number system
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Number Systems (Class XI)
Nita Arora (KHMS)PGT Comp. Sc.
2
Why Binary?
Early computer design was decimal Mark I and ENIAC
John von Neumann proposed binary data processing (1945) Simplified computer design Used for both instructions and data
Natural relationship betweenon/off switches and calculation using Boolean logic
01
NoYes
FalseTrue
OffOn
Number Systems 2-3
Counting and Arithmetic
Decimal or base 10 number system Origin: counting on the fingers “Digit” from the Latin word digitus meaning “finger”
Base: the number of different digits including zero in the number system Example: Base 10 has 10 digits, 0 through 9
Binary or base 2 Bit (binary digit): 2 digits, 0 and 1 Octal or base 8: 8 digits, 0 through 7 Hexadecimal or base 16:
16 digits, 0 through F Examples: 1010 = A16; 1110 = B16
Number Systems 2-4
Keeping Track of the Bits
Bits commonly stored and manipulated in groups 8 bits = 1 byte 4 bytes = 1 word (in many systems)
Number of bits used in calculations Affects accuracy of results Limits size of numbers manipulated by the
computer
Number Systems 2-5
Numbers: Physical Representation
Different numerals, same number of oranges Cave dweller: IIIII Roman: V Arabic: 5
Different bases, same number of oranges 510
1012
123
Number Systems 2-6
Number System
Roman: position independent Modern: based on positional notation (place
value) Decimal system: system of positional notation
based on powers of 10. Binary system: system of positional notation
based powers of 2 Octal system: system of positional notation based
on powers of 8 Hexadecimal system: system of positional
notation based powers of 16
Number Systems 2-7
Positional Notation: Base 10
Sum
Evaluate
Value
Place
340
3 x14 x 10
110
100101
1’s place10’s place
43 = 4 x 101 + 3 x 100
Number Systems 2-8
Positional Notation: Base 10
500
5 x 100
100
102
Sum
Evaluate
Value
Place
720
7 x12 x 10
110
100101
1’s place10’s place
527 = 5 x 102 + 2 x 101 + 7 x 100
100’s place
Number Systems 2-9
Positional Notation: Octal
6248 = 40410
Sum for Base 10
Evaluate
Value
Place
4 x 12 x 86 x 64
416384
808182
1864
64’s place 8’s place 1’s place
Number Systems 2-10
Positional Notation: Hexadecimal
6,70416 = 26,37210
4 x 10 x 167 x 2566 x
4,096
Evaluate
401,79224,576Sum for Base 10
160161162163Place
1162564,096Value
4,096’s place 256’s place 1’s place16’s place
Number Systems 2-11
Positional Notation: Binary
Sum for Base 10
Evaluate
Value
Place
0 x 11 x 21 x 40 x 81 x160 x 321 x 641 x 128
024016064128
1248163264128
2021222324252627
1101 01102 = 21410
Number Systems 2-12
Estimating Magnitude: Binary
1101 01102 = 21410
1101 01102 > 19210 (128 + 64 + additional bits to the right)
Sum for Base 10
Evaluate
Value
Place
0 x 11 x 21 x 40 x 81 x160 x 321 x 641 x 128
024016064128
1248163264128
2021222324252627
Number Systems 2-13
Range of Possible Numbers
R = BK where R = range B = base K = number of digits
Example #1: Base 10, 2 digits R = 102 = 100 different numbers (0…99)
Example #2: Base 2, 16 digits R = 216 = 65,536 or 64K 16-bit PC can store 65,536 different number
values
Number Systems 2-14
Decimal Range for Bit Widths
38+
19+
9+
6
4+
3
2+
1+
0+
Digits
Approx. 2.6 x 1038128
Approx. 1.6 x 101964
4,294,967,296 (4G)32
1,048,576 (1M)20
65,536 (64K)16
1,024 (1K)10
256 8
16 (0 to 15)4
2 (0 and 1)1
RangeBits
Number Systems 2-15
Base or Radix
Base: The number of different symbols required to
represent any given number
The larger the base, the more numerals are required Base 10: 0,1, 2,3,4,5,6,7,8,9 Base 2: 0,1 Base 8: 0,1,2, 3,4,5,6,7 Base 16: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
Number Systems 2-16
Number of Symbols vs. Number of Digits
For a given number, the larger the base the more symbols required but the fewer digits needed
Example #1: 6516 10110 1458 110 01012
Example #2: 11C1628410 4348 1 0001 11002
Number Systems 2-17
Counting in Base 2
Decimal
Number
EquivalentBinary
Number
101 x 211 x 231010
91 x 201 x 231001
81 x 231000
71 x 201 x 211 x 22111
61 x 211 x 22110
51 x 201 x 22101
41 x 22100
31 x 201 x 2111
20 x 201 x 2110
11 x 201
00 x 200
1’s (20)2’s (21)4’s (22)8’s (23)
Number Systems 2-18
Addition
Largest Single DigitProblemBase
1
+0
6
+9
6
+1
6
+3
1Binary
FHexadecimal
7Octal
9Decimal
Number Systems 2-19
Addition
AnswerCarryProblemBase
Carry the 2
Carry the 16
Carry the 8
Carry the 10
10 1
+1Binary
10 6
+AHexadecimal
10 6
+2Octal
10 6
+4Decimal
Number Systems 2-20
Binary Arithmetic
11000001
01101+
101101111111
Number Systems 2-21
Converting from Base 10
256
64
4
2
1164,09665,53616
185124,09632,7688
1281632641282562
01345678 Power
Base
Powers Table
Number Systems 2-22
From Base 10 to Base 2
0
1
0
64
6
42/32= 1
Integer
Remainder
10101
24816322
12345 Power
Base
4210 = 1010102
10/16 = 0
10
10/8 = 1
2
2/4 = 0
2
2/2 = 1
0
0/1 = 0
010
Number Systems 2-23
From Base 10 to Base 2
Most significant bit12 )
( 022 )
42Base 10
101010Base 2
( 152 )
( 0102 )
( 1 212 )
( 0 Least significant bit422 )
Remainder
Quotient
Number Systems 2-24
From Base 10 to Base 16
5,73510 = 166716
103 – 96 = 7
1,639 –1,536 = 103
5,735 - 4,096 = 1,639
Remainder
7103 /16 = 6
1,639 / 256 = 6
5,735 /4,096 = 1
Integer
7661
1162564,09665,53616
01234Power
Base
Number Systems 2-25
From Base 10 to Base 16
5,735Base 10
1667Base 16
016 )
( 1 Most significant bit116 )
( 62216 )
( 635816 )
( 7 Least significant bit5,73516 ) Quotient
Remainder
Number Systems 2-26
From Base 10 to Base 16
8,039Base 10
1F67Base 16
016 )
( 1 Most significant bit116 )
( 153116 )
( 650216 )
( 7 Least significant bit8,03916 ) Quotient
Remainder
Number Systems 2-27
From Base 8 to Base 10
3,584
x 7
512
83
= 3,76310
348128Sum for Base 10
x 3x 6 x 2
1864
808182Power
72638
Number Systems 2-28
From Base 8 to Base 10
= 3,7631072638
+ 3 =
+ 6 =
+ 2 =
3,7633760
464
58
x 8
470
x 8
56
7
x 8
Number Systems 2-29
From Base 16 to Base 2
The nibble approach Hex easier to read and write than binary
Why hexadecimal? Modern computer operating systems and networks
present variety of troubleshooting data in hex format
0111011011110001Base 2
76F1Base 16
Number Systems 2-30
Fractions
Number point or radix point Decimal point in base 10 Binary point in base 2
No exact relationship between fractional numbers in different number bases Exact conversion may be impossible
Number Systems 2-31
Decimal Fractions
Move the number point one place to the right Effect: multiplies the number by the base number Example: 139.010 139010
Move the number point one place to the left Effect: divides the number by the base number Example: 139.010 13.910
Number Systems 2-32
Fractions: Base 10 and Base 2
.008
8 x 1/1000
1/1000
10-3
.2
2 x 1/10
1/10
10-1
Sum
Evaluate
Value
Place
.0009.05
9 x1/10005 x 1/100
1/100001/100
10-410-2
.1010112 = 0.67187510
0 x 1/16
1/16
2-4
0.03125
1 x 1/32
1/32
2-5
0.0156250.125.5Sum
1 x 1/641x 1/80 x 1/41 x 1/2Evaluate
1/641/81/41/2Value
2-62-32-22-1Place
.258910
Number Systems 2-33
Mixed Number Conversion
Integer and fraction parts must be converted separately
Radix point: fixed reference for the conversion Digit to the left is a unit digit in every base B0 is always 1 regardless of the base
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