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has several possible record structures (matrices) and an integer value is used to iden-
tify each type of matrix. The integer value has a fixed number of values; each value
represents a different type of matrix.
7.3.1.4 Real Floating Point Numbers
Floating point numbers draw their notation from the fact that the decimal point can
vary in position, i.e. 1.24567 and 149.243. Their notation is usually the along the
same lines as the scientific notion for real numbers e.g.,
10 3
1.49243
×
where there is a base (b) (which in this case it is base 10), an exponent (e)
(which in this case is -3) and a significand (mantissa) which is the significant
digits 149,243 having a precision of 6 digits. The decimal point is assumed to be
directly after the leftmost digit when reading left to right. But in data and in com-
puter systems the representation of floating point numbers is binary, for example,
1.010x2 1011 . Here the base is b
2 and the exponent value has a binary repre-
sentation along with the significand. Usually the number is normalised in that the
decimal point is assumed to be directly after the left most non-zero digit read-
ing left to right, as this digit is then guaranteed to be 1. This digit can then be
ignored and the significand reduced to 010 (this is what is actually stored in the
data). This normalisation is just a way of making the best use of the bits available
where there are a finite number of bits representing the floating point value and
thus increasing the precision. For example a 24 bit significand can be represented
with 23 bits.
The significand as with integer values can be interpreted as a two's compliment
number, one's compliment number or some other interpretation scheme. The expo-
nent is also usually subject to some interpretation scheme to get a signed integer
value, typically this is a bias scheme where the number is first treated as an unsigned
integer and then some bias is deducted from it. So for an 8 bit exponent with a value
10001101
=
-113.
Also there will be a sign bit (d) to apply to the final number where a 0 may represent
a positive number and a 1 a negative number.
Sometimes some bit patterns in the exponent and the significand are reserves
to represent floating point exceptions. Exceptions can occur during floating point
calculations such as dividing by zero, calculations that would yield an imaginary
number or calculations resulting in a number too large or small to be repre-
sented in the finite range of the floating point type. Most systems of representing
floating point types explicitly state what the bit patterns are reserved for these
exceptions.
The exact location of the bits that correspond to the significand, exponent and
sign bit also needs to be known. Fig. 7.6 shows an IEEE 754 [ 32 ] 32 bit big-endian
and little-endian floating point value (same value). The first bit of the big-endian
representation is the sign bit then it is followed by the exponent (8 bits) and finally
=
141 and a bias (c) of -127 the exponent would be 141-127
=
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