This article gives a brief overview of IEEE floating point and its representation. Discussion of arithmetic implementation may be found in the book mentioned at the bottom of this article. There are several ways to represent real numbers on computers. Fixed point places a radix point somewhere in the middle of the digits, and is equivalent to using integers that represent portions of some unit. Another approach is to use rationals, and represent every number as the ratio of two integers.
Scientific notation represents numbers as a base number and an exponent.
Floating-point solves a number of representation problems. Fixed-point has a fixed window of representation, which limits it from representing very large or very small numbers. Also, fixed-point is prone to a loss of precision when two large numbers are divided. This allows it to represent numbers from 1,000,000,000,000 to 0.0000000000000001 with ease. The sign bit is as simple as it gets.