NUMBER SYSTEMS AND CODES. zPOSITIONAL NUMBER SYSTEMS zARITHMETIC OPERATIONS zCODES.
Number Systems
description
Transcript of Number Systems
Introduction to Computer Engineering by Richard E. Haskell
Number Systems
Module M3.1
Sections 2.1-2.2
Introduction to Computer Engineering by Richard E. Haskell
Number Systems
• Counting in Binary
• Hexadecimal and Octal Numbers
• Positional Notation
• Fractional Numbers
Introduction to Computer Engineering by Richard E. Haskell
Counting in Binary
Position: 8 4 2 1
0 0 0 0 00 0 0 1 10 0 1 0 20 0 1 1 30 1 0 0 40 1 0 1 50 1 1 0 60 1 1 1 71 0 0 0 8
BINARY HEX
Introduction to Computer Engineering by Richard E. Haskell
Counting in Binary
Position: 8 4 2 1
1 0 0 0 81 0 0 1 91 0 1 0 A1 0 1 1 B1 1 0 0 C1 1 0 1 D1 1 1 0 E1 1 1 1 F
BINARY HEX
Introduction to Computer Engineering by Richard E. Haskell
Counting in Binary
128 64 32 16 8 4 2 1
0 0 1 1 0 1 0 0 52
1 0 1 0 0 0 1 1 163
1 1 1 1 1 1 1 1 255
BINARY DEC
Introduction to Computer Engineering by Richard E. Haskell
Table 2.1Hexadecimal and Decimal Conversion
15 Byte 8 7 Byte 015 Char 12 11 Char 8 7 Char 4 3 Char 0Hex Dec Hex Dec Hex Dec Hex Dec0 0 0 0 0 0 0 01 4,096 1 256 1 16 1 12 8,192 2 512 2 32 2 23 12,288 3 768 3 48 3 34 16,384 4 1,024 4 64 4 45 20,480 5 1,280 5 80 5 56 24,576 6 1,536 6 96 6 67 28,672 7 1,792 7 112 7 78 32,768 8 2,048 8 128 8 89 36,864 9 2,304 9 144 9 9A 40,960 A 2,560 A 160 A 10B 45,056 B 2,816 B 176 B 11C 49,152 C 3,072 C 192 C 12D 53,248 D 3,328 D 208 D 13E 57,344 E 3,584 E 224 E 14F 61,440 F 3,840 F 240 F 15
Introduction to Computer Engineering by Richard E. Haskell
Table 2.1Hexadecimal and Decimal Conversion
15 Byte 8 7 Byte 015 Char 12 11 Char 8 7 Char 4 3 Char 0Hex Dec Hex Dec Hex Dec Hex Dec0 0 0 0 0 0 0 01 4,096 1 256 1 16 1 12 8,192 2 512 2 32 2 23 12,288 3 768 3 48 3 34 16,384 4 1,024 4 64 4 45 20,480 5 1,280 5 80 5 56 24,576 6 1,536 6 96 6 67 28,672 7 1,792 7 112 7 78 32,768 8 2,048 8 128 8 89 36,864 9 2,304 9 144 9 9A 40,960 A 2,560 A 160 A 10B 45,056 B 2,816 B 176 B 11C 49,152 C 3,072 C 192 C 12D 53,248 D 3,328 D 208 D 13E 57,344 E 3,584 E 224 E 14F 61,440 F 3,840 F 240 F 15
87C9: 32,768 1,792 192
987C916 = 34,76110
Introduction to Computer Engineering by Richard E. Haskell
Positional Notation
N = P4P3P2P1P0
= P4b4 + P3b3 + P2b2 + P1b1 + P0b0
58410 = 5 x 102 + 8 x 101 + 4 x 100
= 500 + 80 + 4= 584
Introduction to Computer Engineering by Richard E. Haskell
Positional Notation
N = P4P3P2P1P0
= P4b4 + P3b3 + P2b2 + P1b1 + P0b0
101102 = 1 x 24 + 0 x 23 + 1 x 22 + 1 x 21 + 0 x 20
= 16 + 0 + 4 + 2 + 0= 2210
Binary
Introduction to Computer Engineering by Richard E. Haskell
Positional Notation
N = P4P3P2P1P0
= P4b4 + P3b3 + P2b2 + P1b1 + P0b0
3AF16 = 3 x 162 + A x 161 + F x 160
= 3 x 256 + 10 x 16 + 15 x 1= 768 + 160 + 15= 94310
Hex
Introduction to Computer Engineering by Richard E. Haskell
Positional Notation
N = P4P3P2P1P0
= P4b4 + P3b3 + P2b2 + P1b1 + P0b0
4578 = 4 x 82 + 5 x 81 + 7 x 80
= 4 x 64 + 5 x 8 + 7 x 1= 256 + 40 + 7= 30310
Octal
Introduction to Computer Engineering by Richard E. Haskell
Fractional Numbers
N = ...P3P2P1P0 . P-1P-2P-3...
= ... + P3b3 + P2b2 + P1b1 + P0b0
+ P-1b-1 + P-2b-2 + P-3b-3 + ...
375.1710 = 3 x 102 + 7 x 101 + 5 x 100
+ 1 x 10-1 + 7 x 10-2 = 300 + 70 + 5 + 0.1 + 0.07= 375.17
Introduction to Computer Engineering by Richard E. Haskell
Fractional Numbers
N = ...P3P2P1P0 . P-1P-2P-3...
= ... + P3b3 + P2b2 + P1b1 + P0b0
+ P-1b-1 + P-2b-2 + P-3b-3 + ...
1101.112 = 1 x 23 + 1 x 22 + 0 x 21 + 1 x 20 + 1 x 2-1 + 1 x 2-2
= 8 + 2 + 0 + 1 + 1/2 + 1/4= 11.7510
Binary
Introduction to Computer Engineering by Richard E. Haskell
Fractional Numbers
N = ...P3P2P1P0 . P-1P-2P-3...
= ... + P3b3 + P2b2 + P1b1 + P0b0
+ P-1b-1 + P-2b-2 + P-3b-3 + ...
1AB.616 = 1 x 162 + A x 161 + B x 160
+ 6 x 16-1 = 1 x 256 + 10 x 16 + 11 x 1 + 6/16 = 256 + 160 + 11 + 0.375 = 427.37510
Hex
Introduction to Computer Engineering by Richard E. Haskell
Fractional Numbers
N = ...P3P2P1P0 . P-1P-2P-3...
= ... + P3b3 + P2b2 + P1b1 + P0b0
+ P-1b-1 + P-2b-2 + P-3b-3 + ...
173.258 = 1 x 82 + 7 x 81 + 3 x 80
+ 2 x 8-1 + 5 x 8-2
= 1 x 64 + 7 x 8 + 3 x 1+ 2/8 + 5/64
= 64 + 56 + 3 + 0.25 + 0.078125= 123.32812510
Octal
Introduction to Computer Engineering by Richard E. Haskell
Questions
What is the decimal value of 2435?
What is the decimal value of 21.34?