Floating
Points
floating point
-derived from the fact that there is no
fixed number of digits before and after the decimal point.
-that is, the decimal point can float.
fixed-point representations
-representations in which the number of
digits before and after the decimal point is set.
IEE
754 Standard
|
s
|
biased exponent
|
significand
|
s – sign of significand
biased exponent – 8 bits
significand – 23 bits
If we use S for the sign, E for the exponent and M for
the mantissa (significand), then the number stored is equivalent to:
(-1)S x M x 2E
The Sign
The sign of a binary floating-point number is represented by a single bit. A 1 bit indicates a negative number, and a 0 bit indicates a positive number.
The Mantissa
It is useful to consider the way decimal floating-point numbers represent their mantissa. Using -3.154 x 105 as an example, the sign is negative, the mantissa is 3.154, and the exponent is 5. The fractional portion of the mantissa is the sum of each digit multiplied by a power of 10:
.154 = 1/10 + 5/100 + 4/1000
A binary floating-point number is similar. For example, in the number +11.1011 x 23, the sign is positive, the mantissa is 11.1011, and the exponent is 3. The fractional portion of the mantissa is the sum of successive powers of 2. In our example, it is expressed as:
.1011 = 1/2 + 0/4 + 1/8 + 1/16
Or, you can calculate this value as 1011 divided by 24. In decimal terms, this is eleven divided by sixteen, or 0.6875. Combined with the left-hand side of 11.1011, the decimal value of the number is 3.6875. Here are additional examples:
Binary
Floating-Point
|
Base
10 Fraction
|
Base
10 Decimal
|
11.11
|
3 3/4
|
3.75
|
0.00000000000000000000001
|
1/8388608
|
0.00000011920928955078125
|
The last entry in this table shows the smallest fraction that can be stored in a 23-bit mantissa. The following table shows a few simple examples of binary floating-point numbers alongside their equivalent decimal fractions and decimal values:
Binary
|
Decimal Fraction
|
Decimal Value
|
.1
|
1/2
|
.5
|
.01
|
1/4
|
.25
|
.001
|
1/8
|
.125
|
.0001
|
1/16
|
.0625
|
.00001
|
1/32
|
.03125
|
The Exponent
IEEE Short Real exponents are stored as 8-bit unsigned integers with a bias of 127. Let's use the number 1.101 x 25 as an example. The exponent (5) is added to 127 and the sum (132) is binary 10100010. Here are some examples of exponents, first shown in decimal, then adjusted, and finally in unsigned binary:
Exponent
(E)
|
Adjusted
(E + 127) |
Binary
|
+5
|
132
|
10000100
|
0
|
127
|
01111111
|
-10
|
117
|
01110101
|
+128
|
255
|
11111111
|
-127
|
0
|
00000000
|
-1
|
126
|
01111110
|
The binary exponent is unsigned, and therefore cannot be negative. The largest possible exponent is 128-- when added to 127, it produces 255, the largest unsigned value represented by 8 bits. The approximate range is from 1.0 x 2-127 to 1.0 x 2+128.
Normalizing the Mantissa
Before a floating-point binary number can be stored correctly, its mantissa must be normalized. The process is basically the same as when normalizing a floating-point decimal number. For example, decimal 1234.567 is normalized as 1.234567 x 103 by moving the decimal point so that only one digit appears before the decimal. The exponent expresses the number of positions the decimal point was moved left (positive exponent) or moved right (negative exponent).
Similarly, the floating-point binary value 1101.101 is normalized as 1.101101 x 23 by moving the decimal point 3 positions to the left, and multiplying by 23. Here are some examples of normalizations:
Binary Value
|
Normalized As
|
Exponent
|
1101.101
|
1.101101
|
3
|
.00101
|
1.01
|
-3
|
1.0001
|
1.0001
|
0
|
10000011.0
|
1.0000011
|
7
|
You may have noticed that in a normalized mantissa, the digit 1 always appears to the left of the decimal point. In fact, the leading 1 is omitted from the mantissa in the IEEE storage format because it is redundant.
Creating the IEEE Bit Representation
We can now combine the sign, exponent, and normalized mantissa into the binary IEEE short real representation. Using Figure 1 as a reference, the value 1.101 x 20 is stored as sign = 0 (positive), mantissa = 101, and exponent = 01111111 (the exponent value is added to 127). The "1" to the left of the decimal point is dropped from the mantissa. Here are more examples:
Binary Value
|
Biased
Exponent
|
Sign, Exponent, Mantissa
|
-1.11
|
127
|
1 01111111 11000000000000000000000
|
+1101.101
|
130
|
0 10000010 10110100000000000000000
|
-.00101
|
124
|
1 01111100 01000000000000000000000
|
+100111.0
|
132
|
0 10000100 00111000000000000000000
|
+.0000001101011
|
120
|
0 01111000 10101100000000000000000
|
Converting Decimal Fractions to Binary Reals
If a decimal fraction can be easily represented as a sum of fractions in the form (1/2 + 1/4 + 1/8 + ... ), it is fairly easy to discover the corresponding binary real. Here are a few simple examples
Decimal Fraction
|
Factored As...
|
Binary
Real
|
1/2
|
1/2
|
.1
|
1/4
|
1/4
|
.01
|
3/4
|
1/2 + 1/4
|
.11
|
1/8
|
1/8
|
.001
|
7/8
|
1/2 + 1/4 + 1/8
|
.111
|
3/8
|
1/4 + 1/8
|
.011
|
1/16
|
1/16
|
.0001
|
3/16
|
1/8 + 1/16
|
.0011
|
5/16
|
1/4 + 1/16
|
.0101
|
Binary Coded Decimal (BCD)
Binary
Coded Decimal(BCD) is a way to express each of the decimal digits with a
binary
code. There are only ten code groups in the BCD system. The 8421 code is a
type
of BCD code. In BCD each decimal digit , 0 through 9 is represented by a binary
code
of four bits. The designation of 8421 indicates the binary weights of the four
bits
(23,22,21,20).
The largest 4-bit code is 1001. The numbers 1010, 1011, 1100, 1101,
1110,
and 1111 are called forbidden numbers. The following table represents the
decimal
and 8421 equivalent numbers.
Decimal digit 0
1 2 3
4 5 6
7 8 9
BCD 0000
0001 0010 0011
0100 0101 0110
0111 1000 1001
Alphanumeric code
Computers,
printers and the other devices must process both alphabetic and numeric
information. Serial coding systems have been developed to represent
alphanumeric information as a series of 1’s and 0’s. The characters to be coded
are alphabets(26),numerals (10) and special characters such as +,-, /,*, $ etc,
In
order to code a character, string of binary digits is used. In order to ensure
uniformity in coding, two standard codes have been used.
1.
ASCII: American Standard Code for Information Interchange.
2.
EBCDIC: Extended Binary Coded Decimal Interchange Code. It is an 8 bit
code.
ASCII
is 7-bit code of the form X6, X5,
X4,
X3,
X2,
X1,
X0 and
is used to code two types of information. One type is the printable character
such as alphabets, digits and special characters. The other type is known as
control characters which represent the coded information to control the
operation of the digital computer and are not printed.
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