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© 2003-2008BYU
02 NUMBERS Page 2
ECEn 224
Positional Numbers
• What does 5132.13 really mean?• Depends on the number base!• Assuming base 10:
5132.1310 = 5x103 + 1x102 + 3x101 + 2x100 + 1x10-1 + 3x10-2
• Assuming base 6:5132.136 = 5x63 + 1x62 + 3x61 + 2x60 + 1x6-1 + 3x6-2
• We often use a subscript to indicate the base.
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ECEn 224
Positional Number Examples
527.4610 = (5 x 102) + (2 x 101) + (7 x 100) + (4 x 10-1) + (6 x 10-2)
527.468 = (5 x 82) + (2 x 81) + (7 x 80) + (4 x 8-1) + (6 x 8-2)
527.465 = illegal why?
1011.112 = (1 x 23) + (0 x 22) + (1 x 21) + (1 x 20) + (1 x 2-1) + (1 x 2-2)
This works for binary as well…
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ECEn 224
Conversion from Binary
Convert 101011.112 to base 10:
101011.112 = 1x25 + 0x24 + 1x23 + 0x22 + 1x21 + 1x20 + 1x2-1 + 1x2-
2
= 32 + 0 + 8 + 0 + 2 + 1 + ½ + ¼
= 43.7510
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ECEn 224
Convert 11410 to binary:
114
50
18
2
- 64 1x26
- 32 1x25
- 16 1x24
2- 0 0x23
2- 0 0x22
0- 2 1x21
- 0 0x20
0
11410 = 11100102
Rea
d t
his
way
This method also works for fractional numbers.
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ECEn 224
114
An Alternate Method
2
2
2
2
2
2
2
57 R 0
28 R 1
14 R 0
7 R 0
3 R 1
1 R 1
0 R 1
11410 = 11100102
Rea
d t
his
way
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ECEn 224
Converting fractions from base 10 to binary:Convert 0.710 to binary
0.7x 2
(1).4
x 2
(0).8x 2
(1).6x 2
(1).2x 2
(0).4x 2
(0).8
process starts repeating here
0.710 = 0.1 0110 0110 …2R
ead
th
is w
ay
© 2003-2008BYU
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ECEn 224
Convert 114.710 to binary:
We could use the first technique.
114.7- 64 1x26
50.7- 32 1x25
18.7- 16 1x24
2.7 - 0 0x23
2.7 - 0 0x22
2.7 - 2 1x21
0.7 - 0 0x20
0.7 - 0.5 1x2-1
0.2 - 0.0 0x2-2
0.2 - 0.125 1x2-3
0.075 …
Rea
d t
his
way
11210 = 1110010.10...2
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ECEn 224
Convert 114.710 to binary:
Or we could combine the second and third techniques.
11210 = 1110010.10110...2
1142
2
2
2
2
2
2
57 R 0
28 R 1
14 R 0
7 R 0
3 R 1
1 R 1
0 R 1
0.7x 2
(1).4
x 2
(0).8x 2
(1).6x 2
(1).2x 2
(0).4x 2
(0).8
Rea
d t
his
way
© 2003-2008BYU
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ECEn 224
Hexadecimal
• Commonly used for binary data– 1 hex digit 4 binary digits (bits)
• Need more digits than just 0-9– Use 0-9, A-F
• A-F are for 10-15
FA216 = 15x162 + 10x161 + 2x160
FA216 = 1111 1010 0010
Each group of 4 bits 1 hex digit
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ECEn 224
Other Notations For Binary and Hex
• Binary– 101102
– 10110b– 0b10110
• Hexadecimal– 57316
– 0x573– 573h– 16#573
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ECEn 224
Binary Coded Decimal(BCD)
DecimalDigit BCD
0 0 0 0 01 0 0 0 12 0 0 1 03 0 0 1 14 0 1 0 05 0 1 0 16 0 1 1 07 0 1 1 18 1 0 0 09 1 0 0 1
Convert 249610 to BCD Code
2 4 9 6
0 0 1 0 0 1 0 0 1 0 0 1 0 1 1 0
Note this is very different from convertingto binary which yields:
1 0 0 1 1 1 0 0 0 0 0 02
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ECEn 224
Binary Coded Decimal(BCD)
• Why use BCD?• In some applications it may be easier to
work with• Financial institutions must be able to
represent base 10 fractions (e.g., 1/10)– 0.110 = 0.00110011001100…2
– Using BCD ensures that numeric results are identical to base 10 results
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ECEn 224
Binary Codes ASCII Code
• ASCII American Standard Code for Information Interchange
• ASCII is a 7-bit code used to represent letters, symbols, and terminal codes
• There are also Extended ASCII codes, represented by 8-bit numbers
• Terminal codes include such things as:Tab (TAB)Line feed (LF)Carriage return (CR)Backspace (BS)Escape (ESC)And many more!
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ECEn 224
Binary Codes ASCII Code (partial)
Character
c 1 1 0 0 0 1 1d 1 1 0 0 1 0 0e 1 1 0 0 1 0 1f 1 1 0 0 1 1 0g 1 1 0 0 1 1 1h 1 1 0 1 0 0 0I 1 1 0 1 0 0 1j 1 1 0 1 0 1 0k 1 1 0 1 0 1 1l 1 1 0 1 1 0 0m 1 1 0 1 1 0 1n 1 1 0 1 1 1 0o 1 1 0 1 1 1 1p 1 1 1 0 0 0 0q 1 1 1 0 0 0 1
ASCII Code Convert “help” to ASCII
h e l p
1101000 1100101 1101100 1111000
0x68 0x65 0x6C 0x70
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ECEn 224
Binary Codes Gray Code
GrayNumber Code
0 0 0 0 01 0 0 0 12 0 0 1 13 0 0 1 04 0 1 1 05 0 1 1 16 0 1 0 17 0 1 0 08 1 1 0 0 9 1 1 0 110 1 1 1 111 1 1 1 012 1 0 1 013 1 0 1 114 1 0 0 115 1 0 0 0
• Only one bit changes with each number increment
• Not a weighted code
• Useful for interfacing to some physical systems
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ECEn 224
Gray Codes are Not Unique
GrayNumber Code
0 0001 0102 1103 1114 0115 0016 1017 100
GrayNumber Code
0 0001 0012 0113 0104 1105 1116 1017 100
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ECEn 224
Codes - Summary
• Bits are bits…– Modern digital devices represent everything as
collections of bits– A computer is one such digital device
• You can encode anything with sufficient 1’s and 0’s– Text (ASCII)– Computer programs (C code, assembly code,
machine code)– Sound (.wav, .mp3, …)– Pictures (.jpg, .gif, .tiff)