The document discusses different number systems including binary, decimal, octal, and hexadecimal. It explains how to convert between these number systems using methods like partitioning binary numbers into groups for octal and hexadecimal. The document also covers signed numbers, number complements like 1's and 2's complement, detecting overflow, and subtraction of unsigned numbers.

1. DATA REPRESENTATION BY- Ravi Sharma

2. Binary number system- [ 0and 1 ] Radix-2 , e.g.-(101101)2 Decimal number system- [ 0 to 9 ] Radix-10 , e.g.-(243)10 Octal number system- [ 0 to 7 ] Radix-8 , e.g.-(736.4)8 Hexadecimal - [ 0 to 9 and A to F ] Radix-16, e.g.-(F3)16 NUMBER SYSTEMS:

3. Conversion to decimal- A number expressed in base r can be converted to its decimal equivalent by multiplying each coefficient by corresponding power of r and adding . The following is an example of octal to decimal conversion: Conversion

4. Conversion from decimal to ‘r’ : Conversion of decimal integer into a base r is done by successive divisions by r and accumulation of the remainders . The conversion of fraction is done by successive multiplication by r and accumulation of integer so obtained.

5. Conversion from and to binary , octal , hexadecimal- Since 23=8 and 24=16, each octal digits corresponds to three and each hexadecimal corresponds to 4 binary digits . The conversion from binary to octal and hexadecimal is done by partitioning the binary no. into groups of three and four bits respectively .

6. (r-1)’s - - 9’s complement : It follows that the 9’s complement of a decimal no. is obtained by subtracting each digit from 9. e.g.- 9’s complement of 546700 is 999999-546700=453299 -1’s complement: The 1’s complement of a binary no. is obtained by subtracting each digit by 1. e.g.- 1’s complement of 1011001 is 0100110. Complements

7. ( r’s ) – -10’s complement : 10’s complement of a decimal number is obtained by adding 1 to the 9’s complement value. e.g.- 10’s complement of 2389 is 7610+1=7611. -2’s complement : 2’s complement of binary number is obtained by adding 1 to the 1’s complement. e.g. – 2’s complement of 101100 is 010011+1=010100.

8. Subtraction of unsigned numbers

9. Signed Numbers

11. An overflow condition can be detected by observing the carry into the sign bit position and carry out of the sign bit position . If these two carries are not equal an overflow is occurred . carries: 0 1 carries: 1 0 +70 0 1000110 -70 1 0111010 +800 1010000-801 0110000 +150 1 0010110 -150 0 1101010 Overflow

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