Post on 28-Dec-2015
transcript
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Number Systems Lecture 8
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BINARY(BASE 2)
numbers
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DECIMAL(BASE 10)
numbers
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Decimal (base 10) number system consists of 10 symbols or digits
0 1 2 3 4
5 6 7 8 9
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Decimal
The numbers are represented in Units, Tens, Hundreds, Thousands; in other words as 10power.
191=1 x 102 + 9 x 101 + 1 x 100
increasing power
Th.Hund.Tens.Units
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Binary (base 2) number system consists of just two
0 1
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Binary
On the pattern of Decimal numbers one could visualize Binary Representations:
Powers
1012=1 x 22 +0 x 21 + 1 x 20 =510
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Conversion
2 240
2 120 0
2 60 0
2 30 0
2 15 0
2 7 1
2 3 1
2 1 1
= 11110000128 64 32 16 8 4 2 1
= 128 + 64 + 32 + 16 = 240
Decimal to Binary
Binary to Decimal
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Other popular number systems
• Octal– base = 8– 8 symbols (0,1,2,3,4,5,6,7)
• Hexadecimal– base = 16– 16 symbols (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F)
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Codes
• Octal Power
• Hexadecimal
83 82 81 80
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 1 2 3 4 5 6 7 8 9 A B C D E F
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Code conversions
Decimal to Octal
Similar to Dec-to Binary:
Start dividing by 8
and build Octal figures from Remainders:
24010=3608
8 240
8 30 0
8 3 6
0 3
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Code conversionsBinary –to-Octal
3 bits of binary could provide weight of 810, that is equivalent to Octal;i.e:
Bits: b2 b1 b0
Wts: 4 2 1
Bin: 1 1 1 710 and Octal range 0-7
Example: 100111010= 100 111 010
Octal Values: 4 7 2
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Octal Conversionbinary: Octal:
011 010 1102
3 2 6 3268
256 128 64 32 16 8 4 2 1
3 x 82 + 2 x 81 + 6 x 80
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Decimal (base 10) numbers are expressed in the positional notation
4202 = 2x100 + 0x101 + 2x102 + 4x103
The right-most is the least significant digit
The left-most is the most significant digit
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Decimal (base 10) numbers are expressed in the positional notation
4202 = 2x100 + 0x101 + 2x102 + 4x103
1’s multiplier
1
16
Decimal (base 10) numbers are expressed in the positional notation
4202 = 2x100 + 0x101 + 2x102 + 4x103
10’s multiplier
10
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Decimal (base 10) numbers are expressed in the positional notation
4202 = 2x100 + 0x101 + 2x102 + 4x103
100’s multiplier
100
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Decimal (base 10) numbers are expressed in the positional notation
4202 = 2x100 + 0x101 + 2x102 + 4x103
1000’s multiplier
1000
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Binary (base 2) numbers are also expressed in the positional notation
10011 = 1x20 + 1x21
+ 0x22 + 0x23
+ 1x24
The right-most is the least significant digit
The left-most is the most significant digit
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Binary (base 2) numbers are also expressed in the positional notation
10011 = 1x20 + 1x21
+ 0x22 + 0x23
+ 1x24
1’s multiplier
1
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Binary (base 2) numbers are also expressed in the positional notation
10011 = 1x20 + 1x21
+ 0x22 + 0x23
+ 1x24
2’s multiplier
2
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Binary (base 2) numbers are also expressed in the positional notation
10011 = 1x20 + 1x21
+ 0x22 + 0x23
+ 1x24
4’s multiplier
4
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Binary (base 2) numbers are also expressed in the positional notation
10011 = 1x20 + 1x21
+ 0x22 + 0x23
+ 1x24
8’s multiplier
8
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Binary (base 2) numbers are also expressed in the positional notation
10011 = 1x20 + 1x21
+ 0x22 + 0x23
+ 1x24
16’s multiplier
16
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Counting in Decimal0123456789
10111213141516171819
20212223242526272829
30313233343536...
01
1011
100101110111
10001001
101010111100110111101111
10000100011001010011
10100101011011010111110001100111010110111110011101
1111011111
100000100001100010100011100100
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.
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Counting in Binary
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Why binary?Because this system is natural for digital computers
The fundamental building block of a digital computer – the switch – possesses two natural states, ON & OFF.
It is natural to represent those states in a number system that has only two symbols, 1 and 0, i.e. the binary number system
In some ways, the decimal number system is natural to us humans. Why?
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Convert 75 to Binary75237 1218 129 024 122 021 020 1
1001011
remainder
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Check
1001011 = 1x20 + 1x21
+ 0x22 + 1x23
+
0x24 + 0x25
+ 1x26
= 1 + 2 + 0 + 8 + 0 + 0 + 64
= 75
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Convert 100 to Binary100250 0225 0212 126 023 021 120 1
1100100
remainder
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That finishes the - introduction to binary numbers and their conversion to and from decimal numbers