system of mathematics. The real line can be divided into seven regions. They are:
1. Large negative numbers less than -0.999 x 1099.
2. Negative numbers between -0.999 x 1099 and
-0.100 x 10-99.
3. Small negative numbers with magnitude less than
0.100 x 10-99.
4. Zero.
5. Small positive numbers with magnitudes less than
0.100 x 10-99.
6. Positive numbers between 0.100 x 10-99
and 0.999 x 1099.
7. Large positive numbers greater than 0.999 x 1099.
1060 x 1060 = 10120 will cause an overflow
error (region 1 or 7) Similarly underflow error can occur
in the regions 3 or 5 which is not as serious as overflow. Floating-point
numbers do not form a continuum. For example +0.100 x 103
divided by 3 cannot ne expressed exactly. So a rounding
process is used. The table below shows the approximate boundaries
of region 6 for floating-point decimal numbers for various sizes of
fraction and exponent.
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