module 3,Boolean algebra and logic circuits .pptx

module 3,Boolean algebra and logic circuits

1. Boolean Algebra and Logic Circuits- COMBINATIONAL LOGIC Module 3

2. Contents Binary Numbers Number Base Conversions Octal and Hexa decimal numbers Compliments Basic Definitions Axiomatic definition of Boolean Algebra Basic Theorems Properties of Boolean Algebra Boolean Functions Canonical and Standard forms Other logic operations Digital Logic Gates COMBINATIONAL LOGIC: Introduction, D esign procedure, Adders- Half Adder and Full adders

3. Number System : Number System is a way to represent numbers in computer architecture. There are four different types of the number system, such as: Binary number system (base 2) Octal number system (base 8) Decimal number system(base 10) Hexadecimal number system (base 16).

4. In the  number system , each number is represented by its base . If the base is 2 it is a binary number , If the base is 8 it is an octal number, I f the base is 10, then it is called decimal number system and If the base is 16, it is part of the hexadecimal number system.

5. Binary Number System Binary number system is used to define a number in binary system. Binary system is used to represent a number in terms of two numbers only, 0 and 1. The binary number system is used commonly by computer languages like Java, C++. As the computer only understands binary language that is 0 or 1, all inputs given to a computer are decoded by it into series of 0's or 1's to process it further .

6. Octal Number System Octal Number System  has a base of eight and uses the numbers from 0 to 7. The octal numbers, in the  number system , are usually represented by binary numbers when they are grouped in pairs of three. For example, an octal number 12 8  is expressed as 001010 2  in the binary system, where 1 is equivalent to 001 and 2 is equivalent to 010. Octal Number System Base – 8 Octal Symbol – 0, 1, 2, 3, 4, 5, 6 and 7

7. Decimal Number System In the decimal number system, the numbers are represented with base 10. The way of denoting the decimal numbers with base 10 is also termed as decimal notation. This number system is widely used in computer applications. It is also called the base-10 number system which consists of 10 digits, such as, 0,1,2,3,4,5,6,7,8,9 . Each digit in the decimal system has a position and every digit is ten times more significant than the previous digit. Suppose , 25 is a decimal number, then 2 is ten times more than 5 . Some examples of decimal numbers are:- (12)10, (345)10, (119)10, (200)10, (313.9)10

8. A number system which uses digits from 0 to 9 to represent a number with base 10 is the decimal system number . The number is expressed in base-10 where each value is denoted by 0 or first nine positive integers. Each value in this number system has the place value of power 10. It means the digit at the tens place is ten times greater than the digit at the unit place. Let us see some more examples: (92) 10  = 9×10 1 +2×10 (200) 10  = 2×10 2 +0x10 1 +0x10 The decimal numbers which have digits present on the right side of the decimal (.) denote each digit with decreasing power of 10. Some examples are: (30.2) 10 = 30×10 1 +0x10 +2×10 -1 (212.367) 10  = 2×10 2 +1×10 1 +2×10 +3×10 -1 +6×10 -2 +7×10 -3

9. Hexadecimal Number System The  hexadecimal number system  is a type of number system, that has a base value equal to 16. It is also pronounced sometimes as  ‘hex’ . Hexadecimal numbers are represented by only 16 symbols . These symbols or values are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F. Each digit represents a decimal value. For example, D is equal to base-10 13.

10. Decimal to any Base-r Decimal to Binary: Divide by 2 Decimal to Octal: Divide by 8 Decimal to Hexadecimal: Divide by 16 A ny Base-r to Decimal Binary to Decimal : Multiply by powers of 2 Binary to Octal : Multiply by powers of 8 Binary to Hexadecimal : Multiply by powers of 16

11. Binary to Octal: Split the given binary numbers into 3 bits and write the corresponding Octal equivalent. Octal to Binary: Write the corresponding binary numbers equivalent for the given octal numbers. Binary to Hexadecimal: Split the given binary numbers into 4 bits and write the corresponding Hexadecimal equivalent. Hexadecimal to Binary: Write the corresponding binary numbers equivalent for the given Hexadecimal numbers. Octal to Hexadecimal: Firstly convert the given Octal number into Binary and the obtained Binary Value into the Hexadecimal number. Hexadecimal to Octal: Firstly convert the given Hexadecimal number into Binary and the obtained Binary Value into the corresponding Octal number .

12. Data Unit: Bit – a binary digit that can have the value 0 or 1 Nibble – Half of a byte, or 4 bits. Byte – 8 Bits Word – Two bytes or 16 Bites Kilobyte(K) - 2 10 bytes. (1024 Bytes) Megabyte(M) - 2 20 bytes.(1 million – 1,048,576 Bytes) Gigabyte(G) - 2 30 bytes. (1 billion) Terabyte(T) - 2 40 bytes. (1 trillion) Petabyte(P) - 2 50 bytes. Exabyte(P) - 2 60 bytes. Zettbyte (P) – 2 70 bytes. Yottabyte(P) - 2 80 bytes.