Floating Point Representation of Numbers

Numerical inaccuracy in mathematical computations stems from the need to represent real numbers in floating point format.

In floating point format, a fixed number of bits of storage are used to hold the sign, mantissa, and exponent of a real number, usually in base 2.

In IEEE standard floating point on a 32 bit computer, a double precision number consists of

1 bit for the sign,
11 bits for the exponent, and

52 bits for the mantissa.

Some numbers, called machine numbers, can be exactly represented in floating point. Others can't and must be rounded to the nearest machine number, introducing a small error.