MODULE –I NUMBER SYSTEM Digital Design Amit Kumar Assistant Professor SCSE, Galgotias University,...
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Transcript of MODULE –I NUMBER SYSTEM Digital Design Amit Kumar Assistant Professor SCSE, Galgotias University,...
M O D U L E – I
N U M B E R S Y S T E M
Digital Design
Amit Kumar
Assistant Professor
SCSE,
Galgotias University, Greater Noida
•Introduction to Number System• Types of Number System• Binary• Octal• Decimal• Hexadecimal• Conversion from one number system to another
Outline
Introduction to Number SystemA number system defines how a number can be
represented using distinct symbols. A number can be represented differently in different systems. For example, the two numbers (2A)16 and (52)8 both refer to the same quantity, (42)10, but their representations are different.
Several number systems have been used in the past and can be categorized into two groups: positional and non-positional systems. Our main goal is to discuss the positional number systems, but we also give examples of non-positional systems.
Types of Number Systems
System Base SymbolsUsed by humans?
Used in computers?
Decimal 10 0, 1, … 9 Yes No
Binary 2 0, 1 No Yes
Octal 8 0, 1, … 7 No No
Hexa-decimal
16 0, 1, … 9,A, B, … F
No No
POSITIONAL NUMBER SYSTEMS
In a positional number system, the position a symbol occupies in the number determines the value it represents. In this system, a number represented as:
has the value of:
in which S is the set of symbols, b is the base (or radix).
S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
The decimal system (base 10)
The word decimal is derived from the Latin root decem (ten). In this system the base b = 10 and we use ten symbols
The symbols in this system are often referred to as decimal digits or just digits.
Integers
Figure 1 Place values for an integer in the decimal system
Example 1
The following shows the place values for the integer +224 in the decimal system.
Note that the digit 2 in position 1 has the value 20, but the same digit in position 2 has the value 200. Also note that we normally drop the plus sign, but it is implicit.
Example 2
The following shows the place values for the decimal number −7508. We have used 1, 10, 100, and 1000 instead of powers of 10.
Note that the digit 2 in position 1 has the value 20, but the same digit in position 2 has the value 200. Also note that we normally drop the plus sign, but it is implicit.
( ) Values
Reals
Example 3
The following shows the place values for the real number +24.13.
The word binary is derived from the Latin root bini (or two by two). In this system the base b = 2 and we use only two symbols,
The binary system (base 2)
S = {0, 1}
The symbols in this system are often referred to as binary digits or bits (binary digit).
Integers
Figure 2 Place values for an integer in the binary system
Example 4
The following shows that the number (11001)2 in binary is the same as 25 in decimal. The subscript 2 shows that the base is 2.
The equivalent decimal number is N = 16 + 8 + 0 + 0 + 1 = 25.
Reals
Example 5
The following shows that the number (101.11)2 in binary is equal to the number 5.75 in decimal.
The word hexadecimal is derived from the Greek root hex (six) and the Latin root decem (ten). In this system the base b = 16 and we use sixteen symbols to represent a number. The set of symbols is
The hexadecimal system (base 16)
S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}
Note that the symbols A, B, C, D, E, F are equivalent to 10, 11, 12, 13, 14, and 15 respectively. The symbols in this system are often referred to as hexadecimal digits.
Integers
Figure 3 Place values for an integer in the hexadecimal system
Example 6
The following shows that the number (2AE)16 in hexadecimal is equivalent to 686 in decimal.
The equivalent decimal number is N = 512 + 160 + 14 = 686.
The word octal is derived from the Latin root octo (eight). In this system the base b = 8 and we use eight symbols to represent a number. The set of symbols is
The octal system (base 8)
S = {0, 1, 2, 3, 4, 5, 6, 7}
Integers
Figure 4 Place values for an integer in the octal system
Example 7
The following shows that the number (1256)8 in octal is the same as 686 in decimal.
Note that the decimal number is N = 512 + 128 + 40 + 6 = 686.
Table 1 shows a summary of the four positional number systems discussed in this chapter.
Summary of the four positional systems
Table 2 shows how the number 0 to 15 is represented in different systems.
Decimal Binary Octal Hex00 0000 00 001 0001 01 102 0010 02 203 0011 03 304 0100 04 405 0101 05 506 0110 06 607 0111 07 708 1000 10 809 1001 11 910 1010 12 A11 1011 13 B12 1100 14 C13 1101 15 D14 1110 16 E15 1111 17 F
2.23
We need to know how to convert a number in one system to the equivalent number in another system. Since the decimal system is more familiar than the other systems, we first show how to covert from any base to decimal. Then we show how to convert from decimal to any base. Finally, we show how we can easily convert from binary to hexadecimal or octal and vice versa.
Conversion
The possibilities:
Hexadecimal
Decimal Octal
Binary
Any base to decimal conversion
Figure 5 Converting other bases to decimal
Example 8
The following shows how to convert the binary number (110.11)2 to decimal: (110.11)2 = 6.75.
Example 9
The following shows how to convert the hexadecimal number (1A.23)16 to decimal.
Note that the result in the decimal notation is not exact, because 3 × 16−2 = 0.01171875. We have rounded this value to three digits (0.012).
Example 10
The following shows how to convert (23.17)8 to decimal.
This means that (23.17)8 ≈ 19.234 in decimal. Again, we have rounded up 7 × 8−2 = 0.109375.
Decimal to any base
Figure 6 Converting other bases to decimal (integral part)
Figure 7 Converting the integral part of a number in decimal to other bases
Example 11
The following shows how to convert 35 in decimal to binary. We start with the number in decimal, we move to the left while continuously finding the quotients and the remainder of division by 2. The result is 35 = (100011)2.
Example 12
The following shows how to convert 126 in decimal to its equivalent in the octal system. We move to the right while continuously finding the quotients and the remainder of division by 8. The result is 126 = (176)8.
Example 13
The following shows how we convert 126 in decimal to its equivalent in the hexadecimal system. We move to the right while continuously finding the quotients and the remainder of division by 16. The result is 126 = (7E)16
Figure 8 Converting the fractional part of a number in decimal to other bases
Figure 9 Converting the fractional part of a number in decimal to other bases
Example 14
Convert the decimal number 0.625 to binary.
Since the number 0.625 = (0.101)2 has no integral part, the example shows how the fractional part is calculated.
Example 15The following shows how to convert 0.634 to octal using a maximum of four digits. The result is 0.634 = (0.5044)8. Note that we multiple by 8 (base octal).
Example 16
The following shows how to convert 178.6 in decimal to hexadecimal using only one digit to the right of the decimal point. The result is 178.6 = (B2.9)16 Note that we divide or multiple by 16 (base hexadecimal).
Example 17
An alternative method for converting a small decimal integer (usually less than 256) to binary is to break the number as the sum of numbers that are equivalent to the binary place values shown:
Example 18A similar method can be used to convert a decimal fraction to binary when the denominator is a power of two:
The answer is then (0.011011)2
Binary-hexadecimal conversion
Figure 10 Binary to hexadecimal and hexadecimal to binary conversion
Example 19
Show the hexadecimal equivalent of the binary number (110011100010)2.
SolutionWe first arrange the binary number in 4-bit patterns:
100 1110 0010
Note that the leftmost pattern can have one to four bits. We then use the equivalent of each pattern shown in Table 2.2 on page 25 to change the number to hexadecimal: (4E2)16.
Example 20
What is the binary equivalent of (24C)16?
SolutionEach hexadecimal digit is converted to 4-bit patterns:
2 → 0010, 4 → 0100, and C → 1100
The result is (001001001100)2.
Binary-octal conversion
Figure 10 Binary to octal and octal to binary conversion
Example 21
Show the octal equivalent of the binary number (101110010)2.SolutionEach group of three bits is translated into one octal digit. The equivalent of each 3-bit group is shown in Table 2.2 on page 25.
The result is (562)8.
101 110 010
Example 22
What is the binary equivalent of for (24)8?
SolutionWrite each octal digit as its equivalent bit pattern to get
2 → 010 and 4 → 100
The result is (010100)2.
Octal-hexadecimal conversion
Figure 12 Octal to hexadecimal and hexadecimal to octal conversion
Example 23Find the minimum number of binary digits required to store decimal integers with a maximum of six digits.
Solutionk = 6, b1 = 10, and b2 = 2. Then
x = ék × (logb1 / logb2)ù = é6 × (1 / 0.30103)ù = 20.
The largest six-digit decimal number is 999,999 and the largest 20-bit binary number is 1,048,575. Note that the largest number that can be represented by a 19-bit number is 524287, which is smaller than 999,999. We definitely need twenty bits.
NONPOSITIONAL NUMBER SYSTEMS
Although non-positional number systems are not used in computers, we give a short review here for comparison with positional number systems. A non-positional number system still uses a limited number of symbols in which each symbol has a value. However, the position a symbol occupies in the number normally bears no relation to its value—the value of each symbol is fixed. To find the value of a number, we add the value of all symbols present in the representation.
In this system, a number is represented as:
and has the value of:
There are some exceptions to the addition rule we just mentioned, as shown in Example 24.
Example 24Roman numerals are a good example of a non-positional number system. This number system has a set of symbols S = {I, V, X, L, C, D, M}. The values of each symbol are shown in Table 2.3
To find the value of a number, we need to add the value of symbols subject to specific rules (See the textbook).
Example 24
The following shows some Roman numbers and their values.
(Continued)
More on Base Conversion
Binary to Decimal
Hexadecimal
Decimal Octal
Binary
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
Example
1010112 => 1 x 20 = 11 x 21 =
20 x 22 =
01 x 23 =
80 x 24 =
01 x 25 =
32
4310
Bit “0”
Octal to Decimal
Hexadecimal
Decimal Octal
Binary
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
Example
7248 => 4 x 80 = 42 x 81 = 167 x 82 = 448
46810
Hexadecimal to Decimal
Hexadecimal
Decimal Octal
Binary
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
Example
ABC16 => C x 160 = 12 x 1 = 12 B x 161 = 11 x 16 = 176 A x 162 = 10 x 256 = 2560
274810
Decimal to Binary
Hexadecimal
Decimal Octal
Binary
Decimal to Binary
Technique Divide by two, keep track of the remainder First remainder is bit 0 (LSB, least-significant bit) Second remainder is bit 1 Etc.
Example
12510 = ?22 125 62 12 31 02 15 12 7 12 3 12 1 12 0 1
12510 = 11111012
Octal to Binary
Hexadecimal
Decimal Octal
Binary
Octal to Binary
Technique Convert each octal digit to a 3-bit equivalent binary
representation
Example
7058 = ?2
7 0 5
111 000 101
7058 = 1110001012
Hexadecimal to Binary
Hexadecimal
Decimal Octal
Binary
Hexadecimal to Binary
Technique Convert each hexadecimal digit to a 4-bit equivalent
binary representation
Example
10AF16 = ?2
1 0 A F
0001 0000 1010 1111
10AF16 = 00010000101011112
Decimal to Octal
Hexadecimal
Decimal Octal
Binary
Decimal to Octal
Technique Divide by 8 Keep track of the remainder
Example
123410 = ?8
8 1234 154 28 19 28 2 38 0 2
123410 = 23228
Decimal to Hexadecimal
Hexadecimal
Decimal Octal
Binary
Decimal to Hexadecimal
Technique Divide by 16 Keep track of the remainder
Example
123410 = ?16
123410 = 4D216
16 1234 77 216 4 13 = D16 0 4
Binary to Octal
Hexadecimal
Decimal Octal
Binary
Binary to Octal
Technique Group bits in threes, starting on right Convert to octal digits
Example
10110101112 = ?8
1 011 010 111
1 3 2 7
10110101112 = 13278
Binary to Hexadecimal
Hexadecimal
Decimal Octal
Binary
Binary to Hexadecimal
Technique Group bits in fours, starting on right Convert to hexadecimal digits
Example
10101110112 = ?16
10 1011 1011
2 B B
10101110112 = 2BB16
Octal to Hexadecimal
Hexadecimal
Decimal Octal
Binary
Octal to Hexadecimal
Technique Use binary as an intermediary
Example
10768 = ?16
1 0 7 6
001 000 111 110
2 3 E
10768 = 23E16
Hexadecimal to Octal
Hexadecimal
Decimal Octal
Binary
Hexadecimal to Octal
Technique Use binary as an intermediary
Example
1F0C16 = ?8
1 F 0 C
0001 1111 0000 1100
1 7 4 1 4
1F0C16 = 174148
Exercise – Convert ...
Decimal Binary OctalHexa-
decimal
33
1110101
703
1AF
Exercise – Convert …
Decimal Binary OctalHexa-
decimal
33 100001 41 21
117 1110101 165 75
451 111000011 703 1C3
431 110101111 657 1AF
Answer
Fractions
Binary to decimal
pp. 46-50
10.1011 => 1 x 2-4 = 0.06251 x 2-3 = 0.1250 x 2-2 = 0.01 x 2-1 = 0.50 x 20 = 0.01 x 21 = 2.0 2.6875
Fractions
Decimal to binary
p. 50
3.14579
.14579x 20.29158x 20.58316x 21.16632x 20.33264x 20.66528x 21.33056
etc.11.001001...
Exercise – Convert ...
Decimal Binary OctalHexa-
decimal
29.8
101.1101
3.07
C.82
Exercise – Convert …
Decimal Binary OctalHexa-
decimal
29.8 11101.110011… 35.63… 1D.CC…
5.8125 101.1101 5.64 5.D
3.109375 11.000111 3.07 3.1C
12.5078125 1100.10000010 14.404 C.82
Answer
Thank you