View source for Real numbers and numerical precision

== Overview ==

An important aspect of computational physics is the numerical precision involved. To design a
good algorithm, one needs to have a basic understanding of propagation of inaccuracies and
errors involved in calculations. There is no magic recipe for dealing with underﬂow, overﬂow,
accumulation of errors and loss of precision, and only a careful analysis of the functions
involved can save one from serious problems.

Since we are interested in the precision of the numerical calculus, we need to understand
how computers represent real and integer numbers. Most computers deal with real numbers
in the binary system, or octal and hexadecimal, in contrast to the decimal system that we
humans prefer to use. The binary system uses 2 as the base, in much the same way that the
decimal system uses 10. Since the typical computer communicates with us in the decimal system, but works internally in e.g., the binary system, conversion procedures must be executed
by the computer, and these conversions involve hopefully only small roundoff errors

Computers are also not able to operate using real numbers expressed with more than a
ﬁxed number of digits, and the set of values possible is only a subset of the mathematical
integers or real numbers. The so-called word length we reserve for a given number places a
restriction on the precision with which a given number is represented. This means in turn,
that for example ﬂoating numbers are always rounded to a machine dependent precision,
typically with 6-15 leading digits to the right of the decimal point. Furthermore, each such
set of values has a processor-dependent smallest negative and a largest positive value.
Why do we at all care about rounding and machine precision? 

== Example: Loss of precision due to cancellations of nearly equal numbers ==

 x^2