1. Chapter 1: Binary Systems
Topics in this Chapter:
Digital Systems
Numbering Systems
Number Base Conversions
Complements
Arithmetic Operations
Binary Coded Decimal (BCD)
Binary Storage and Registers
Binary Logic
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2. Digital Systems
Discrete quantities: weekly salaries, income taxes, letters or alphabet, digits
Digital systems: digital telephones, digital cameras, digital computers and etc.
Digital systems have the ability to manipulate discrete quantities.
Some digital systems can operate with extreme reliability by using error-correcting code. For example: DVD
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3. Numbering Systems Decimal numbering systems:
A decimal digit can have the values of 0,1,2,3,4,5,6,7,8 and 9. The number of combination of 2 decimal digits is In general for any numbering system the number of combination of n digits can be determined by: Bn
where B is the base of numbering system and n is the number of digits to be combined.
For example for two decimal digits the largest number is 99 so the two decimal digits end at 99. Using the formula of Bn the total number of combinations for two decimal digits would be 102 = 100
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4. Numbering Systems A decimal number such as 7,392.42 is equal to:
7 * * * * * * 10-2
In general for any numbering system
an an-1 an-2…… a1 a0. a-1 a-2……. a-m
the equal decimal number is:
an* rn + an-1* rn-1 + ….a1* r + a0 + a-1* r-1+… am* rm
where aj is the coefficient
and
the number expressed in a base-r system
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5. Octal Numbering System
The Octal numbering system is a base-eight numbering system with eight digits of
0,1,2,3,4,5,6,7
Decimal Octal
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6. Octal Numbering System
For example to count in octal the digits combine after reaching a count of 7
1,..7,10,11,12,…,15,16,17,20,21,…,75,76,77,100
for two octal digits the largest number is 77 so the two octal digits end after at 77. Using the formula of Bn the total number of combinations for two octal digits would be 82 = 64
To find the decimal number equal to an octal number:
(127.4)8= 1* * * * 8-1 = (87.5)10
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7. The Hexadecimal Numbering System
It is a base-sixteen numbering system. That is there are 16 digits in this system:
Hexadecimal Decimal
…… …… A B C D E F
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8. The Hexadecimal Numbering System
For example to count in hexadecimal
…F,10,11,12,…19,1A,1B,1C,1D,1E,1F,20,
21,….99,9A,…,9F,A0,A1…..,FE,FF,100
for two hexadecimal digits the largest number is FF so the two hexadecimal digits end after at FF. Using the formula of Bn the total number of combinations for two hexadecimal digits would be 162 = 256.
To find the decimal number equal to a hexadecimal number:
(B65F)16= 11 * * * * 160 = (46,687)10
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9. The Binary Numbering System
Only two digits required {1,0} 21 20
Base 10 Equivalent 1 2 3
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10. The Binary Numbering System
For four binary digits the largest number is 1111 so the four binary digits end at Using the formula of Bn the total number of combinations for four binary digits would be 24 = 16.
To find the decimal number equal to a binary number:
(110101)2 = = (53)10
(0.1101)2 =(1*.5 +1* * *.0625 )10
=(0.8125) 10
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11. Adding Binary Numbers Same rule as decimal Example: + 1 10 1 1 11 10
1 1 1
carry
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12. Multiplication of Binary numbers
Multiplication Table Example: * 1
1 1 1 *
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13. Number Base Conversions
For converting decimal number to binary number REPETETIVE DIVISION is used as follows:
The following algorithm generates the binary digits.
Q = decimal number
While Q is not equal to 0 do the following
Binary digit = Q mod 2
Q = Q / 2 (quotient)
End While
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14. Conversion Example Convert (58)10 to (?)2 58 mod 2 = 0 29
1 mod 2 = 1 1 Ans: (111010)2
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15. Conversion Example
To convert a fraction, it multiples by 2 to give an integer and a fraction and only the fraction again multiplies by 2. This process continues until the faction becomes zero or until the number of digits have sufficient accuracy:
(0.6875) 10 = ( ? ) 2
* 2 =
* 2 =
*2 =
* 2 =
== result is (0.1011)2
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16. Hex to Binary Converting from Hex to Binary is easy:
Every hex digit becomes 4 binary digits
(1AF5) 16
=( ) 2
=( )2
(306.D)16
= ( )2
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17. Binary to Hex Just as simple, reverse to process
( ) 2
=( ) 2
=(3955D) 16
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18. Octal to Binary Converting from Octal to Binary is trivial:
Every octal digit becomes 3 binary digits
(175 ) 8
=( ) 2
=( ) 2
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19. Binary to Octal Just as simple, reverse to process
( ) 2
=( ) 2
=(312535) 8
Using the hex and octal equivalent instead of binary numbers are more convenient and less prone to errors.
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20. Complements
One of the use of complements is the simplifying subtraction operation. We want to be able to perform subtraction through the addition operation.
There are two types of complements for each base-r system:
r’s complement ( radix complement )
(r-1)’s complement ( diminished radix complement)
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21. (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
For example for decimal numbers since r is 10,
(r-1)’s complement of a number N is:
( 10n – 1) – N
For example: 9’s complement of is
( ) = =
Or 9’s complement of 2389 is
(104 -1) – 2389 = = 7610
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22. (r-1)’s Complements
1’s complement of a number N, is defined as (2n – 1 ) – N
For example for it is equal to :
( 27 – 1 ) – =
Therefore, 1’s complement of binary numbers is formed by changing 1’s to 0’s and 0’s to 1’s
For example 1’s complement of is
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23. (r’s complement) (r’s) complement for number N in base r with n digits
is defined as rn – N
also it is clear r’s complement= ( r-1)’s complement +1
Therefore we can use (r-1)’s complements of the previous examples to calculate their r’s complement :
10’s complement of 2389= = 7611
2’s complement = ’s complement
010100
2’s complement of by using rn – N definition generates same result:
2n – N =
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24. Subtraction with complements (Unsigned Numbers)
For doing the subtraction on two n digits unsigned numbers M and N the in base r the following algorithm can be used:
1- Add M to r’s complement of N
2- If M>=N subtract rn form the result
3- if M<N place a negative sign “-” in front of r’s complement of the result
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25. Subtraction with complements (Unsigned Numbers)
For example
M =
10’s complement of N =
30718
Since N is > M the result is
- (10’s complement 30718) =-69282
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26. Subtraction with complements (Unsigned Numbers)
For Binary numbers the algorithm is same. See the examples in the book
Subtraction also can be done by (r-1)’s complement. In that case if M>=N, after discarding the end carry one should be added to result:
For example for X = Y= :
2’s compl ’s compl.
X =
+ Y =
The problem is how we can simplify the subtraction for signed numbers? For example how about 2- (-1)?
For doing that first we should represent a signed number and then devise an algorithm for the arithmetic operations of the signed numbers. See the next slides…
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27. Signed Binary Numbers
Positive integer numbers can be represented by unsigned numbers. However we need a notation for negative numbers:
A negative binary number assumed to be signed or unsigned. For example can be considered
unsigned binary = 9 or
signed binary = +9
Or can be considered
unsigned binary = 25 or
signed binary = -9
If the representation of the signed numbers uses 0 for “+” and 1 for “-” , this system is called Signed-magnitude convention
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28. Signed Binary Numbers
Another system to represent negative numbers is signed complement system
For example
+9 is in two systems
but -9 is
in signed-magnitude
in signed-1’s-complement (complementing all bits include sign bit or subtracting from 2n -1)
in signed-2’s-complement
( 2’complement of all bits or subtracting from 2n )
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29. Signed Binary Numbers
Representation method only matters when we are talking about negative numbers
All negative numbers have 1 in leftmost bit
The signed magnitude is mostly used in ordinary arithmetic
The 1’s complement is mostly used in logical operations
The 2’s complement is mostly used in computer arithmetic
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30. Decimal signed-2’s signed-1’s signed complement complement magnitude
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31. Arithmetic Addition (Signed Numbers)
The addition of binary numbers in signed-magnitude system follows the same rules as ordinary arithmetic. (sign of larger number)
For adding the numbers in 2’s complement form the result can be obtained by adding the two numbers including their sign bits. The carry of the sign bits is discarded and negative results are automatically in 2’s complement form. It means in order to get the negative result we must find the 2’s complement form of the result including the sign bit.
When adding 2 numbers of same sign (pos or neg) in 2’s complement form, if the result cannot be shown in the available bits then the overflow occurs that indicates the result is wrong. Generally when the sign of the result shown in available bits is different from the sign of the numbers, overflow has been occurred. See the next slide.
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32. Arithmetic Addition (2’s Comp)
Two positive numbers
Neg result
wrong, overflow
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33. Arithmetic Addition (2’s Comp.)
For example:
(5)
Discarded final carry. Note that sign bit also participates in the process
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34. Arithmetic Addition (2’s Comp.)
For example:
=> (2’s
comp.)
it is equal to 7 so the result is -7
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35. Arithmetic Addition (2’s Comp.)
For example:
=> (2’s comp.)
it is equal to 13 so the result is -13
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36. Arithmetic Subtraction
In 2’s complement format it is very simple :
Take the 2’s complement of sabtrahend (the second number) including the sign bit and add it to minuend (the first number) including the sign bit and discard a carry out of sign bit
By taking 2’s complement of the subtrahend its sign can be changed. Thus, we can change subtraction to addition operation.
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37. Arithmetic Subtraction (2’s Comp)
For example:
(+9)
-(+4) is changed to
(+9)
+(-4)
(+5) discard carry
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38. Arithmetic Subtraction (2’s Comp)
For example:
(-4)
-(+9) is changed to
(-4)
(2’s comp means magnitude is 13) discard carry => (-13)
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39. Arithmetic Subtraction
For example:
=> is changed to
=> discard carry
 is +7
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40. Binary Coded Decimal (BCD)
4 bits used to encode one decimal digit
For example (4321)10 =
The problem is BCD can not be used for conversion from decimal to binary and from binary to decimal. Because there is no decimal digits for 1010, 1011,1100,1101,1110,1111
For example : ( )2= (72?)
there is no decimal digits (0..9) for 1100
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41. BCD Arithmetic Add 2 BCD numbers using regular binary addition
Check each nibble ( 4-bit) if result is greater than 9 add 6 to it
If there is carry between 2 nibble or coming from 2th nibble add 6
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42. BCD Arithmetic Example: 27 0010 0111 + 34 0011 0100 -----------------
> 9
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43. BCD Arithmetic Example: 1 carry 59 0101 1001 + 39 0011 1001
0110
(98)
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44. BCD Arithmetic 1 1 carry 98 1001 1000 + 89 1000 1001 ---------------
(187)
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45. 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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46. ASCII – Characters Hex Equiv. Binary Character 00 00000000 NULL 07
Bell 09
Horizontal tab 0A
Line feed 0D
Carriage return 20
Space (blank)
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47. ASCII – Characters Hex Equiv. Binary Character 30 00110000 31 0011000131 1 39 9 41 A 42 B 61 a 62 b
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48. Error-Detecting Code
To detect the error in data communication, an eighth bit is added to ASCII character to indicate its parity.
A parity bit is an extra bit included with a message to make the total number of 1’s 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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49. Error-Detecting Code Example: even parity odd parity
ASCII A =
ASCII T =
The method of error checking with even parity detects any odd combinations of errors in each character that is transmitted. An even combination of errors is undetected with this method.
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50. Binary Storage and Registers
Register is a 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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53. Binary Logic
Binary logic or Boolean algebra deals with variables that take on two discrete values.
Binary logic consists of binary variables (e.g. A,B,C, x,y,z and etc.) that can be 1 or 0 and logical operations such as:
AND: x.y=z or xy=z (see AND truth table)
OR: x + y =z (see OR truth table)
NOT: x’ =z (not x is equal to z)
Note that binary logic is different from binary arithmetic. For example in binary logic =1 (1OR1) but in binary arithmetic = 10 (1 PLUS 1)
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54. Logic Gates
Logic gates are electronic circuits that operate on one or more input signals to produce an output signal.
Electrical signal can be voltage.
Voltage-operated circuits respond to two separate levels equal to logic 1 or 0.
Logical gates can be considered as a block of hardware that produced the equivalent of logic 1 or logic 0 output signals if input logical requirement are satisfied
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