Chapter 3Chapter 3

Number System and Number System and CodesCodes

Decimal and Binary Decimal and Binary NumbersNumbers

Decimal and Binary Decimal and Binary NumbersNumbers

Converting Decimal to Converting Decimal to BinaryBinary

1.Sum of powers of 2

Converting Decimal to Converting Decimal to BinaryBinary

1.Repeated Division

Binary Numbers and Binary Numbers and ComputersComputers

Hexadecimal NumbersHexadecimal Numbers

Converting decimal to Converting decimal to hexadecimalhexadecimal

Converting binary to Converting binary to hexadecimalhexadecimal

Converting hexadecimal to Converting hexadecimal to binarybinary??

Hexadecimal numbersHexadecimal numbers

Binary arithmeticBinary arithmetic

Binary additionBinary addition

Representing Integers with Representing Integers with binarybinary

Some of challenges:-Some of challenges:- Integers can be positive or negativeIntegers can be positive or negative Each integer should have a unique Each integer should have a unique

representationrepresentation The addition and subtraction should be The addition and subtraction should be

efficient.efficient.

Representing a positive Representing a positive numbersnumbers

Representing a negative Representing a negative numbers using Sign-numbers using Sign-Magnitude notationMagnitude notation

-5 = 1101 4-bits sign-manitude-55 =10110111 8-bits sign-

magnitude

11’’s Complements Complement

The 1’s complement representation The 1’s complement representation of the positive number is the same of the positive number is the same as sign-magnitude.as sign-magnitude. +84 = 01010100+84 = 01010100

11’’s Complements Complement

The 1’s complement representation The 1’s complement representation of the negative number uses the of the negative number uses the following rule:-following rule:- Subtract the magnitude from 2Subtract the magnitude from 2nn-1-1

For example:For example: -36 = ???-36 = ???

+36 = 0010 0100+36 = 0010 0100

11’’s Complements Complement

Example :-Example :- - 57- 57

+57 = 0011 1001+57 = 0011 1001 -57 = 1100 0110-57 = 1100 0110

Converting to decimal Converting to decimal formatformat

22’’s Complements Complement

For negative numbers:-

Subtract the magnitude from 2n. Or

Add 1 to the 1’s complement

ExampleExample

Convert to decimal valueConvert to decimal value

Positive values:- Positive values:- 0101 1001 = +890101 1001 = +89

Negative valuesNegative values

Two's Complement Arithmetic

Adding Positive Integers in 2's Complement Form

Overflow in Binary Addition

Overflow in Binary Addition

Overflow in Binary Addition

Overflow in Binary Addition

Adding Positive and Negative Integers in 2's

ComplementForm

Adding Positive and Negative Integers in 2's

ComplementForm

Subtraction of Positive and Negative Integers

Digital Codes

Binary Coded Decimal (BCD)

BCD

BCD

4221 Code

Gray Code

In pure binary coding or 8421 BCD then counting from 7 (0111) to 8 (1000) requires 4 bits to be changed simultaneously.

Gray coding avoids this since only one bit changes between subsequent numbers

Binary –to-Gray Code Conversion

Gray –to-Binary Conversion

Gray –to-Binary Conversion

The Excess-3- Code

ParityParity

The method of parity is widely used as The method of parity is widely used as a method of error detection.a method of error detection. Extar bit known as parity is added to data Extar bit known as parity is added to data

wordword The new data word is then transmitted.The new data word is then transmitted.

Two systems are used:Two systems are used: Even parity: the number of 1’s must be Even parity: the number of 1’s must be

even.even. Odd parity: the number of 1’s must be odd.Odd parity: the number of 1’s must be odd.

ParityParity

Example:Example:

Even Even ParityParity

Odd Odd parityparity

11001110011110011110010110010

11110111100111100111101111101

11000110000110000110001110001