meljun cortes's - number system lecture
TRANSCRIPT
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Number Systems (Class XI)
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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 relationsh
ip betweenon/off switches and
calculation using Boolean logic
01
NoYes
FalseTrue
OffOn
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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 includingzero in the number system Example: Base 10 has 10 digits, 0 through 9
Binaryorbase 2
Bit(binary digit): 2 digits, 0 and 1
Octalorbase 8: 8 digits, 0 through 7 Hexadecimal or base 16:
16 digits, 0 through F Examples: 1010 = A16; 1110 = B16
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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
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Number Systems 2-5
Numbers: Physical Representation
Different numerals,same number of oranges Cave dweller: IIIII
Roman: V
Arabic: 5
Different bases, samenumber of oranges
510 1012
123
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Number Systems 2-6
Number System
Roman: position independent
Modern: based on positional notation (placevalue)
Decimal system: system ofpositional notationbased on powers of 10.
Binary system: system ofpositional notationbased powers of 2
Octal system: system ofpositional notation based
on powers of 8 Hexadecimal system: system ofpositional
notation based powers of 16
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Number Systems 2-7
Positional Notation: Base 10
Sum
Evaluate
Value
Place
340
3 x14 x 10
110
100101
1s place10s place
43 = 4 x 101 + 3 x 100
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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
1s place10s place
527 = 5 x 102 +2 x 101+ 7 x 100
100s place
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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
64s place 8s place 1s place
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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
4,096s place 256s place 1s place16s place
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Number Systems 2-11
Positional Notation: Binary
Sum forBase 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
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Number Systems 2-12
Estimating Magnitude: Binary
110101102 = 21410
110101102 > 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
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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 (099)
Example #2: Base 2, 16 digits
R = 216= 65,536 or 64K 16-bit PC can store 65,536 different number
values
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Number Systems 2-14
Decimal Range forBit 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
2568
16 (0 to 15)4
2 (0 and 1)1
RangeBits
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Number Systems 2-15
Base or Radix
Base:
The number of different symbols required to
represent any given number
The largerthe base, the more numerals arerequired
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
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Number Systems 2-16
Number of Symbolsvs. Number ofDigits
For a given number, the largerthe base
the more symbols required
but the fewerdigits needed
Example #1:
6516 10110 1458 110 01012
Example #2:
11C16 28410 4348 1 0001 11002
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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
1s (20)2s (21)4s (22)8s (23)
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Number Systems 2-18
Addition
Largest Single DigitProblemBase
1
+0
6
+9
6
+1
6
+3
1Binary
FHexadecimal
7Octal
9Decimal
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Number Systems 2-19
Addition
AnswerCarryProblemBase
Carry the 2
Carry the 16
Carry the 8
Carry the 10
101
+1Binary
106
+AHexadecimal
106
+2Octal
106
+4Decimal
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Number Systems 2-20
Binary Arithmetic
11000001
01101+1011011
11111
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Number Systems 2-21
Converting from Base 10
256
64
4
2
1164,09665,53616
185124,09632,7688
1281632641282562
01345678Power
Base
Powers Table
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Number Systems 2-22
From Base 10 to Base 2
0
1
0
64
6
42/32
= 1
Integer
Remainder
10101
24816322
12345Power
Base
4210 = 1010102
10/16
= 0
10
10/8
= 1
2
2/4
= 0
2
2/2
= 1
0
0/1
= 0
010
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Number Systems 2-23
From Base 10 to Base 2
Most significant bit12 )( 022 )
42Base 10
101010Base 2
( 152 )
( 0102 )
( 1212 )
( 0 Least significant bit422 )
Remainder
Quotient
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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
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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
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Number Systems2-26
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
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Number Systems2-27
From Base 8 to Base 10
3,584
x 7
512
83
= 3,76310
348128Sum for
Base 10
x 3x 6x 2
1864
808182Power
72638
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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
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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
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Number Systems 2-30
Fractions
Number point orradix 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
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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
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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
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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