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Chapter 3
Number
Representation
©Brooks/Cole, 2003
Number representation
It is not efficient to represent numbers in the same way as
symbols representation.
For example representing 67854 as symbols using ASCII
requires 5 bytes each representing a digit. And it is obvious
That as the number of digits increases the number of bytes
Increases.
Therefore we need a more efficient way to represent
numbers.
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3.1
DECIMAL
AND
BINARY
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Figure 3-1
Decimal system
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Figure 3-2
Binary system
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3.2
CONVERSION
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Figure 3-3
Binary to decimal conversion
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Figure 3-4
Decimal to binary conversion
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3.4
INTEGER
REPRESENTATION
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Figure 3-5
Range of integers
•Integers are whole numbers without a fraction.
For example,134 is an integer but 134.987 is not.
• Integers can be positive and negative.
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Figure 3-6
Taxonomy of integers
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Unsigned integers
• The maximum unsigned integer depends on the
number of bits allocated in the computer to store an
unsigned integer.
• For N number of bits, the maximum integer to be
represented is (2N -1)
# of Bits
--------8
16
Range
------------------------------------0
255
0
65,535
©Brooks/Cole, 2003
Table 3.2 Example of storing unsigned integers in
two different computers
Decimal
8-bit allocation
16-bit allocation
-----------7
234
258
24,760
1,245,678
-----------00000111
11101010
overflow
overflow
overflow
-----------------------------0000000000000111
0000000011101010
0000000100000010
0110000010111000
overflow
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Applications (unsigned)
• Counting
• Addressing
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Signed integer:
1.Sign and magnitude
• In this format one bit is needed to represent the
sign of integer number (0 for positive integer
and 1 for negative)
• For N number of bits allocated for the signed
number, the range of integer numbers is given
by:
-(2N-1 – 1) …… (2N-1 -1)
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Sign-and-magnitude integers
# of Bits
---------8
16
32
Range
------------------------------------------------------+127
-127
-0 +0
+32767
-32767
-0 +0
+2,147,483,647
-2,147,483,647 -0 +0
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Sign and Magnitude Representation
1. Convert the magnitude to binary
2. Add 0s to the left to make the total number of
bits N-1.
3. If number is positive add 0 to the left; if
number is negative add 1 to the left
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Table 3.4 Example of storing sign-and-magnitude integers
in two computers
Decimal
8-bit allocation
16-bit allocation
-----------+7
-124
+258
-24,760
-----------00000111
11111100
overflow
overflow
-----------------------------0000000000000111
1000000001111100
0000000100000010
1110000010111000
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Sign and Magnitude Interpertation
1. Ignore the first (liftmost) bit,
2. Change the following bits to decimal.
3. Attach + or – sign based on the left most bit.
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Advantages, Disadvantages, and
Application (sign and magnitude)
1. Disadvantages: two zeros
2. Advantages: easy transformation
3. Application: data communication.
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Signed integer:
2. One’s Complement
• In this format the negative number is the
complement of the positive number, for
example -7 is represented as the binary
complement of +7.
• The number’s complement is obtained by
changing all 0s to 1s and all 1s to 0s.
• The left most digit is assigned for the number’s
sign (0 for positive, 1 for negative).
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Range of one’s complement integers
For N number of bits allocated for the signed
number, the range of integer numbers is given by:
-(2N-1 – 1) …… (2N-1 -1)
# of Bits
--------8
16
32
Range
-------------------------------------------------------127
-0 +0
+127
-32767
-0 +0
+32767
-2,147,483,647 -0 +0
+2,147,483,647
©Brooks/Cole, 2003
One’s Complements
Representation
1. Convert the magnitude to binary
2. If number is positive, add 0s to the left to
make the total number of bits N.
3. If number is negative add complement the
positive representation (step 2)
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Table 3.6 Example of storing one’s complement integers in
two different computers
Decimal
8-bit allocation
16-bit allocation
-----------+7
-7
+124
-124
+24,760
-24,760
-----------00000111
11111000
01111100
10000011
overflow
overflow
-----------------------------0000000000000111
1111111111111000
0000000001111100
1111111110000011
0110000010111000
1001111101000111
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One’s Complement Interpretation
• If the leftmost 0 (positive number)
1. Change the entire number to decimal
2. Put (+) in front of the number.
• If the leftmost is 1 (negative number)
1. Complement the entire number.
2. Change the entire number to decimal.
3. Put (-) in front of the number
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Disadvantages, and Application
(One’s Complement)
1. Disadvantages: two zeros
2. Application: error detection and correction.
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Signed integer:
3. Two’s Complement
• Two’s complement solves the problems of two
zeros.
• Two’s complement is widely used
representation of integers today.
• For N number of bits allocated for the signed
number, the range of integer numbers is given
by: -(2N-1) …… (2N-1 -1)
• The left most digit shows the number’s sign (0
for positive, 1 for negative).
©Brooks/Cole, 2003
Range of two’s complement integers
# of Bits
--------8
16
32
Range
------------------------------------------------------0
+127
-128
0
+32,767
-32,768
0
+2,147,483,647
-2,147,483,648
©Brooks/Cole, 2003
Two’s Complements
Representation
1. Convert the magnitude to binary
2. Add 0s to the left to make the total number of
bits equal to N.
3. If the number is positive no action is needed.
If the number is negative, complement after
the first 1 from the right of the positive
representation.
©Brooks/Cole, 2003
Table 3.8 Example of storing two’s complement integers in
two different computers
Decimal
8-bit allocation
16-bit allocation
-----------+7
-7
+124
-124
+24,760
-24,760
-----------00000111
11111001
01111100
10000100
overflow
overflow
-----------------------------0000000000000111
1111111111111001
0000000001111100
1111111110000100
0110000010111000
1001111101001000
©Brooks/Cole, 2003
Two’s Complement Interpretation
• If the leftmost 0 (positive number)
1. Change the entire number to decimal
2. Put (+) in front of the number.
• If the leftmost is 1 (negative number)
1. Complement after the first 1 from the right.
2. Change the entire number to decimal.
3. Put (-) in front of the number
©Brooks/Cole, 2003
Note:
Two’s complement can be achieved by reversing all
bits except the rightmost bits up to the first 1
(inclusive). If you two’s complement a positive
number, you get the corresponding negative number.
If you two’s complement a negative number, you get
the corresponding positive number. If you two’s
complement a number twice, you get the original
number.
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Application (Tow’s Complement)
Two’s complement is used in today’s computer
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Table 3.9 Summary of integer representation
Contents of
Memory
-----------0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
Unsigned
-----------0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Sign-andMagnitude
--------+0
+1
+2
+3
+4
+5
+6
+7
-0
-1
-2
-3
-4
-5
-6
-7
One’s
Complement
--------+0
+1
+2
+3
+4
+5
+6
+7
-7
-6
-5
-4
-3
-2
-1
-0
Two’s
Complement
-------+0
+1
+2
+3
+4
+5
+6
+7
-8
-7
-6
-5
-4
-3
-2
-1
©Brooks/Cole, 2003
Signed integer:
Excess System
• Excess system is used to store signed integer
numbers in computer.
• In Excess system a positive number called
magic number is used for this format.
• The magic number is either 2N-1 or 2N-1-1. For
example if N = 8 the magic number is either
128 (called excess_128) or 127 (called
excess_127)
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Excess System Representation
1. Add the magic number to the integer.
2. Change the result to binary
3. Add 0s to the left to make total number of bits
equal to N.
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Excess System Interpretation
1. Change the number to decimal
2. Subtract the magic number form the integer.
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Excess System Range
1. For Magic number equal to 2N-1 the range is
- (2N-1)….. +(2N-1-1)
# of Bits
--------8
16
32
Range
------------------------------------------------------0
+127
-128
0
+32,767
-32,768
0
+2,147,483,647
-2,147,483,648
©Brooks/Cole, 2003
Excess System Range
1. For Magic number equal to 2N-1 -1 the range is
- (2N-1 -1)….. +(2N-1)
# of Bits
--------8
16
32
Range
------------------------------------------------------0
+128
-127
0
+32,768
-32,767
0
+2,147,483,648
-2,147,483,647
©Brooks/Cole, 2003
Advantages, Disadvantages and
Application (Excess System)
• Advantages: Easy transformation.
• Disadvantages: operation on excess system is
very complicated.
• Applications: used in storing exponential
values of fraction.
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FLOATING-POINT
REPRESENTATION
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Floating Point Number
• Floating point number is a number containing
an integer and a fraction
• For example in floating point number 21.345,
the integer part is 21 and the fraction is 0.345
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Floating Point Representation
1. Convert the integer part to binary
2. Convert the fraction part to binary by
repetitively multiply the fraction part by 2.
The binary digit is the integer part of the
result of multiplication. Stop whenever the
fraction part becomes 0 or when you reach
the maximum number of bits you can use.
3. Put a decimal point between the two parts.
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Figure 3-7
Changing fractions to binary
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Floating Point Interpretation
1. Convert the integer part to decimal
2. Convert the fraction part to decimal by:
1. multiplying each digit by its weight.
2. Add altogether
3. Put a decimal point between the two parts.
©Brooks/Cole, 2003
Figure 3-3
0
Binary floating to decimal conversion
.
2-1
2-2
2-3
2-4
2-5
2-6
2-7
0 + 0.25 + 0 + 0.0625 + 0.03125 +0 +0.0078125
0.3515625
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Normalization
• Normalization is a standard way to write floating
point binary number.
• Normalization is achieved by moving the decimal
point so that there is only one 1 to the left of it.
• To indicate how many digits the decimal point has
moved, multiply the number by 2e which indicates
the direction and number of digits moved.
• If :
– e is positive, decimal point moved to the left
– e is negative, decimal point moved to the right
©Brooks/Cole, 2003
Table 3.10 Example of normalization
Original
Original
Number
Number
Move
---------------------------------6
+1010001.1101
+1010001.1101
+26
2
-111.000011
-111.000011
-22
6
+0.00000111001
+0.00000111001
+2-6
3
-001110011
-0.001110011
-2-3
Normalized
-----------x 1.01000111001
x 1.11000011
x 1.11001
x 1.110011
©Brooks/Cole, 2003
Sign, Exponent and Mantissa
• After the number normalization you store only
three pieces of information about the number:
sign, exponent and mantissa.
– Sign is the number sign (stored in 1 bit)
– Exponent (power of 2) shows direction and
magnitude of decimal point movement.
– Mantissa is the bits to the right of the decimal
point.
©Brooks/Cole, 2003
Normalization (example)
For the number: + 1000111.0101
Normalized Number:
+ 26 x 1.0001110101
Sign: +
Exponent: 6
Mantissa: 0001110101
©Brooks/Cole, 2003
Figure 3-8
IEEE standards for floating-point representation
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Single Precision Representation
•
•
•
•
Store the sign as 0 (positive) or 1 (negative)
Store the exponent (power of 2) as Excess_127
Store the mantissa as an unsigned integer.
Add 0s to the right of the mantissa to make the
total number of mantissa bits equal to 23
©Brooks/Cole, 2003
Table 3.11 Example of floating-point representation
-22
+2-6
-2-3
Number
-----------x 1.11000011
x 1.11001
x 1.110011
Sign
---1
0
1
Exponent
----------10000001
01111001
01111100
Mantissa
------------------------------11000011000000000000000
11001000000000000000000
11001100000000000000000
©Brooks/Cole, 2003
Floating point Interpretation
(Single Precision)
• The sign is the leftmost bit
• Change the next 8 bits to decimal and subtract
127 from it.
• add 1 to the left of the mantissa (next 23 bits)
and discard the zeros to the left of it.
• Move the decimal point to its original position.
• Convert the integer and fraction to decimal
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Hexadecimal Notation
• Floating point numbers can be written using
hexadecimal notation.
• For example :
81.125, using IEEE standards is
01000010101000111001000000000000
And in Hexadecimal notation:
x42A39000
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