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Operations on Numbers
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Binary Addition
Binary addition is very simple.
Same rule as decimal + 0 1
0 0 1
1 1 10
1 1 1 1 0 1
+ 1 0 1 1 1
--------------------- 0
1
0
1
1
1 1 1 1
1 1 0 0
carries
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Multiplication of Binary numbers
Much the same as decimal multiplication, except that the
multiplication operations are much simpler
1 0 1 1 1
X 1 0 1 0
-----------------------
0 0 0 0 0
1 0 1 1 1
0 0 0 0 0
1 0 1 1 1
-----------------------
1 1 1 0 0 1 1 0
* 0 1
0 0 0
1 0 1
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Complements
Common use of complement: subtraction operation.
Perform subtraction through the addition operation.
Two types of complements for each base-r system:
rs complement ( radix complement )
(r-1)s complement ( diminished radix complement)
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(r-1)s Complements
(r-1)s complement for a number N with n digits in base-r
numbering system is defined as:
(rn -1 ) N
Example #1: Find the 9s complement of 12389:
= (105 -1 ) - 12389
= 99999 -12389 = 87610
Example #2: Find the 9s complement of 1234:
= (104 -1) 1234 = 9999 -1234
= 8765
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(r-1)s Complements
Example #3: Find the 1s complement of 1011001:
= (27 1 ) 101 1001
= 111 1111 101 1001 = 010 0110
Notice that the 1s complement of binary numbers is formed by
changing 1s to 0s and 0s to 1s
Example: 1s complement of 0001111 is 1110000.
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(rs complement)
(rs) complement for number N in base r with n digits is defined
as rn N
Also, rs complement= ( r-1)s complement +1
Example #1: Find the 10s complement of 2389:
=9999 - 2389 = 7610 + 1 = 7611
Example #2: Find the 2s complement of 10 1100:
= 11 1111 - 10 1100 = 01 0011 + 1
= 010100
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Subtraction with complements
(Unsigned Numbers)
Use the following algorithms:
1. Add M to rs complement of N
2. If M>=N, then subtract the sum to rn to form the
result.
3. if M
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Unsigned Example #1:
Do the operation: 3250 - 72532
M = 03250
10s complement of N = + 27468
------------
30718
Since N is > M the result is:
- (10s complement 30718) =-69282
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Unsigned Example #2:
Perform the operation using 2s complement: 1010100 - 1000011
Solution:
X = 1010100 = 101 0100 (copy)
- Y = 1000011 =+ 011 1101 (2s)
1 001 0001
- 1 000 0000 001 0001
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Signed Binary Numbers
01001 can be considered unsigned binary = 9 or signed binary =
+9
But, 11001 can be considered unsigned binary = 25 or signed
binary = -9
Signed numbers uses 0 for + and 1 for - , this system is called
Signed-magnitude convention
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Signed Binary Numbers
All negative numbers have 1 in leftmost bit
Signed magnitude is mostly used in ordinary arithmetic.
The 1s complement is mostly used in logical operations.
The 2s complement is mostly used in computer arithmetic.
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Decimal signed-2s signed-1s signed
complement complement magnitude
-------- ----------- ------------- -----------
+ 7 0111 0111 0111
+2 0010 0010 0010
+1 0001 0001 0001
+0 0000 0000 0000
-0 --- 1111 1000
-1 1111 1110 1001
-2 1110 1101 1010
-7 1001 1000 1111
-8 1000 ------- ------
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Binary Coded Decimal (BCD)
Uses 4 bits to encode one decimal digit
Example: (4321)10 = 0100 0011 0010 0001
Invalid digits: 1010, 1011,1100,1101, 1110,and 1111.
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BCD Arithmetic Rules:
1. Add 2 BCD numbers using regular binary addition.
2. Check each nibble (4-bit), if result is greater than 9, then
add 6 to it.
3. If there is a carry between 2 nibble or coming from 2th
nibble add 6.
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BCD Addition
Example: 27 0010 0111
+ 34 0011 0100
------ -----------------
61 0101 1011 > 9
+ 0110 + 6
----------------
0110 0001 (61)
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BCD Addition
Example: 1 carry
59 0101 1001
+ 39 0011 1001
----------------
1001 0010
0110 (+6)
----------------
1001 1000 (98)
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BCD Addition
1 1 carry
98 1001 1000
+ 89 1000 1001
---------------
1 0010 0001
+ 0110 0110 (+6 6)
---------------
1 1000 0111 (187)
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Character Representation
ASCII American Standard Code for Information Interchange
128 characters (7 bits required)
Contains:
Control characters (non-printing)
Printing characters (letters, digits, punctuation)
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ASCII Characters
Hex Equiv. Binary Character
00 00000000 NULL
07 00000111 Bell
09 00001001 Horizontal tab
0A 00001010 Line feed
0D 00001110 Carriage return
20 00100000 Space (blank)
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ASCII Characters
Hex Equiv. Binary Character
30 00110000 0
31 00110001 1
39 00111001 9
41 01000001 A
42 01000010 B
61 01100001 a
62 01100010 b
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Error-Detecting Code
To detect the error in data communication, an 8th bit is added
to ASCII character to indicate its parity.
Parity bit - extra bit included with a message to make the total
number of 1s either even or odd
The 8-bit characters included parity bits (with even parity) are
transmitted to their destination. If the parity of received
character is not even it means at least one bits has been
changed.
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Binary Storage and Registers
Register - group if binary cells that are responsible for
storing and holding the binary information.
Register transfer operation is transferring binary operation
from one set of registers to another set of registers.
Digital logic circuits process the binary information stored in
the registers.
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Exercises
Perform the following operation using a) radix complement b)
diminished radix complement
1. 20910-12010 = _____ 10
2. 101001112 100011102 = _________2
3. AB1916 1EB316 =________ 16
