1 number systemslecture 8. 2 binary (base 2) numbers
TRANSCRIPT
1
Number Systems Lecture 8
2
BINARY(BASE 2)
numbers
3
DECIMAL(BASE 10)
numbers
4
Decimal (base 10) number system consists of 10 symbols or digits
0 1 2 3 4
5 6 7 8 9
5
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
6
Binary (base 2) number system consists of just two
0 1
7
Binary
On the pattern of Decimal numbers one could visualize Binary Representations:
Powers
1012=1 x 22 +0 x 21 + 1 x 20 =510
8
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
9
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)
10
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
11
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
12
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
13
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
14
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
15
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
17
Decimal (base 10) numbers are expressed in the positional notation
4202 = 2x100 + 0x101 + 2x102 + 4x103
100’s multiplier
100
18
Decimal (base 10) numbers are expressed in the positional notation
4202 = 2x100 + 0x101 + 2x102 + 4x103
1000’s multiplier
1000
19
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
20
Binary (base 2) numbers are also expressed in the positional notation
10011 = 1x20 + 1x21
+ 0x22 + 0x23
+ 1x24
1’s multiplier
1
21
Binary (base 2) numbers are also expressed in the positional notation
10011 = 1x20 + 1x21
+ 0x22 + 0x23
+ 1x24
2’s multiplier
2
22
Binary (base 2) numbers are also expressed in the positional notation
10011 = 1x20 + 1x21
+ 0x22 + 0x23
+ 1x24
4’s multiplier
4
23
Binary (base 2) numbers are also expressed in the positional notation
10011 = 1x20 + 1x21
+ 0x22 + 0x23
+ 1x24
8’s multiplier
8
24
Binary (base 2) numbers are also expressed in the positional notation
10011 = 1x20 + 1x21
+ 0x22 + 0x23
+ 1x24
16’s multiplier
16
25
Counting in Decimal0123456789
10111213141516171819
20212223242526272829
30313233343536...
01
1011
100101110111
10001001
101010111100110111101111
10000100011001010011
10100101011011010111110001100111010110111110011101
1111011111
100000100001100010100011100100
.
.
.
Counting in Binary
26
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?
27
Convert 75 to Binary75237 1218 129 024 122 021 020 1
1001011
remainder
28
Check
1001011 = 1x20 + 1x21
+ 0x22 + 1x23
+
0x24 + 0x25
+ 1x26
= 1 + 2 + 0 + 8 + 0 + 0 + 64
= 75
29
Convert 100 to Binary100250 0225 0212 126 023 021 120 1
1100100
remainder
30
That finishes the - introduction to binary numbers and their conversion to and from decimal numbers