lecture 7 how computers process data (number systems) presented by md. mahbubul alam, phd 1
TRANSCRIPT
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Lecture 7How computers process data (Number Systems)
PRESENTED BY MD. MAHBUBUL ALAM, PHD 1
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Common Number SystemsCommon Number Systems
System Base SymbolsUsed by humans?
Used in computers?
Decimal 10 0, 1, … 9 Yes No
Binary 2 0, 1 No Yes
Octal 8 0, 1, … 7 No No
Hexa-decimal
16 0, 1, … 9,A, B, … F
No No
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Quantities/Counting (1 of 3)Quantities/Counting (1 of 3)
Decimal Binary OctalHexa-
decimal
0 0 0 0
1 1 1 1
2 10 2 2
3 11 3 3
4 100 4 4
5 101 5 5
6 110 6 6
7 111 7 7
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Quantities/Counting (2 of 3) Quantities/Counting (2 of 3)
Decimal Binary OctalHexa-
decimal
8 1000 10 8
9 1001 11 9
10 1010 12 A
11 1011 13 B
12 1100 14 C
13 1101 15 D
14 1110 16 E
15 1111 17 F
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Quantities/Counting (3 of 3) Quantities/Counting (3 of 3)
Decimal Binary OctalHexa-
decimal
16 10000 20 10
17 10001 21 11
18 10010 22 12
19 10011 23 13
20 10100 24 14
21 10101 25 15
22 10110 26 16
23 10111 27 17 Etc.
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Conversion Among BasesConversion Among Bases
The possibilities:
Hexadecimal
Decimal Octal
Binary
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Quick Example
2510 = 110012 = 318 = 1916
Base
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Decimal to Decimal (just Decimal to Decimal (just for fun)for fun)
Hexadecimal
Decimal Octal
Binary
Next slide…
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12510 => 5 x 100 = 52 x 101 = 201 x 102 = 100
125
Base
Weight
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Binary to DecimalBinary to Decimal
Hexadecimal
Decimal Octal
Binary
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Binary to DecimalBinary to Decimal
Technique◦ Multiply each bit by 2n, where n is the “weight” of the
bit◦ The weight is the position of the bit, starting from 0 on
the right◦ Add the results
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ExampleExample
1010112 => 1 x 20 = 11 x 21 = 20 x 22 = 01 x 23 = 80 x 24 = 01 x 25 = 32
4310
Bit “0”
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Octal to DecimalOctal to Decimal
Hexadecimal
Decimal Octal
Binary
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Octal to DecimalOctal to Decimal
Technique◦ Multiply each bit by 8n, where n is the “weight” of the
bit◦ The weight is the position of the bit, starting from 0 on
the right◦ Add the results
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ExampleExample
7248 => 4 x 80 = 42 x 81 = 167 x 82 = 448
46810
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Hexadecimal to DecimalHexadecimal to Decimal
Hexadecimal
Decimal Octal
Binary
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Hexadecimal to Decimal
Technique◦ Multiply each bit by 16n, where n is the “weight” of the
bit◦ The weight is the position of the bit, starting from 0 on
the right◦ Add the results
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ExampleExample
ABC16 => C x 160 = 12 x 1 = 12 B x 161 = 11 x 16 = 176 A x 162 = 10 x 256 = 2560
274810
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Decimal to BinaryDecimal to Binary
Hexadecimal
Decimal Octal
Binary
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Decimal to Binary
Technique◦ Divide by two, keep track of the remainder◦ First remainder is bit 0 (LSB, least-significant bit)◦ Second remainder is bit 1◦ Etc.
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ExampleExample
12510 = ?22 125 62 12 31 02 15 12 7 12 3 12 1 12 0 1
12510 = 11111012
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Octal to BinaryOctal to Binary
Hexadecimal
Decimal Octal
Binary
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Octal to BinaryOctal to Binary
Technique◦ Convert each octal digit to a 3-bit equivalent binary
representation
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ExampleExample
7058 = ?2
7 0 5
111 000 101
7058 = 1110001012
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Hexadecimal to Binary
Hexadecimal
Decimal Octal
Binary
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Hexadecimal to Binary Technique
◦ Convert each hexadecimal digit to a 4-bit equivalent binary representation
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Example10AF16 = ?2
1 0 A F
0001 0000 1010 1111
10AF16 = 00010000101011112
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Decimal to Octal
Hexadecimal
Decimal Octal
Binary
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Decimal to Octal Technique
◦ Divide by 8◦ Keep track of the remainder
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Example123410 = ?8
8 1234 154 28 19 28 2 38 0 2
123410 = 23228
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Decimal to Hexadecimal
Hexadecimal
Decimal Octal
Binary
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Decimal to Hexadecimal
Technique◦ Divide by 16◦ Keep track of the remainder
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Example123410 = ?16
123410 = 4D216
16 1234 77 216 4 13 = D16 0 4
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Binary to Octal
Hexadecimal
Decimal Octal
Binary
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Binary to Octal Technique
◦ Group bits in threes, starting on right◦ Convert to octal digits
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Example10110101112 = ?8
1 011 010 111
1 3 2 7
10110101112 = 13278
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Binary to Hexadecimal
Hexadecimal
Decimal Octal
Binary
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Binary to Hexadecimal Technique
◦ Group bits in fours, starting on right◦ Convert to hexadecimal digits
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Example
10101110112 = ?16
10 1011 1011
2 B B
10101110112 = 2BB16
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Octal to Hexadecimal
Hexadecimal
Decimal Octal
Binary
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Octal to Hexadecimal Technique
◦ Use binary as an intermediary
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Example10768 = ?16
1 0 7 6
001 000 111 110
2 3 E
10768 = 23E16
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Hexadecimal to Octal
Hexadecimal
Decimal Octal
Binary
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Hexadecimal to Octal Technique
◦ Use binary as an intermediary
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Example1F0C16 = ?8
1 F 0 C
0001 1111 0000 1100
1 7 4 1 4
1F0C16 = 174148
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Exercise – Convert ...
Don’t use a calculator!
Decimal Binary Octal
Hexa-decimal
33
1110101
703
1AF
Skip answer Answer
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Exercise – Convert …
Decimal Binary Octal
Hexa-decimal
33 100001 41 21
117 1110101 165 75
451 111000011 703 1C3
431 110101111 657 1AF
Answer
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Binary Arithmetic: Addition & Binary Arithmetic: Addition & SubtractionSubtraction
X Y X+Y
0 0 0
0 1 1
1 0 1
1 1 10
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X Y X-Y
0 0 0
0 1 1
1 0 1
1 1 0
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Binary Arithmetic: Binary Arithmetic: Multiplication & DivisionMultiplication & Division
X Y X*Y
0 0 0
0 1 0
1 0 0
1 1 1
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Boolean AlgebraBoolean Algebra• The digital circuits present in a digital computer
are designed using a mathematical discipline known as Boolean Algebra.
• It describes the relationship between the inputs and outputs of a digital circuit.
• Boolean Algebra was named in honor of Gorge Boole, an English Mathematician, who had proposed the basic principles of this.
• Objective:• Boolean Algebra is used mainly by design engineers in
order to obtain the required output by using least number of logic gates.
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ComponentsComponents
Like any other algebra, Boolean Algebra also uses variables and operations.
◦ A Boolean variable has only two possible values which is either true (1) or false (0)
◦ Basic Boolean operations are: AND, OR and NOT
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Basic Logical OperationsBasic Logical OperationsAll these three basic logical operations can be represented symbolically as
◦ A AND B = A . B◦ A OR B = A + B◦ NOT A = A’
These operations can be defined in a form known as Truth Table, which s a list of all possible input values and the output for each input combination.
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Truth Table for AND Truth Table for AND OperatorOperator
Truth Table for a 2-input AND Operator is as follows
A B Y = A . B
0 0 0
0 1 0
1 0 0
1 1 1
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Truth Table for OR OperatorTruth Table for OR Operator
Truth Table for a 2-input OR Operator is as follows
A B Y = A + B
0 0 0
0 1 1
1 0 1
1 1 1
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Truth Table for NOT Truth Table for NOT OperatorOperator
Truth Table for NOT Operator is as follows
A Y = A’
0 1
1 0
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Logic GateLogic Gate In electronics, a logic gate is an idealized or physical device implementing a Boolean function; that is, it performs a logical operation on one or more logical inputs, and produces a single logical output.
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Thank you
(Courtesy: Dept. of IT, York University)
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