2.3 Floating Point Numbers
The ﬁxed point number representation, which we explored in Section 2.2, has a ﬁxed position for the radix point, and a ﬁxed number of digits to the left and right of the radix point. A ﬁxed point representation may need a great many digits in order to represent a practical range of numbers. For example, a computer that can represent a number as large as a trillion1 maintains at least 40 bits to the left of the radix point since 240 ≈ 1012. If the same computer needs to represent one trillionth, then 40 bits must also be maintained to the right of the radix point, which results in a total of 80 bits per number.
In practice, much larger numbers and much smaller numbers appear during the course of computation, which places even greater demands on a computer. A great deal of hardware is required in order to store and manipulate numbers with 80 or more bits of precision, and computation proceeds more slowly for a large number of digits than for a small number of digits. Fine precision, however, is generally not needed when large numbers are used, and conversely, large numbers do not generally need to be represented when calculations are made with small numbers. A more efﬁcient computer can be realized when only as much precision is retained as is needed.