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In scientific language we use the term ``precision'' to describe the
number of significant digits in a number and the term ``accuracy'' to
describe the confidence we have in the number. In statistical terms,
accuracy can be described by a standard deviation. Here we
concentrate on precision.

A computer represents numbers with a limited number of binary
bits. The `float`

single precision type is stored in 32
bits and the `double`

double precision type, in 64 bits.
Most computers use the IEEE standard storage format, which represents
a number in binary scientific notation with a sign, mantissa and
exponent. What is a mantissa? In decimal notation the number
has a mantissa of 0.512, an exponent of 5 and a negative
sign. Single precision has room for a mantissa of approximately 8
significant decimal digits and double precision, approximately 15. By
shifting decimal places and changing the exponent, we can always
arrange for the mantissa to start with ``0.'', followed by a nonzero
digit. The significant digits are then counted to the right of the
decimal point. For example, in decimal language the number
.0012345678 would be written in single precision as
and would have eight significant decimal digits. The allowed
range for the exponent, expressed as a power of ten is approximately
in single precision and in double precision. So a
number that is precisely represented with an infinite number of digits
must be rounded to a finite number of digits in the computer.

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Carleton DeTar
2017-09-05