Chapter 1 _ Number System a

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HomeChapter 1 : Number System and Binary Code (Part 1)

Remember These:

There are two input signals : analog signals have infinite

number of distinct values and are continuous, while digital

signals have finite number of distinct values and are

discrete in nature.

Types bf number systems are : decimal, binary, octal,

hexadecimal.

Codes are representation if digital in specified format which

include symbols, alphabets etc.

There are two logic levels in digital system : high (1) and

low (0).

There are two logic systems positive and negative.

Number system is a set of rules and symbols to represent

numbers. It can be weighted or non-weighted.

Number of values that a character or digit can assume is

called Radix or Base.

Decimal system has radix 10. Leftmost digit is MSD and

rightmost digit is LSD.

Binary system has radix 2 and two binary digits one 1 and

0. Its weight is expressed as a power of 2.

The smallest unit of information is called bit (0 and 1).

Binary representation of four bits is called a Nibble.

A byte is a combination of 8-binary bits.

A word is a combination of 16-binary bits.

Octal numbers system has radix 8 and the digits are (0 to

7). Its weight is expressed in power of 8.

Hexadecimal has radix 16. The digits are 0 to 9 in

continuation with letters A to F Its weight is expressed in

power of 16.

ls complement of a binary number is written by simply

replacing all 0s by 1 and all 1s by 0.

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2s complement is one increment of ls complement.

Numbers without +ve / -ve sign are unsigned numbers.

Numbers represented by sign magnitude are signed

numbers.

BCD represent as 4 bit binary code; also known as 8421

code.

Excess 3 code is obtained by addition of three Le. (0011)2

to BCD and is set

complementary,

Gray codes are reflected codes in which the successive

coded characters differ in only bit position.

Alphanumeric codes are represented by letters, symbols

and numbers. These are ABC codes, EBCDIC codes and ICII

code.

Q 1. What is number system? What are its types? Give

example for each type of number system.

Ans. Number system : It is a set of rules and symbols, used

to represent number, The number system can be classified

into weighted. or positional and non-weighted or non-

positional systems. Most of the number systems are of

weighted type.

The knowledge of number system is very essential

because the design and of a computer is dependent upon the

number systems. Few important points related to number

system are:

1. Base or Radix: It is defined as the number of different

symbols used in the number system. The number of values

that a character or digit can assume is called the Radix Base of

the system.

2. The largest value of a digit is always less than the

Radix or Base : If Radix r Base is. represented by r or b,

then the largest value of a digit is given by (r 1) or (b 1)

For e.g. The largest digit in decimal number system is (10

1) = 9. Where, 10 is the radix of decimal number system.

Types of Number System : Following table shows the

various, number system with their radix (r) or base (b).


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1. The Decimal Number System : The number system

having radix or base 10 called as decimal number system.

The number which we make use in our life is called decimal

number system.

The decimal system has the base value of 10. So, its

maximum or largest value of digit is (r 1) = 10 1 = 9,

where r = radix or base. Decimal position values as powers of10 are as shown:

2. Binary Number System : The number system having

radix or base 2 is called as binary number system.

As the radix or base value of binary number system is 2,

so its maximum value of digit

(r 1) = (2 1) = 1,where r is radix

or base.

The two binary digits are 1 and 0. in binary system each

binary digit is known as bit and has its own weight or value.

Its weight is expressed as a power of 2.

3. Octal Number System : The number system which

make use of radix 8 is known as octal number system. As the

radix or base of octal number system is 8, so its maximum or

largest value of a digit is

(r 1) = (8 1) = 7 where r is radix or base.

It make use of first eight digits of decimal number systemi.e. 0, 1, 2, 3, 4, 5, 6 and 7.

Thus, 8 and 9 digits never come in octal number system.

Octal positions values as a power of 8 are as shown:


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4. Hexadecimal Number System: The number system

having radix or base 16 is called as hexadecimal number

system.. In short these are known as hex system. The

number of values assumed by each digit are 0 through 9 andletters A, B, C, D, E and F. Thus the sixteen possible values

are

0, 1; 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F,

Here A represents 10

B represents 1.1

C represents 12

D represents .13

E represents 14

F represents 15The largest value or maximum value for the system is

(r 1) = (16 - 1) = 15

For e.g.

CONVERSION OF DIFFERENT

NUMBERS

(I) CONVERSION FROM

BINARY TO DECIMAL

Q 1. Convert the binary number (110)2 to its decimal

equivalent.


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Solution.

Q 2. Convert the binary number (1011.01) to its decimal

equivalent.Solution.

Q 3. Convert the following

(11011.111)2 = (?)10

Solution.

(II) CONVERSION FROM

OCTAL TO DECIMAL

Q 4. Convert the following:

(71)8 = (?)10

Solution.


(521.63)8 = (?)10

Solution.

Q 6. Convert the following:

(385.24)8 = (?)10


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Solution.

(III) CONVERSION FROM

HEXADECIMAL TO DECIMAL


(3A)16 > (?)10

Solution.


(2B.48)16 > (?)10

Solution.


(4C8.2) 16 > (?)10

Solution.

(IV) CONVERSION FROM

DECIMAL TO BINARY

Q 10. Convert (14) 10 > (?)2

Solution.


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Q 11. (204)10 > (?)2

Solution.

Q 12. (0.6875) 10 > (?)2

Solution.

(V) CONVERSION DECIMAL TO

OCTAL

Q 13. Convert (241)10 = (?)8

Solution.

Q 14. Convert (0.6234) 10 > (?)8

Solution.


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Q 15. Convert (305.6875) 10 = (?)8

Solution.

(VI) CONVERSION FROM

DECIMAL TO HEX.

Q 16. Convert (80) 10 = (?)16

Solution.

Q 17. Convert (0.122)10 = (?)16

Solution.

(0.122)10 (0.1F3B64)16

Q 18. Convert (7825.760) 10 (?)16

Solution.


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(VII) CONVERSION FROM

BINARY TO OCTAL

Q 19. Convert (10110)2 (?)8

Solution.

Q 20. Convert (11010010) 2 (?)8

Solution.

(11010010)2 = (322)8

Q 21. Convert (0.1011011) 2 (?)8Solution.

(0.1011011)2 (0.554)8

(VIII) CONVERSION FROM BINARY TO HEXADECIMAL

Q 22. Convert (1010111) 2 (?)16

Solution.

(1010111) 2 = (57)16

Q 23. Convert (1010 1111 1011 0010) 2 (?)16

Solution.

(1010 1111 1011 0010) 2 = (AFB2)16

Q 24. Convert (10110110.101111001) 2 (?)16


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Solution.

(10110110.101111001) 2 = (B6.BC8)16

(IX) CONVERSION FROM OCTAL TO HEXADECIMAL

Q 25. Convert (436) 8 (?)16

Solution.

(X) CONVERSION FROM HEXADECIMAL TO OCTAL

Q 26. Convert (1AF)16 (?)8

Solution.

Q 27. Convert (3CFB.2E) 16 = (?)8

Solution.

Q 28. Convert (68.4B) 16 = (?)8

Solution.


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Q 29 What is the largest decimal number that can be represented by a 16 bit

binary word?

Ans.

=15 x 4096 + 15 x 256 + 15 x 16 + 15

= 61440 + 3840 + 240 + 15

= (65535)10

Thus, 65535 is the largest decimal number that can be represented by a 16 bit binary word.

Q 30. Convert the decimal number 39.75 to octal.

Ans. (39.75)10 = (?)8

Integer part

(39)10 = (47)8

Fractional part

0.75 x 8 = 6.0

(0.75)10 = (0.6)8

(39.75)10 = (47.6)8

Q 31. Define bit, byte and nibble.

Ans.1. Bit : Bit is an abbreviation of the binary digit and it is the smallest unit of

information. It is either 0 or 1.

2. Byte : A byte is a combination of 8-binary bits A byte contains two nibbles. It is used in

case of representation of memory.

3. Nibble : Binary representation of four bits is called a nibble. In case of BCD i.e. binary

coded decimal and hexadecimal numbers nibble is used as both are four bit numbers.

Q 32. Find the complement of

Ans.

Its complement is given by:

Q 33. What do you mean by weighted code? Give example.

Ans. Weighted codes are those codes which make use of weighted sum method i.e. which

obey the principle of positional weight. In weighted codes each position of the number

has a specific weight

Weighted codes are:

1. Binary codes

2. BCD codes. For example: 8421, 2421, 4221, 5311, 7421, etc.


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Chapter 1 _ Number System a