Number System - · PDF fileBase or Radix of this number system is 2 (two) ... 8 43 6 8 5 3 5...
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Transcript of Number System - · PDF fileBase or Radix of this number system is 2 (two) ... 8 43 6 8 5 3 5...
COMPUTER FUNDAMENTALS
Fundamentals
Number System
Conversion’s
1’s Complement & 2’s Complement
Binary Codes
Logic Gates
Boolean Algebra
Adder
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NUMBER SYSTEM
Decimal Number System
Binary Number System
Octal Number System
Hexadecimal Number System
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DECIMAL NUMBER SYSTEM
Symbols
Base or Radix - ten
In general,
dn dn-1 dn-2 …. d1 d0 . d-1 d-2 …..d-k
Given by –
(dn x 10 n) + (dn-1 x 10n-1) + (dn-2 x 10n-2)+….+ (d1 x 10 1)
+ (d0 x 100) . (d-1 x 10-1) + (d-2 x 10-2)+…..+(d-k x 10 -k)
Example –
9258.147 = (9 x 103) + (2 x 102) + (5 x 101) + (8 x 100). (1 x 10-1) +
( 4 x 10-2) + (7 x 10-3)
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0 1 2 3 4 5 6 7 8 9
Digits
BINARY NUMBER SYSTEM
Symbols
Base or Radix of this number system is 2 (two)
In general,
dn dn-1 dn-2 …. d1 d0 . d-1 d-2 …..d-k
Its decimal equivalent is –
(dn x 2 n) + (dn-1 x 2n-1) + (dn-2 x 2n-2)+….+ (d1 x 21) + (d0 x 20) . (d-1
x 2-1) + (d-2 x 2-2)+…..+(d-k x 2-k)
Example –
1011.001 = (1 x 23) + (0 x 22) + (1 x 21) + (1 x 20). (0 x 2-1) + ( 0 x
2-2) + (1 x 2-3)
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0 1
Bits
OCTAL NUMBER SYSTEM Symbols
Base or Radix of this number system is 8 (eight)
In general,
dn dn-1 dn-2 …. d1 d0 . d-1 d-2 …..d-k
Its decimal equivalent is –
(dn x 8n) + (dn-1 x 8n-1) + (dn-2 x 8n-2)+….+ (d1 x 81) +
(d0 x 80) . (d-1 x 8-1) + (d-2 x 8-2)+…..+(d-k x 8-k)
Example –
7256.147 = (7 x 83) + (2 x 82) + (5 x 81) + (6 x 80). (1 x 8-1) + ( 4
x 8-2) + (7 x 8-3)
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0 1 2 3 4 5 6 7
HEXADECIMAL NUMBER SYSTEM Symbols
Base or Radix of this number system is 16 (sixteen)
In general,
dn dn-1 dn-2 …. d1 d0 . d-1 d-2 …..d-k
Its decimal equivalent is –
(dn x 16n) + (dn-1 x 16n-1) + (dn-2 x 16n-2)+….+ (d1 x 161) +
(d0 x 160) . (d-1 x 16-1) + (d-2 x 16-2)+…..+(d-k x 16-k)
Example –
92A8.14D = (9 x 163) + (2 x 162) + (10 x 161) + (8 x 160). (1 x 16-1)
+ ( 4 x 16-2) + (13 x 16-3) 7
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0 1 2 3 4 5 6 7 8 9 A B C D E F
10 11 12 13 14 15
CONVERSION Decimal to Binary
Decimal to Octal
Decimal to Hexadecimal
Binary to Decimal
Binary to Octal
Binary to Hexadecimal
Octal to Decimal
Octal to Binary
Octal to Hexadecimal
Hexadecimal to Decimal
Hexadecimal to Binary
Hexadecimal to Octal 8
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DECIMAL TO BINARY
Convert (39)10
Given function 39
Divide 39 by 2 2 39
2 19 1
2 9 1
2 4 1
2 2 0
2 1 0
1
Result (39)10 = (100111)2
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DECIMAL TO BINARY
Convert (0.75)10
Double-dabble method
Given function 0.75
Multiply 0.75 by 2 1.50
Multiply 0.50 by 2 1.00
Result (0.75)10 = (0.11)2
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DECIMAL TO BINARY (SECOND METHOD )
28 27 26 25 24 23 22 21 20
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256 128 64 32 16 8 4 2 1
0 1
Convert (169)10
Test 256 < 169 True / False
Test 128 < 169 (169-128 = 41 )
Test 64 < 41, 32 <41 (41-32 = 9)
Test 9 < 16, 8 < 9 (9-8=1)
0 1 0 1
0
0 1 0 1 0 1 0 0 1
TEST YOUR KNOWLEDGE
(0.15)10
= (.0010011)2 7 bit fraction
2 * 0.15 = 0.3
2 * 0.3 = 0.6
2 * 0.6 = 1.2
2 * 0.2 = 0.4
2 * 0.4 = 0.8
2 * 0.8 = 1.6
2 * 0.6 = 1.2
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DECIMAL TO OCTAL
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Convert (39)10
Given function 39
Divide 39 by 8 8 39
8 4 7
4
Result (39)10 = (47)8
Cross Check
=(4*81) + (7*80)
=(4*8) + (7*1)
DECIMAL TO OCTAL
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Convert (350)10
Given function 350
Divide 350 by 8 8 350
8 43 6
8 5 3
5
Result (350)10 = (536)8
= (5 * 82)+(3 * 81)+(6 * 80)
= 320 + 24 + 6
= 350
DECIMAL TO OCTAL
Convert (0.93)10 to Octal
0.93 * 8 7.44
0.44 * 8 3.52
0.52 * 8 4.16
0.16 * 8 1.28
Hence (0.93)10 = (0.7341)8
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DECIMAL TO HEXADECIMAL
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Convert (350)10
Given function 350
Divide 350 by 16 16 350
16 21 14
16 1 5
1
Result (350)10 = (15E)16
= (1 * 162)+(5 * 161)+(14 * 160)
= (256) + (80) + 14
= 350
BINARY TO DECIMAL
(10101)2
Step 1
Step 2
Step 3
Result
(10101)2 = (21)10 17
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1 0 1 0 1
1 x 24 0 x 23 1 x 22 0 x 21 1 x 20
16 0 4 0 1
16 + 4 + 1 = (21)10
B2D CONT...
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Convert (11101.001)2 to decimal
Step 1
Step 2
Step 3
Result
(11101.001)2 = (16+8+4+1+0.125)10 = (29.125)10
1 1 1 0 1 . 0 0 1
1 x 24 1 x 23 1 x 22 0 x 21 1 x 20 . 0 x 2-1 0 x 2-2 1 x 2-3
24 23 22 0 20 . 0 0 2-3
16 8 4 0 1 . 0 0 0.125
BINARY TO DECIMAL (SECOND METHOD)
1 0 0 1 0 1 1
1*2+0=2
2*2+0=4
4*2+1=9
9*2+0=18
18*2+1=37
37*2+1 =75
Result (1001011)2 = (75)10
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BINARY TO OCTAL
Convert (110101.101010)2 to Octal
Step 1
Step 2
Step 3
Step 4 Result :- (65.52)8
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1 1 0 1 0 1 . 1 0 1 0 1 0
1 1 0 1 0 1 . 1 0 1 0 1 0
G-2 G-1 G-1 G-2
1 1 0 1 0 1 . 1 0 1 0 1 0
6 5 . 5 2
B2O CONT…
Convert (10101111001.0111)2 to Octal
Step 1
Step 2
Step 3
Step 4 Result :- (2571.34)8
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1 0 1 0 1 1 1 1 0 0 1 . 0 1 1 1
0 1 0 1 0 1 1 1 1 0 0 1 . 0 1 1 1 0 0
G-4 G-3 G-2 G-1 . G-1 G-2
0 1 0 1 0 1 1 1 1 0 0 1 . 0 1 1 1 0 0
2 5 7 1 . 3 4
BINARY TO HEXADECIMAL Convert (110101.101010)2 to Hexadecimal
Step 1
Step 2
Step 3
Step 4 Result :- (35.A8)16
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1 1 0 1 0 1 . 1 0 1 0 1 0
0 0 1 1 0 1 0 1 . 1 0 1 0 1 0 0 0
G2 G1 G1 G2
0 0 1 1 0 1 0 1 . 1 0 1 0 1 0 0 0
3 5 . A 8
OCTAL TO DECIMAL
Convert (4057.06)8 to Decimal
= 4 * 83 + 0 * 82 + 5 * 81 + 7 * 80 + 0 * 8-1 + 6 * 8-2
= 2048 + 0 + 40 + 7 + 0 +0.0937 = (2095.0937)10
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OCTAL TO BINARY Convert (4057.06)8 to Binary
Step 1
Step 2
Result
(4057.06)8 = (100000101111.000110)2
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4 0 5 7 . 0 6
1 0 0 0 0 0 1 0 1 1 1 1 . 0 0 0 1 1 0
4 0 5 7 . 0 6
OCTAL TO HEXADECIMAL
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Convert (4057.06)8 to Hexadecimal
Step 1 : Group three bit
Step 2 : place the bits
Step 3 : Group 4 bit each and place digit
Result
(4057.06)8 = (82F.18)16
4 0 5 7 . 0 6
1 0 0 0 0 0 1 0 1 1 1 1 . 0 0 0 1 1 0
4 0 5 7 . 0 6
1 0 0 0 0 0 1 0 1 1 1 1 . 0 0 0 1 1 0 0 0
8 2 15 . 1 8
HEXADECIMAL TO DECIMAL
?
Self compute
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HEXADECIMAL TO BINARY
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Convert (4057.06)16 to Binary
Step 1
Step 2
Result
(4057.06)8 = (100000001010111.00000110)2
4 0 5 7 . 0 6
0 1 0 0 0 0 0 0 0 1 0 1 0 1 1 1 . 0 0 0 0 0 1 1 0
4 0 5 7 . 0 6
HEXADECIMAL TO OCTAL
?
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THE END
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