explain what binary has in common with other systems includi

explain what binary has in common with other systems including decimal and Babylonian. Explain what 2\'s complement and 1\'s compliment are about, how to add, subtract, multiply and divide in binary. Show how digital logic gates use binary inputs and form binary outputs based on a TRUTH TABLE. Show how binary can be easily converted into hexadecimal?

Solution

PART 1 :

Common features in binary, decimal and babylonian number system -

1. All three i.e. binary, decimal and babylonian number systems use positional or place value notations to encode the magnitude or value.

Place value notation - A digit with static face value may have different place value.

Exapmple: Digit \'1\' has face value equal to \'1\' but it has place value -

a.            1 * 10 = 10 in the number \'12\'

b.            1 * 100 = 100 in the number \'102\'

and so on.

2. Each one of them follows a base system for representation of numbers. i.e. Base 2 for binary, Base 10 for decimal and base 60 for babylonian.

3. All three of them are very much in use in contemporary world.

                a. BINARY NUMBER SYSTEM predominantly used in computers

                b. DECIMAL NUMBER SYSTEM forms a major and most common part of our educational curriculam since it is used in daily life calculations.

                c. BABYLONIAN NUMBER SYSTEM since has a base 60 so is used to count angles and time.

PART 2:

2\'s complement and 1\'s complement -

Both 2\'s complement and 1\'s complement are concepts relating to binary representaion of numbers.

1\'s complement represents the negative of original binary number. It is obtained by inverting all the digits of the original number i.e. replacing 1 by 0 and vice versa.

2\'s complement is majorly used for representation and storage of signed numbers in computers. 2\'s complement of a binary number is obtained by adding 1 to the 1\'s complement of the same number .

Example- Suppose the number whose 1\'s complement and 2\'s complement has to be calculated is

10011 B so following the above given procedure, we get

PART 3:

Add, subtract, multiply and divide in binary –

1.ADD - We can add two binary numbers just like we add two decimal numbers.

Example: Suppose we have two binary numbers 11001B and 00101B. The sum of these two numbers would be: 11110B

2. SUBTRACT - We can subtract two binary numbers by simply taking 2’s complement (as explained in PART 2) of subtrahend and adding it to minuend. Example: Suppose we have two binary numbers 10100B and 10101B. The difference between these two numbers would be: 11111B

3. MULTIPLY - We can multiply two binary numbers by using the concept of partial product which stores the value of intermediary sums as well as the final product. Rules for multiplication would be:

Example:

X = Multiplicand, Y = Multiplier, P = Partial product

X = 0000 1111                           P         .

Y = 0000 1011                     0000 0000

      0000 1111                      0000 1111

      0001 1110                      0010 1101

    0000 0000                      0010 1101

      0111 1000                      1010 0101

      1010 0101

4. DIVIDE – The division process is exact complementary of multiplication.

Here it is shown with example:

11010                                    Quotient = Q

Divisor B= 10001               0111000000                         Dividend = A

0110                                       5 bits of A< B, qoutient has 5 bits

011100                                  6 bits of A greater than equal to B

- 10001                                 Shift right B and subtract; enter 1 in Q

-010110                                7 bits of remainder greater than equal to B

-- 10001                                Shift right B and subtract; enter 1 in Q

--001010                               Remainder < B; enter 0 in Q; Shift right B

---010100                             Remainder is greater than equal to B

----10001                              Shift right B and subtract; enter 1 in Q

----000110                           Remainder < B; enter 0 in Q

-----00110                            Final remainder

PART 4:

How digital logic gates use binary inputs and form binary outputs based on a TRUTH TABLE -

1.AND LOGIC GATE – Output would be high if and only if both inputs are high

INPUTS

OUTPUT

A

B

Q

0

0

0

0

1

0

1

0

0

1

1

1

2.OR LOGIC GATE - Output would be high if either of the input is high.

INPUTS

OUTPUT

A

B

Q

0

0

0

0

1

1

1

0

1

1

1

1

3.NOT LOGIC GATE - Output is simple inversion of the input.

INPUT

OUTPUT

A

out

0

1

1

0

All other logic gates are derived from above given three basic logic gates

PART 5:

How binary can be easily converted into hexadecimal

BINARY number can be converted into hexa decimal number with help of decimal representation of number as given below

Example -

BINARY NUMBER: 10110B = 1*(2^4)+ 0*(2^3)+ 1*(2^2)+ 1*(2^1)+ *(2^0)=16+0+4+2+0=22D

DECIMAL NUMBER: 22D=1*(16^1)+6*(16^0)=16H

Hence, 10110B = 16H.

INPUTS	OUTPUT
A	B	Q
0	0	0
0	1	0
1	0	0
1	1	1