1. Why binary numbers are used with digital circuits?

Digital circuits can only work with electrical signals or pulses.A high pulse is equivalent to 1 and a low pulse is equivalent to 0.The Binary system has two digits only they are 0 and 1.

2. What is the advantage of Octal and Hexadecimal number systems over binary system?

In Octal and Hexadecimal system,large numbers can be represented in short form.

3. Explain the Addition of Binary Numbers:

To add two binary numbers, we should write the numbers one below the other with their least significant bits (LSB) aligned. If the numbers have fractional part, then the binary points must be aligned. Then we can start addition with the LSB by using the following rules.

 0   +   0     =   0

 0   +   1     =   1

 1   +   1     =    10 (answer is 1 and carry is 0)

1    +   1   +   1   =  11 (answer is 1 and carry is also 1).

4. Explain the  Subtraction of Binary Numbers:

The principles of decimal subtraction can be applied here also. If the subtrahend (the lower digit- the number to be subtracted) is larger than the minuend (the upper digit- the number from which it is subtracted), it is necessary to borrow a 1 from the next left bit of the first number. A small binary number can be subtracted from a larger number by using the following rules:

 0   -   0   = 0

 1   -   0   = 1

 1   -   1   = 0

 0   -   1   = 1 (borrowing a  1 from the left place)

Example.

Subtract   01110₂   from   1010₂

5. Explain the Sign and Magnitude Representation of Numbers.

In the signed-magnitude representation, a number consists of a magnitude string and a symbol indicating the sign of the number. The sign symbol is at the MSB. The rest of the bits form the magnitude and are
interpreted similarly to unsigned numbers.

The Most Significant Bit (MSB) is used to represent the sign of a number. IF MSB holds the value 0, represents positive (+) sign and if it holds 1 that means the sign is negative (-).

Example : A computer with 1 byte (8 bits) word size. Here the MSB (8th bit) is reserved for indicating the sign. Therefore the maximum magnitude which can be used is of 7 bits. So an 8 bit word can represent total 28- 1 = 255 numbers (-277 to + 127). In sing and magnitude representation the number + 12 is represented as.

6. What is 1's and 2's complement representation of numbers ?

All arithmetic operations are done internally on computer in the form of addition.The subtraction performed by an additive approach known as complementary subtraction.

1's Complement Representation 

1's complement represents positive numbers by their binary equivalents(called true forms) and negative numbers by their complements.In this representation total bits can be used for representing the magnitude, because there is no sign bit. –ve numbers are represented by writing the complements of the number. 1’s complement can be obtained by changing 1 by 0 and 0 by 1.

Example :- in 8 bit representation :

                       + 13 is 0000 1101

                       - 13 is 1111 0 010 . ( 1’s complement form ).

2’s complement representation

2's complement method represents positive numbers in their binary equivalent (that is ,true forms) and negative in 2's complement form.This representation also uses the whole bits for representing the magnitude. This is the more consistent negative number representation used in computer 2’s complement of a number can be obtained by adding with 1’s complement of a number. 

Example : - in 8 bit representation.

              + 13 is 0000 1101

               - 13 is 11 11 00 11 . ( 2’s complement form ).

7. Write a note on Real number representation in binary.

Real numbers are represented as exponents and mantissa. For example, suppose we want to represent 67.54, at first it is written in normalized form as 0.6754 x 102

Here 0.6754 is the Mantissa and ‘2’ is the exponent. The same method is applied in binary also, Binary numbers are represented on 10 bit mantissa and 6 bit exponent.
In the mantissa part first bit is assumed to be sign bit. Decimal point is assumed to be the right side first bit.

8. Find the 2's complement of 100101 in 8- bit format.

The given number in 8 - bit format is :    00100101

1's Complement  of this number is    11011010

Adding 1 to it 3 + 1

2's Complement  is       00100101

                                      11011010

                                      +            1

                                   ___________

                                       11011011

9. Subtract (101101)₂ from (1110011)₂using 1's complement method.

Subtrahend in 7 bit form is         0101101

The minuend is                          1110011

1's complement of subtrahend  1010010

                                                   _________

Adding minuend                        11000101  

Adding the carry 1                                    1

                                                   _________

                                                     1000110

10. Subtract (101101)₂ from (1110011)₂using 2's compliment method.

Making the number of bits same, the subtrahend become 0101101.

1's compliment of this number is 10110011.

2's compliment of this number is 1010011.

Adding 1010011 to 1110011, we get: 11000110.

Removing the MSB of this number we get: 1000110

Thus the final answer is (1000110)₂

11. If (144)x = (64)y, Find x and y.

The value of x will be smaller than y, because the number of digits in the first number is more than that in the second number. Neither of these number is binary. By applying trial and error method with the other bases, we can find that the value of x is 8 and that of y is 16.

12. Express the number - 17 in 2's compliment form.

Binary equivalent of 17 is 10001. Setting into 8-bit format, we get 00010001.

1's compliment of this number is 11101110.

2's compliment is 11101111, which represents-17.

13. Consider the following number : - 94. What are the methods that are used to represent this negative number in computer arithmetic ?

For this the method for binary representation of integers are used.

(i)  Sign and magnitude method
(ii) 1’s complement method
(iii) 2’s complement method

14. The price of certain product in a multinational company are shown like 2A, B2, D, BA, F8 and EE5.

                    a. Which number system is used in this type of coding?

                    b. How many digits are allowed in this system

                    c.Convert any one of the above codes into decimal system.

Answer :

          a. Hexadecimal number system

          b. 16 digits are allowed in this system.

          c. (2A)_{16   = 2 x 16 ¹ + A x 16 ⁰} 

                           = 2 x 16 + 10 x 1

                           = 32 +10

                           = 42