Transcript m5zn_d45f00ca48164aa

Number systems, Operations, and Codes
1- Decimal Numbers
The decimal number system has ten digits .
These are : 0 , 1 , 2 , 3, 4 , 5 , 6 , 7 , 8 , 9 .
The decimal number
system has the base = 10
Example -1-
Example -2-
Example -3- : Express the decimal number 897.9 as a sum of the value
of each digit.
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2- Binary Numbers
The binary number system has two digits (bits) .
These are : 0 , 1.
The binary number system
has the base = 2
The weight of a bit
increases from right to left
in a binary whole number
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Decimal to Binary Conversion
Method : To get the binary number for a given decimal number ,
divide decimal number by 2 until the quotient is 0 .
Remainders form the binary number .
Example -1-
Example -2- :
convert the
following decimal
numbers into
binary :
19
-
45
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Binary to Decimal Conversion
Method : Add the weights of all “1”s in a binary number to get
the decimal values
Example -1-
Example -2-
Example -3- : convert the following binary numbers
10101110 , 11.011101 into decimal number ?
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Convert Decimal Fraction to binary
Method : Repeated multiplication by 2 until fractional part is zero
Example -1-
Example -2- : convert the following decimal numbers into binary :
0.375
0.559
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3- Octal Numbers
The Octal number system has 8 digits.
These are : 0 , 1, 2 , 3, 4 , 5 , 6, 7
The Octal system has the base = 8
Assignment:
1. How could we convert Octal to decimal ? give an example .
2. How could we convert Decimal to Octal ? give an example .
3. How could we convert Binary to Octal ? give an example .
4. How could we convert Octal to Binary ? give an example .
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Convert Decimal to Octal

Divide the given decimal number by base 8 and write the
reminders starting first reminder from right to left
Example: (57)10 = (?)8
Answer: (71)8
Or using polynomial of weights, like: … 83 82 81 80
512 64 8 1
Example: (413)10 =
0 6
3 5
Where: (635)8 = 5.80 + 3.81 + 6.82 =
5.1 + 3.8 + 6.64 = 5 + 24 + 384 =
(413)10
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4- Hexadecimal Numbers
The hexadecimal number system has 16 digits.
These are : 0 , 1, 2 , 3, 4 , 5 , 6, 7 , 8 , 9, A , B , C , D , E , F
The hexadecimal system has
the base = 16
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Convert Decimal to Hexadecimal

Divide the given decimal number by base 16 and write
the reminders starting first reminder from right to left
Example: (57)10 = (?)16
Answer: (39)16
Or using polynomial of weights, like: … 162 161 160
256 16
1
Example: (413)10 =
1
9
D
Where: (19D)16 = D.160 + 9.161 + 1.162 =
13.1 + 9.16 + 1.256 =
(413)10
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Convert between different Number systems



Base of binary system is 2 = 21
Base of Octal System is 8 = 23
Base of Hexadecimal system is 16 = 24

It means each Hexadecimal digit ~
4 binary digits, and each Octal digit ~
3 binary digits
Examples: Convert the binary value 110011000101100
To the corresponding Octal and hexadecimal value
110 011 000 101 100 = 63054 Octal
110 0110 0010 1100 = 662C Hexadecimal!
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Hexadecimal Numbers (cont.)
Binary to hexadecimal
Method:
•
Break the binary number into 4-bit groups starting at the rightmost bit , Then:
•
Replace each 4-bit group with the equivalent hexadecimal symbol.
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Hexadecimal Numbers (cont.)
hexadecimal to Binary
Method :
Replace each hexadecimal symbol with the appropriate four bits.
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Hexadecimal Numbers (cont.)
hexadecimal to Decimal
Method
Convert the hexadecimal to binary then convert from binary to decimal.
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Hexadecimal Numbers (cont.)
Decimal to hexadecimal
Method :
Repeated division of a decimal number by 16 .
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Arithmetic Operations
1st and 2nd Complement
1st complement :
Method : Invert each bit to get the 1st complement
Example -1-
Example -2- : Determine the first complement of the following binary
00011010
-
11110111 - 10001101
2nd complement :
Method -1- :
2nd complement = 1st complement
+
1
Example -115
Arithmetic Operations
1st and 2nd Complement (cont.)
2nd complement (cont.) :
Method -2- : Change all
the bits to the left of the
least significant 1 to gets
the 2nd complement
Example -1- : Determine the 2nd complement of the following binary
00010110
- 11111100
-
10010001
Application
Example
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