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Basic Computer Concepts Number Systems Concepts. “Deka” – Greek word denoting a value equal. Ø. to ten Table 1: Decimal Example 2,048 TABLE 2: The value denotation of a number “place” or position. v. Given a base-n number system, the value denotation of a position m is: m - PowerPoint PPT Presentation
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Digit ValueDenotation
NumericValueDenotation
2 thousands 1,000
0 hundreds 100
4 tens 10
8 ones 1
Computer Organization
Basic Computer Concepts
Number Systems ConceptsØ“Deka” – Greek word denoting a value equal
to tenTable 1: Decimal Example 2,048
TABLE 2: The value denotation of a number“place” or position
vGiven a base-n number system, the valuedenotation of a position m is:
mValue denotation
11
21 x n
31 x n x n
41 x n x n x n
M1 x n x n . . . x n (n multiplied M-1 times)0
Basic Computer Concepts
Number Systems ConceptsExample A:ØFinding the value of a number in any number
systemGiven a base-n system and a number PQRST, the
value is given by:T x 1+ S x n+ R x n x n+ Q x n x n x n+ P x n x n x n x n
Example B:ØGiven a base-8 system and a number 56472,the value is given by:
Value2x1= 27 x 8 = 564 x 64 = 2566 x 512 = 3,0725 x 4,096 = 20,480
Step2x1
+7x8+4x8x8+6x8x8x8+5x8x8x8x8Total: 23,866
Computer Organization
Computer Organization
Basic Computer Concepts
Number Systems Concepts
TABLE 3: General method for obtaining the valuedenotation of a base-n numeral
Value denotation Vm
The value denotation of a position m in a base-nnumber system is:
value denotation Vm = n (m-1)
Decimal value N of a base-n numeralThe decimal value N of a number
pkpk-1pk-2, . .,p2p1in a base-n number system is given by:
numerical value N = i=1?i=k pi Vi
= i=1 ? i=k pi n(i-1)
Computer Organization
Basic Computer Concepts
Number Systems Concept
Examples:
1. Derive the decimal value of 6409Solution:From Table 3:
6409 = 0 x 90 + 4 x 91 + 6 x 92
= 0 x1+ 4 x9+ 6 x9x9= 0 + 36 + 486
= 52210
2. Derive the decimal value of 64012Solution:From Table 3:
64012 = 0 x 120 + 4 x 121 + 6 x 122
= 0 x 1 + 4 x 12 + 6 x 12 x 12= 0 + 48 + 864
= 91210
Computer Organization
Basic Computer Concepts
Number Systems Concept
Using the notation of Table 3:
3. Derive the decimal value of 64016
Solution:From Table 3:
64016 = i=1?i=3 pi 16(i-1)= 0 x 160 + 4 X 161+ 6 X 162
Value = 0 + 64 + 1536 = 1600
Computer Organization
Basic Computer Concepts
Hexadecimal, Octal, &Binary Systems
1. Derive the decimal value of 110111012Solution:110111012 = 1 x 20 + 0 x 21 + 1 x 22
+ 1 x 23+ 1 x 24 + 0 x 25
+ 1 x 26+ 1 x 27
Value = 1 + 0 + 4 + 8 + 16 + 0 + 64 + 128
= 22110
2. Derive the decimal value of 541760238Solution:
541760238 = 3 x 80 + 2 x 81+ 0 x 82
+ 6 x 83 + 7 x 84 + 1 x 85
+ 4 x 86 + 5 x 87
Value = 3 + 16 + 0 + 3072 + 28672+ 32768 + 1048576 + 10485760
= 1159886710
Computer Organization
Basic Computer Concepts
Hexadecimal, Octal, &Binary Systems
3. Derive the decimal value of 24CF16Solution:
24CF16 = F x 160 + C x 161 + 4 x 162 + 2 x 163
Value = 15 + 192 + 1024 + 8192
= 94231010
4
83
82
81
80
8
4,096 512 64 8 1
0 5 1 - -
4
83
82
81
80
8
4,096 512 64 8 1
0 5 - - -
Computer Organization
Basic Computer Concepts
Converting: Decimal toAnother Number System
Ø
Ø
Given a decimal number M, to convert to anumber system with base - n:
As an example, let M = 2,654 and n = 8
4
83
82
81
80
8
4096 512 64 8 1
0 5 1 3 6
4
83
82
81
80
8
4096 512 64 8 1
0 5 1 3
Computer Organization
Basic Computer Concepts
Converting: Decimal toAnother Number System
2,65410 = 51368
Computer Organization
Basic Computer Concepts
Converting: Decimal toAnother Number System
Derive the octal value of 10112Solution:
a. Convert 10112 to decimal:10112 = 1 x 20 + 1 x 21 + 0 x 22 + 1 x 23
decimal value = 1 + 2 + 0 + 8 = 1110
b. Convert 1110 to octal:
: 11/8 = 1 remainder 3: 3/1 = 3 remainder 0octal value = 138
Computer Organization
Basic Computer Concepts
Converting: Decimal toAnother Number System
Shortcut Method
Derive the octal value of 101110112Solution:Group the binary digits into three :
101110112 = 10 111 0112 = 010 111 0112
421Now 010= 2
421111= 7421011 = 3
Therefore
010 111 0112 = 2738
Computer Organization
Basic Computer Concepts
Converting: Decimal toAnother Number System
Shortcut Method
Derive the hexadecimal value of 101110112Solution:Group the binary digits into four :
101110112 = 1011 10112
8 421Now 1 0 1 1 = 11
84211 0 1 1= 11
But 1116 = B; therefore101110112 = BB16
