Computers are incapable of representing arbitrary real numbers exactly. This is due to two intractable facts of real numbers:
Uncountably Infinite Domain: There are an infinite number of real numbers.
Uncountably Infinite Precision: Some real numbers require infinite precision.
Since computers use a finite number of bits, computer architects must settle on capturing a finite number of real numbers at a finite level of precision. This is done by fixing a mapping between bit patterns and real numbers called a representation. A representation makes a tradeoff between the quantity of representable numbers and the level of precision.
