Learn to enjoy every minute of your life.Only I can change my life.: August 2013

Monday, August 26, 2013

Logic Gates

The logic gates have one or many input and only one output.The gates used to form integrated circuits (IC's) for digital circuit with combination and sequential of gates.this gates are of mainly 6 types .The three are basic gates and others are the prepared from basic gates called as derived gates.They are as follow:-
I ) Basic gates :-
                               i ) OR gate
                               ii) AND gate
                               iii)NOT gate
II) Derived gates :-
                              i ) NOR gate
                              ii) NAND gate
                              iii)EX-OR gate
The Logic gates have the following details:
1)Logic:-It is the statement.
2)Truth table:-It shows the input provided and output get.It is combination of input provided making rows in the table with inputs and output column.
3)Boolean equation :-It is the mathematical representation with the equation.
4)Symbol:-It is the logical symbol of the logic gate.

1) OR gate

    Logic :- If A is true or B is true then Y is true

    Boolean Equation :- Y=A+B

   Truth table:

A	B	Y
0	0	0
0	1	1
1	0	1
1	1	1

Symbol:-

2) AND gate

    Logic :- If A is true and B is true then Y is true

    Boolean Equation :- Y=A.B or Y=AB

   Truth table:

A	B	Y
0	0	0
0	1	0
1	0	0
1	1	1

Symbol:-

3) NOT gate

    Logic :- If A is true then Y is false and A is false then Y is true

    Boolean Equation :- Y=A.B or Y=AB

   Truth table:

A	Y
0	1
1	0

Symbol :-

4) NAND gate
Truth table:

A	B	Y
0	0	1
0	1	1
1	0	1
1	1	0

Boolean Equation :- Y=A.B

Symbol:-

5) NOR gate
Boolean Equation :- Y=A+B
Truth table:

A	B	Y
0	0	1
0	1	0
1	0	0
1	1	0

Symbol :-

6)EX-OR gate:
Y=A.B+A.B
Logic :When A is true or Bis true but not both then Y is true.
Truth table:

A	B	Y
0	0	0
0	1	1
1	0	1
1	1	0

Symbol :-

De Morgan's Law :-
First law :-
Statement:-The compliment of sum is equal to product of compliments.

The NOR gate is equivalent to bubbled NAND gate.
Boolean Equation :- A+B = A.B
Second Law :-
Statement:-The compliment of product is equal to sum of compliments.

The NAND gate is equivalent to bubbled OR gate.
Boolean Equation :- A.B = A+B

Half Adder:-
For addition of two bits half adder is used .
Truth table:

A	B	Carry	SUM
0	0	0	0
0	1	0	1
1	0	0	1
1	1	1	0

From the above table we can conclude that gates used in half adder are AND gate for carry and EX-OR gate for Sum.

Monday, August 5, 2013

Number System

Number System
The number system of a number is determined by the Radix of a number.Radix means the base of the number which is obtained by the number of symbols present in that number System.They are as follow:-

Sr.No.	Number System	Radix	Symbols
1.	Binary number System	2	0,1
2.	Octal Number System	8	0,1,2,3,4,5,6,7
3.	Decimal Number System	10	0,1,2,3,4,5,6,7,8,9
4.	Hexadecimal Number System	16	0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F

1.Binary Number System:- The number System with Radix 2 are said to be binary numbers.It contains only 2 symbols 0 and 1.For Examples (1110)₂,(1001)₂,(1101)₂,(1010)₂,(1010.111)₂,etc.
2.Octal Number System:- The number System with Radix 8 are said to be octal numbers.It contains 8 symbols 0,1,2,3,4,5,6 and 7.For Examples (257)₂,(346)₈,(412)₈,(728)₈,(72.39)₈,etc.
3.Decimal Number System:- The number System with Radix 10 are said to be decimal numbers.It contains 10 symbols 0,1,2,3,4,5,6,7,8 and 9.For Examples (123)₁₀,(786)₁₀,(157)₁₀,(928)₁₀,(28.625)₁₀,etc.
4.Hexadecimal Number System:- The number System with Radix 16 are said to be hexadecimal numbers.It contains only 16 symbols 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E and F.Here A represent for number 10,B represents for number 11,C represents for number 12,D represents for number 13,E represents for number 14,F represents for number 15,For Examples (1ABC)₁₆,(25DE)₁₆,(85C.3D)₁₆,(0AB)₁₆,etc.
Note that for hexadecimal number should start with symbols of numbers only that is 0,1,2,3,4,5,6,7,8,9 and not with A,B,C,D,E,F .If there is a possibility of number to star with alphabets then write 0(zero before that number example (0AB)₁₆,(0ADE)_16,etc.

Conversion of numbers Systems:
# Convert Binary to Decimal:- To convert the binary number into decimal we have to multiply each binary number with 2 and then take the raise to power from Least Significant Bit (LSB) side starting with 0 and so on.
1.(1010)₂ = 1*2³ + 0*2²+ 1*2¹ + 0*2⁰
                  = 1*8 + 0*4 +1*2 + 0*1
                  = 8 + 0 + 2 + 0
                  =(10)₁₀
2.(101)₂    = 1*2² + 0*2¹ + 1*2⁰
                 = 1*4 + 0*2 + 1*1
               = 4 + 0 +1
                 = (5)₁₀
3.(1001)₂ = 1*2³ + 0*2²+ 0*2¹ + 1*2⁰
                  = 1*8 + 0*4 +0*2 + 1*1
                  = 8 + 0 + 0 +1
                  =(9)₁₀
_4..(1111)₂ = 1*2³ + 1*2²+ 1*2¹ + 1*2⁰
                  = 1*8 + 1*4 +1*2 + 1*1
                  = 8 + 4 + 2 +1
                  =(15)₁₀
But for fractional numbers that is for the numbers with decimal point we have to multiply fractional numbers with 2 raise to power of -1 from decimal point.
1. (1111.101) =1*2³ + 1*2²+ 1*2¹ + 1*2⁰+ 1*2^-1+ 0*2^-2+ 1*2^-3
                    = 1*8 + 1*4 + 1*2 + 1*1+ 1*1/2¹+ 0*1/2²+ 1*1/2²
                      = 1*8 + 1*4 + 1*2 + 1*1+ 1*1/2 + 0*1/4 + 1*1/8
                      = 1*8 + 1*4 +1*2 + 1*1+1*0.5 + 0*0.25 + 1*0.125
= 8 + 4 + 2 + 1 + 0.5 + 0 + 0.125
                    =(15.625)₁₀
# Convert Decimal to Binary :- To convert the decimal number to binary divide the number by 2 to get integers note down the remainders in front of them simultaneously and then write remainders in reverse order from bottom to top.
1.(157)₁₀
Solution : -

2	157	Remainders
2	78	1
2	39	0
2	19	1
2	9	1
2	4	1
2	2	0
2	1	1
	0	1

Answer=(11011101)₂

2.(9)₁₀
Solution : -

2	9	Remainders
2	4	1
2	2	0
2	1	0
	0	1

Answer=(1001)₂
Note:Binary counting can be done by taking only those values of numbers from similarly like starting from decimal numbers only those numbers which contains symbols 0 and 1.That is 0,1,2,3,4,5,6,7,8,9,10,11,12,13,............,99,100,101,~~102~~,~~103~~,....,110,111,...and so on.Cancel the numbers that contains symbols other than 0 and 1.So the counting will be:

Decimal Numbers	Binary numbers	Hexadecimal Numbers
0	0	0
1	1	1
2	10	2
3	11	3
4	100	4
5	101	5
6	110	6
7	111	7
8	1000	8
9	1001	9
10	1010	A
11	1011	B
12	1100	C
13	1101	D
14	1110	E
15	1111	F

# Convert octal to decimal:- To convert the octal number into decimal we have to multiply each octal number with 8 and then take the raise to power from Least Significant Bit (LSB) side starting with 0 for integers and so on.

1. (157)₈ = 1*8² + 5*8¹ + 7*8⁰
              = 1*64 + 5*40 + 7*1
              = 64 + 200 +7
              = (271)₁₀
2.(459)₈ = 4*8² + 5* 8¹ + 9*8⁰
              = 4*64+5*8+9*1
              =256 + 40 +9
              =(305)₁₀
But for fractional numbers that is for the numbers with decimal point we have to multiply fractional numbers with 8 raise to power of -1 from decimal point and so on -2,-3,.. .
1.(351.123)₈= 3*8² + 5*8¹ + 1*8⁰ + 1*8^-1 + 2*8^-2 + 3*8^-3
                        = 3*64 + 5*8 + 1*1 + 1*1/8¹ + 2*1/8² + 3*1/8³
                        =3*64 + 5*8 + 1*1 + 1*1/8 + 2* 1/64 + 3* 1/512
                        =192 + 40 + 1 + 1*0.125 + 2*0.015625 + 3* 0.001953125
                        = 192 + 40 + 1 + 0.125 + 0.03125 + 0.005859375
                        = (233.162109375)₁₀
# Convert decimal to octal:- To convert the decimal number to octal divide the number by 8 to get integers note down the remainders in front of them simultaneously and then write remainders in reverse order from bottom to top.
1.(157)₁₀
Solution : -