No human system of numeration can give a unique representation to every real number; there are just too many of them. So it is conventional to use approximations. For instance, the assertion that pi is 3.14159 is, strictly speaking, false, since pi is actually slightly larger than 3.14159; but in practice we sometimes use 3.14159 in calculations involving pi because it is a good enough approximation of pi.
One approach to representing real numbers, then, is to specify some
epsilon and to say that a real number
can be approximated by any number in the range from
x + epsilon. Then, if a system of
numeration can represent selected numbers that are never more than twice
epsilon apart, every real number has a representable
approximation. For instance, in the United States, the prices of stocks
are given in dollars and eighths of a dollar, and rounded to the nearest
eighth of a dollar; this corresponds to a tolerance of one-sixteenth of a
dollar. In retail commerce, however, the conventional tolerance is half a
cent; that is, prices are rounded to the nearest cent. In this case, we
can represent a sum of money as an whole number of cents, or equivalently
as a number of dollars that is specified to two decimal places.
Such a representation, in which each real number is represented by a numeral for an approximation to some fixed number of decimal places, is called a fixed-point representation. However, fixed-point representations are unsatisfactory for most applications involving real numbers, for two reasons
0.01, even though the former is more than twice as large as the latter. Moreover, both 4/1000 and -7/10000 would be represented as
0.00, and they don't even have the same sign! If one is counting dollars, these differences are probably irrelevant, but there are a lot of scientific and technological applications (navigation systems, for instance) in which they are critical.
These two considerations really have the same underlying cause: When approximating some real number, what we usually know is some number of significant digits at the beginning of the number. A fixed-point system of representation conceals some of what we know when the number is small, because some of these significant digits are too far to the right of the decimal point, and demands more than we know when the number is large, because the number of digits demanded by the representation system is greater than the number that we can provide.
Scientists and engineers long ago learned to cope with this problem by
using scientific notation, in which a number is expressed as the
product of a mantissa and some power of ten. The mantissa is a
signed number with an absolute value greater than or equal to one and less
than ten. So, for instance, the speed of light in vacuum is
2.99792458 x 10^8 meters per second, and one can specify only
the digits about which one is completely confident.
Using scientific notation, one can easily see both that
10^-2 is more than twice as large as
6 x 10^-3, and
that both are close to
1 x 10^-2; and one can easily
4 x 10^-3 and
-7 x 10^-4 as small
numbers of opposite sign. The rules for calculating with
scientific-notation numerals are a little more complicated, but the
benefits are enormous.
The three varying components of a number represented in scientific notation are (1) the sign of the number (positive or negative), (2) the absolute value of the mantissa and (3) the exponent on the power of ten. A system of numeration for real numbers that is adapted to computers will typically store the same three data (a sign, a mantissa, and an exponent) into an allocated region of storage. By contrast with fixed-point representations, these computer analogues of scientific notation are described as floating-point representations.
The exponent does not always indicate a power of ten; sometimes powers of
sixteen are used instead, or, most commonly of all, powers of two. The
numerals will be somewhat different depending how this choice is made. For
instance, the real number -0.125 will be expressed as
10^-1 if powers of ten are used, or as
-2 x 16^-1 if
powers of sixteen are used, or as
-1 x 2^-3 if powers of two
are used. The absolute value of the mantissa is, however, always greater
than or equal to 1 and less than the base of numeration.
The particular system used on MathLAN computers (and, I believe, in Java) was formulated and recommended as a standard by the Institute of Electrical and Electronics Engineers (IEEE) and is the most commonly used numeration system for computer representation of real numbers. Actually, their standard includes several variants of the system, depending on how much storage is available for a real number. We'll discuss two of these variants, both of which use binary numeration and powers of 2: the IEEE single-precision representation, which fits in thirty-two bits, and the IEEE double-precision representation, which occupies sixty-four bits. We'll begin with single-precision numbers.
In the IEEE single-precision representation of a real number, one bit is
reserved for the sign, and it is set to
for a positive
number and to
1 for a negative one. A representation of the
exponent is stored in the next eight bits, and the remaining twenty-three
bits are occupied by a representation of the mantissa of the number.
The exponent, which is a signed integer in the range from -126 to 127, is represented neither as a signed magnitude nor as a twos-complement number, but as a biased value. The idea here is that the integers in the desired range of exponents are first adjusted by adding a fixed ``bias'' to each one. The bias is chosen to be large enough to convert every integer in the range into a positive integer, which is then stored as a binary numeral. The IEEE single-precision representation uses a bias of 127.
For example, the exponent -5 is represented by the eight-bit pattern
01111010, which is the binary numeral for 122, since -5 + 127
= 122. The least allowable exponent, -126, is represented by
(since -126 + 127 = 1);
the greatest allowable exponent, 127, is represented by the
binary numeral for 254,
11111110. Note that neither
11111111 are allowed as
Since the base of numeration is two, the mantissa is always a number greater than or equal to one and less than two. Its fractional part is represented, using binary numeration, as a sum of the negative powers of two. For instance, the binary numeral for 21/16 is 1.0101 -- one, plus no halves, plus one quarter, plus no eighths, plus one sixteenth. (The ``decimal point'' in this case is actually a ``binary point,'' separating the digit in the units place from the digits representing multiples of negative powers of two.)
Somewhat surprisingly, this means that approximations are used for many real numbers that can be represented exactly in decimal numeration. For instance, 7/5 is 1.4 exactly in decimal numeration, but the .4 part cannot be expressed as a sum of powers of two; 7/5 has an infinite binary expansion 1.011001100110011001100..., just as in decimal numeration a fraction like 27/11 leads to an infinite decimal expansion 2.4545454545.... In a single-precision representation, the expansion is rounded off at the twenty-third digit after the binary point. Thus 7/5 is not actually stored as 7/5, but (in single precision) as the very close approximation 11744051/8388608, which can be expressed as a sum of powers of two.
Only the part of the mantissa that comes after the binary point is actually stored, since the bit to the left of the binary point is completely predictable (it's always 1, since the mantissa is always greater than or equal to one and less than two). This suppressed digit at the beginning of the mantissa is called the ``hidden bit.''
Here's an example. Consider the thirty-two-bit word
The number represented by this sequence of bits is negative (because the
sign bit is
1). It has an exponent of 8 (because the exponent
10000111, form the binary numeral for 135, and therefore
represent the exponent 8 when the bias of 127 is removed). Its mantissa is
expressed by the binary numeral
where the initial
1 is the hidden bit and the remaining digits
are taken from the right end of the word. This last binary numeral
expresses the number 75/64 (one, plus no halves, plus no quarters, plus one
eighth, plus no sixteenths, plus one thirty-second, plus one sixty-fourth).
So the complete number is -(75/64) x 2^8, which is -300.0.
Conversely, let us find the IEEE single-precision representation of, say,
5.75. The sign bit is
, since the number is positive.
5.75 is 23/4; to express this as the product of a power of two and a
mantissa greater than or equal to one and less than two, one must factor
out 2^2: 23/4 = (23/16) x 2^2, so the exponent is 2. It will be stored,
with a bias of 127, as
10000001 (the binary numeral for 129).
The complete mantissa, extended to twenty-three digits after the binary
1.01110000000000000000000. We line up the sign bit,
the biased representation of the exponent, and the digits following the
binary point in the mantissa and get
The greatest number that has an exact IEEE single-precision representation is 340282346638528859811704183484516925440.0 (2^128 - 2^104), which is represented by
and the least is the negative of this number, which has the same
representation except that the sign bit is
The alert reader will have noticed that there is a serious gap in this
scheme, as so far described: What about 0.0? It is not possible to
represent 0.0 as the product of a power of two and a mantissa
greater than or equal to one, so none of the representations described
above will do. However, not all the possible settings of thirty-two
switches have been used for numbers. Recall that the exponents permitted
in IEEE single-precision reals range from -126 to +127, so that the binary
numerals for the biased exponents range from
11111110. We haven't yet used any of the bit patterns in
which the exponent bits are all zeroes or all ones.
In the IEEE system, the all-zero exponent is used for numbers that are
very close to zero -- closer than 2^-126, which is the least of
the positive reals that can be represented in the part of the system
described above (it is
Such numbers are expressed in a slightly different form of scientific
notation: The exponent is held fixed at -126, and the mantissa is a number
greater than or equal to zero and less than one. So, for
instance, the mantissa used for 3 x 2^-129 is
0.011 (3 x
2^-129 is a quarter plus an eighth of 2^-126). Once again, only the part
of the mantissa that follows the binary point is stored explicitly, so the
representation of 3 x 2^-129 is
(sign positive, exponent -126, mantissa
0.01100000000000000000000). Mantissas less than one are said
to be ``unnormalized'' (because the ``normal form'' is the one in which the
mantissa is greater than or equal to one and less than the base of
numeration), so an all-zero exponent indicates an unnormalized number.
Unnormalized numbers are stored less accurately than normalized ones (since
there are fewer significant digits in the mantissa), but without this
special convention for the all-zero exponent it would not be possible to
represent them at all, and the designers of the IEEE standard felt that a
degraded approximation is better than none.
This convention allows zero to be represented in two different ways. In
one, the sign bit is
, the exponent bits are
00000000, and the visible bits of the mantissa are
00000000000000000000000 -- yielding
0.00000000000000000000000 x 2^-126, or 0. The other representation is
the same, except that the sign bit is
1. (So, as in
signed-magnitude representations, there is a ``non-negative'' and a
The least positive real number that can be represented exactly in this way is 2^-149, which is stored as
When the exponent bits are all ones, the value represented is not a real number at all, but a conventional signal of a computation that has gone wrong, either by going above the greatest representable real or below the least, or by attempting some undefined arithmetic operation, such as dividing by zero or taking the logarithm of a negative number. For instance, the thirty-two bits
represent ``positive infinity'', a pseudo-number that indicates that some
unrepresentably large quantity was generated by an arithmetic operation.
Changing the sign bit to
1 yields a representation of negative
infinity, an indication of a similar problem at the other end of the range.
If some of the bits of the mantissa are
1s, the pseudo-number
is called a NaN (``Not a Number''); trying to compute 0/0, for
instance, typically produces a NaN. It should be clear that positive
infinity, negative infinity, and NaN are not real numbers,
although some language implementations will try to do something sensible
if they appear in places that are normally occupied by real numbers. The
value of such attempts at recovery is questionable, however, since the
appearance of a pseudo-number is supposed to be a danger signal to the
programmer and usually results from a programming error.
IEEE double-precision representations are quite similar. A
double-precision real begins with a sign bit, with the same interpretation
as in a single-precision representation. The next eleven bits are used for
the exponent, which is an integer in the range from -1022 to +1023; a bias
of 1023 is added to the exponent, and the result is stored in a binary
numeral (the smallest is
00000000001, the largest
11111111110). The remaining fifty-two bits are used for the
mantissa, and as above only the digits following the binary point of the
mantissa are actually stored; the
1 that precedes the binary
point is once again a ``hidden bit.'' As in single-precision
representations, the all-zero exponent is used for unnormalized numbers and
(with an all-zero mantissa) for 0, and the all-one exponent is used for the
pseudo-numbers positive infinity, negative infinity, and NaN. The greatest
real number that can be represented exactly as a double-precision real is
2^1024 - 2^971, and the least positive real that can be so represented is
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