lecture 3 computer number system
DESCRIPTION
Lecture 3 Computer Number System. 0 (00) 1 (01) 2 (10) 3 (11). OFF. ON. Number System. The Binary Number System To convert data into strings of numbers, computers use the binary number system. Humans use the decimal system (“ deci ” stands for “ten”). - PowerPoint PPT PresentationTRANSCRIPT
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Lecture 3Lecture 3 Computer Number Computer Number SystemSystem
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Number SystemThe Binary Number System•To convert data into strings of numbers, computers use the binary number system.
•Humans use the decimal system (“deci” stands for “ten”).
•Elementary storage units inside computer are electronic switches. Each switch holds one of two states: on (1) or off (0).
• We use a bit (binary digit), 0 or 1, to represent the state.
ON OFF
0 (00)
1 (01)
2 (10)
3 (11)
• The binary number system works the same way as the decimal system, but has only two available symbols (0 and 1) rather than ten (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9).
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Number SystemBits and Bytes• A single unit of data is called a bit, having a value of 1 or 0. • Computers work with collections of bits, grouping them to represent larger pieces of data,
such as letters of the alphabet.• Eight bits make up one byte. A byte is the amount of memory needed to store one
alphanumeric character.• With one byte, the computer can represent one of 256 different symbols or characters.
1 0 1 1 0 0 1 01 0 0 1 0 0 1 01 0 0 1 0 0 1 11 1 1 1 1 1 1 1
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Common Number Systems
System Base Symbols
Decimal 10 0, 1, … 9
Binary 2 0, 1
Octal 8 0, 1, … 7
Hexa-decimal
16 0, 1, … 9,
A, B, … F
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Quantities/Counting (1 of 2)
Decimal Binary Octal
Hexa-decimal
0 0 0 0
1 1 1 1
2 10 2 2
3 11 3 3
4 100 4 4
5 101 5 5
6 110 6 6
7 111 7 7
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Quantities/Counting (2 of 2)
Decimal Binary Octal
Hexa-decimal
8 1000 10 8
9 1001 11 9
10 1010 12 A
11 1011 13 B
12 1100 14 C
13 1101 15 D
14 1110 16 E
15 1111 17 F
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Conversion Among Bases
• The possibilities:
Hexadecimal
Decimal Octal
Binary
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Quick Example
2510 = 110012 = 318 = 1916
Base
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Decimal to Decimal (just for fun)
Hexadecimal
Decimal Octal
Binary
Next slide…
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12510 => 5 x 100 = 52 x 101 = 201 x 102 = 100
125
Base
Weight
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Binary to Decimal
Hexadecimal
Decimal Octal
Binary
![Page 12: Lecture 3 Computer Number System](https://reader036.vdocuments.us/reader036/viewer/2022062315/56814d05550346895dba3485/html5/thumbnails/12.jpg)
Binary to Decimal
• Technique– Multiply each bit by 2n, where n is the “weight” of
the bit– The weight is the position of the bit, starting from
0 on the right– Add the results
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Example
1010112 => 1 x 20 = 11 x 21 = 20 x 22 = 01 x 23 = 80 x 24 = 01 x 25 = 32
4310
Bit “0”
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Octal to Decimal
Hexadecimal
Decimal Octal
Binary
![Page 15: Lecture 3 Computer Number System](https://reader036.vdocuments.us/reader036/viewer/2022062315/56814d05550346895dba3485/html5/thumbnails/15.jpg)
Octal to Decimal
• Technique– Multiply each bit by 8n, where n is the “weight” of
the bit– The weight is the position of the bit, starting from
0 on the right– Add the results
![Page 16: Lecture 3 Computer Number System](https://reader036.vdocuments.us/reader036/viewer/2022062315/56814d05550346895dba3485/html5/thumbnails/16.jpg)
Example
7248 => 4 x 80 = 42 x 81 = 167 x 82 = 448
46810
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Hexadecimal to Decimal
Hexadecimal
Decimal Octal
Binary
![Page 18: Lecture 3 Computer Number System](https://reader036.vdocuments.us/reader036/viewer/2022062315/56814d05550346895dba3485/html5/thumbnails/18.jpg)
Hexadecimal to Decimal
• Technique– Multiply each bit by 16n, where n is the “weight”
of the bit– The weight is the position of the bit, starting from
0 on the right– Add the results
![Page 19: Lecture 3 Computer Number System](https://reader036.vdocuments.us/reader036/viewer/2022062315/56814d05550346895dba3485/html5/thumbnails/19.jpg)
Example
ABC16 => C x 160 = 12 x 1 = 12 B x 161 = 11 x 16 = 176 A x 162 = 10 x 256 = 2560
274810
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Decimal to Binary
Hexadecimal
Decimal Octal
Binary
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Decimal to Binary
To convert a whole number to binary, use successive division by 2 until the quotient is 0. The remainders form the answer, with the first remainder as the least significant bit (LSB) and the last as the most significant bit (MSB).
(43)10 = (101011)2
2 43 2 21 rem 1 LSB
2 10 rem 1 2 5 rem 0 2 2 rem 1 2 1 rem 0 0 rem 1 MSB
Repeated Multiplication-by-2 Method (for fractions)
Repeated Division-by-2 Method (for whole number)
To convert decimal fractions to binary, repeated multiplication by 2 is used, until the fractional product is 0 (or until the desired number of decimal places). The carried digits, or carries, produce the answer, with the first carry as the MSB, and the last as the LSB.
(0.3125)10 = (.0101)2
Carry
0.31252=0.625 0 MSB
0.6252=1.25 1
0.252=0.50 0
0.52=1.00 1 LSB
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Example12510 = ?2
2 125 62 12 31 02 15 12 7 12 3 12 1 12 0 1
12510 = 11111012
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Octal to Binary
Hexadecimal
Decimal Octal
Binary
![Page 24: Lecture 3 Computer Number System](https://reader036.vdocuments.us/reader036/viewer/2022062315/56814d05550346895dba3485/html5/thumbnails/24.jpg)
Octal to Binary
• Technique– Convert each octal digit to a 3-bit equivalent
binary representation
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Example7058 = ?2
7 0 5
111 000 101
7058 = 1110001012
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Hexadecimal to Binary
Hexadecimal
Decimal Octal
Binary
![Page 27: Lecture 3 Computer Number System](https://reader036.vdocuments.us/reader036/viewer/2022062315/56814d05550346895dba3485/html5/thumbnails/27.jpg)
Hexadecimal to Binary
• Technique– Convert each hexadecimal digit to a 4-bit
equivalent binary representation
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Example10AF16 = ?2
1 0 A F
0001 0000 1010 1111
10AF16 = 00010000101011112
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Decimal to Octal
Hexadecimal
Decimal Octal
Binary
![Page 30: Lecture 3 Computer Number System](https://reader036.vdocuments.us/reader036/viewer/2022062315/56814d05550346895dba3485/html5/thumbnails/30.jpg)
Decimal to Octal
• Technique– Divide by 8– Keep track of the remainder
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Example123410 = ?8
8 1234 154 28 19 28 2 38 0 2
123410 = 23228
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Decimal to Hexadecimal
Hexadecimal
Decimal Octal
Binary
![Page 33: Lecture 3 Computer Number System](https://reader036.vdocuments.us/reader036/viewer/2022062315/56814d05550346895dba3485/html5/thumbnails/33.jpg)
Decimal to Hexadecimal
• Technique– Divide by 16– Keep track of the remainder
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Example123410 = ?16
123410 = 4D216
16 1234 77 216 4 13 = D16 0 4
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Binary to Octal
Hexadecimal
Decimal Octal
Binary
![Page 36: Lecture 3 Computer Number System](https://reader036.vdocuments.us/reader036/viewer/2022062315/56814d05550346895dba3485/html5/thumbnails/36.jpg)
Binary to Octal
• Technique– Group bits in threes, starting on right– Convert to octal digits
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Example10110101112 = ?8
1 011 010 111
1 3 2 7
10110101112 = 13278
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Binary to Hexadecimal
Hexadecimal
Decimal Octal
Binary
![Page 39: Lecture 3 Computer Number System](https://reader036.vdocuments.us/reader036/viewer/2022062315/56814d05550346895dba3485/html5/thumbnails/39.jpg)
Binary to Hexadecimal
• Technique– Group bits in fours, starting on right– Convert to hexadecimal digits
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Example10101110112 = ?16
10 1011 1011
2 B B
10101110112 = 2BB16
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Octal to Hexadecimal
Hexadecimal
Decimal Octal
Binary
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Octal to Hexadecimal
• Technique– Use binary as an intermediary
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Example10768 = ?16
1 0 7 6
001 000 111 110
2 3 E
10768 = 23E16
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Hexadecimal to Octal
Hexadecimal
Decimal Octal
Binary
![Page 45: Lecture 3 Computer Number System](https://reader036.vdocuments.us/reader036/viewer/2022062315/56814d05550346895dba3485/html5/thumbnails/45.jpg)
Hexadecimal to Octal
• Technique– Use binary as an intermediary
![Page 46: Lecture 3 Computer Number System](https://reader036.vdocuments.us/reader036/viewer/2022062315/56814d05550346895dba3485/html5/thumbnails/46.jpg)
Example1F0C16 = ?8
1 F 0 C
0001 1111 0000 1100
1 7 4 1 4
1F0C16 = 174148
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Exercise – Convert ...
Don’t use a calculator!
Decimal Binary Octal
Hexa-decimal
33
1110101
703
1AF
Skip answer Answer
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Exercise – Convert …
Decimal Binary Octal
Hexa-decimal
33 100001 41 21
117 1110101 165 75
451 111000011 703 1C3
431 110101111 657 1AF
Answer
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Thank you