Transcript Number Systems - SNGCE DIGITAL LIBRARY

Number Systems
Topics
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The Decimal Number System
The Binary Number System
Converting from Binary to Decimal
Converting from Decimal to Binary
The Hexadecimal Number System
(r-1)’s and r’s compliment
Number Systems
 The on and off states of the capacitors in RAM can
be thought of as the values 1 and 0, respectively.
 Therefore, thinking about how information is
stored in RAM requires knowledge of the binary
(base 2) number system.
BINARY NUMBER SYSTEM
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The Binary Number System
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The binary number system is also known as base 2.
The values of the positions are calculated by taking
2 to some power.
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Why is the base 2 for binary numbers?
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Because we use 2 digits, the digits 0 and 1.
The Binary Number System
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The binary number system is also a positional
numbering system.
Instead of using ten digits, 0 - 9, the binary system
uses only two digits, 0 and 1.
Example of a binary number and the values of the
positions:
1 0 0 1 1 0 1
26 25 24 23 22 21 20
Converting from Binary to Decimal
1 0 0 1 1 0 1
26 25 24 23 22 21 20
20 = 1
21 = 2
22 = 4
23 = 8
24 = 16
25 = 32
26 = 64
decimal=(77)10
1 X 20 = 1
0 X 21 = 0
1 X 22 = 4
1 X 23 = 8
0 X 24 = 0
0 X 25 = 0
1 X 26 = 64
DECIMAL NUMBER SYSTEM
The Decimal Number System
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The decimal number system is also known as base
10.
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The values of the positions are calculated by taking
10 to some power.
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Why is the base 10 for decimal numbers?
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Because we use 10 digits, the digits 0 through 9.
The Decimal Number System
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The decimal number system is a positional number
system.
Example:
4210 = 1 0 1 0 1 0 2
The Decimal Number System
Converting From Decimal to Binary
 Make a list of the binary place values up to the number
being converted.
 Perform successive divisions by 2, placing the
remainder of 0 or 1 in each of the positions from right
to left.
 Continue until the quotient is zero
OCTAL NUMBER SYSTEM
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 Octal Number System (Base-8)
The octal number system uses EIGHT values
to represent numbers.
The values are,0 1 2 3 4 5 6 7.
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Convert octal to decimal.
Example: (264)8
Each column represents a power of 8,
264 becomes .....
4 * 80 = 4
6 * 81 = 48
2 * 82 = 128
adding the results together gives 18010
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 DECIMAL TO OCTAL
Example: convert (177)10 to octal
177 / 8 = 22 remainder is 1
22 / 8 = 2 remainder is 6
2 / 8 = 0 remainder is 2
Answer = (2 6 1)8
 Note: the answer is read from bottom to top as
(261)8, the same as with the binary case.
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Working with Large Numbers
0101000010100111 = ?
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Humans can’t work well with binary numbers; there
are too many digits to deal with.
Memory addresses and other data can be quite
large. Therefore, we sometimes use the
hexadecimal number system.
The Hexadecimal Number System
 The hexadecimal number system is also known as
base 16. The values of the positions are calculated by
taking 16 to some power.
 Why is the base 16 for hexadecimal numbers ?
 Because we use 16 symbols, the digits 0 through
9 and the letters A through F.
The Hexadecimal Number System
Binary Decimal
0
1
10
11
100
101
110
111
1000
1001
Hexadecimal
Binary Decimal
Hexadecimal
0
0
1010
10
A
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1011
1100
1101
1110
1111
11
12
13
14
15
B
C
D
E
F
The Hexadecimal Number System
 Example of a hexadecimal number and the values of
the positions:
3 C 8 B 0 5 1
166 165 164 163 162 161 160
 Conversion of hex to decimal ( base 16 to base10)
 Example: convert (F4C)16 to decimal
=>(F x 162) + (4 x 161) + (C x 160)
=> (15 x 256) + (4 x 16) + (12 x 1)
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 Conversion of decimal to hex ( base 10 to base 16)
 Example: convert (4768)10 to hex.
=> 4768 / 16 = 298 remainder 0
=>298 / 16 = 18 remainder 10 (A)
=>18 / 16 = 1 remainder 2
=>1 / 16 = 0 remainder 1
 Answer: (1 2 A 0)16
 Note: the answer is read from bottom to top , same
as with the binary case.
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Octal to hexadecimal conversion
 The conversion is made in two steps using binary as an
intermediate base. Octal is converted to binary and then
binary to hexadecimal, grouping digits by fours, which
correspond each to a hexadecimal digit.
 For instance, convert (1057)8 to hexadecimal:
 To binary:
1
0
5
7
001
000
101
111
 Then to hexadecimal:
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0010
0010
1111
2
2
F
23
REPRESENTATION OF NUMBERS
COMPLEMENT OF NUMBERS
 Two types of complements for base R number
system:
 R's complement and
 (R-1)'s complement
 The (R-1)'s Complement
Subtract each digit of a number from (R-1)
FOR DECIMAL NUMBERS (r-1)=9 and for binary numbers
(r-1)=1
Example:
 9's complement of 83510 is 16410
 1's complement of 10102 is 01012(bit by bit complement
operation)
 The (r – 1)’s complement of octal or hexadecimal numbers
are obtained by subtracting each digit from 7 or F (decimal
15) respectively.
 (r’s) Complement
The r’s complement is obtained by adding 1 to the (r – 1)’s
complement since rn – N = [(rn – 1) – N] + 1. Thus the 10’s
complement of the decimal 2389 is 7610 + 1 = 7611 and is
obtained by adding 1 to the 9’s complement value. The 2’s
complement of binary 101100 is 010011 + 1 = 010100 and
is obtained by adding 1 to the 1’s complement value.
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Add 1 to the low-order digit of its (R-1)'s complement
 Example
- 10's complement of 83510 is 16410 + 1 = 16510
- 2's complement of 10102 is 01012 + 1 = 01102