Transcript Number Systemes n Computer Codesx

Chapter 2
NUMBER SYSTEM AND
COMPUTER CODES
Prelude
• Fingers, sticks, and other things for counting were
not enough!
• Counting large numbers
• Count in groups
Evolution of the
number system
Number systems
A set of values used to represent quantity
• Non-Positional Number Systems
• count with their fingers, stones and pebbles
• difficult to perform arithmetic operations
• No zero, difficult to calculate large numbers
• E.g. the Roman number system
• Positional Number Systems
• Finite number of symbols to represent any
numbers
• Symbol and it’s position defines a number
• Decimal, binary, octal, hexadecimal
ASCII- American standard for Information Interchange
Base or radix
• Number of unique digits
Number Systems - Decimal
• The decimal system is a base-10 system.
• There are 10 distinct digits (0 to 9) to
represent any quantity.
• For an n-digit number, the value that each
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digit represents depends on its weight or
position.
The weights are based on powers of 10.
1024 = 1*103 + 0*102 + 2*101 + 4*100
= 1000 + 20 + 4
Number Systems - Binary
• The binary system is a base-2 system.
• There are 2 distinct digits (0 and 1) to
represent any quantity.
• For an n-digit number, the value of a digit in
each column depends on its position.
• The weights are based on powers of 2.
10112 = 1*23 + 0*22 + 1*21 + 1*20
=8+2+1 =1110
Number Systems - Octal
• Octal and hexadecimal systems provide a
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•
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shorthand way to deal with the long strings of
1’s and 0’s in binary.
Octal is base-8 system using the digits 0 to 7.
To convert to decimal, you can again use a
column weighted system
75128 = 7*83 + 5*82 + 1*81 + 2*80 =
391410
An octal number can easily be converted to
binary by replacing each octal digit with the
corresponding group of 3 binary digits
75128 = 1111010010102
Number Systems - Hexadecimal
• Hexadecimal is a base-16 system.
• It contains the digits 0 to 9 and the letters A
to F (16 digit values).
• The letters A to F represent the unit values 10
to 15.
• This system is often used in programming as a
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condensed form for binary numbers (0x00FF,
00FFh)
To convert to decimal, use a weighted system
with powers of 16.
Example- Value of 2001 in Binary, Octal
and Hexadecimal
Example- Conversion: Binary
Hexadecimal
Octal
Converting decimal to binary, octal and
hexadecimal
• To convert from
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•
decimal to a different
number base such as
Octal, Binary or
Hexadecimal involves
repeated division by
that number base
Keep dividing until
the quotient is zero
Use the remainders in
reverse order as the
digits of the
converted number
Repeated Divide by 2
BaseN to Decimal Conversions
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04/03/10
Multiply each digit by increasing powers of the base
value and add the terms
Example: 101102 = ??? (decimal)
Binary Addition
4 Possible Binary Addition Combinations:
(1)
0
(2)
0
+0
+1
Carry
Sum
00
01
(3)
1
(4)
1
Ex 1,2,3
+0
+1
For Exam
01
10
• Similar to decimal operation
• Leading zeroes are frequently dropped.
Binary Subtraction
Just like subtraction in any other base
Minuend
Subtrahend
Difference
10110
-10010
00100
And when a borrow is needed. Note that the
borrow gives us 2 in the current bit position.
Ex 1,2
For Exam
And a full example
And more ripple -
Octal/Hex addition/subtraction
Octal Addition
1 1 1
Carries
5 4 7 1 Augends
+ 3 7 5 4 Addend
11445 Sum
Hexadecimal Addition
1 0
5
+ D
1 2
1
B
0
C
1
A 9
5 8
0 1
Carries
Augend
Addend
Sum
Octal Subtraction
Hexadecimal Subtraction
6 10
7 4
- 5 6
1 6
4 10
5 1
4 3
0 6
Borrows
Minuend
Subtrahend
Difference
+
9 10
A 5
5 8
4 D
A 10
B 9
0 D
A C
Borrows
Minuend
Subtrahend
Difference
BCD
Binary-coded decimal, or BCD, is a method of
using binary digits to represent the decimal
digits 0 through 9. A decimal digit is
represented by
four binary digits …
The binary combinations 1010 to
1111 are invalid and are not used.
ASCII Code
"ask-key“- common code for microcomputer
Standard ASCII character set
• 128 decimal numbers ranging (0-127)
• Assigned to letters, numbers, punctuation marks,
and the most common special characters.
The Extended ASCII Character Set
• also consists of 128 decimal numbers (128-255)
• representing additional special, mathematical,
graphic, and foreign characters.
Groups of 32 characters
EBCDIC - Extended Binary Coded
Decimal Interchange Code
• It is an 8 bit character encoding
• Used on IBM mainframes and AS/400s.
• It is descended from punched cards
• The first four bits are called the zone
• category of the character
• Last four bits are the called the digit
• identify the specific character
There are a number of different versions of
EBCDIC, customized for different countries.
Assignments
IOA, IA, GA, Case !@#$
Binary
Multiplication
1 1 0 1 0 Multiplicand
x 1 0 1 0 Multiplier
00000
11010
00000
11010
1 0 0 0 0 0 1 0 0 Product
Division
1 1 0 Quotient
Divider 1 0 0 1 1 1 1 1 0 1 Dividend
1001
1100
1001
1 1 1 Remainder