Number System

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r’s and r-1’s COMPLEMENTS USED FOR

SIMPLIFY SUBTRACTION OPERATIONS

TWO TYPES Decimal Binary Octal Hexadecimal

r’ s complement 10’s 2’s 8’s 16’s r - 1’s complement 9’s 1’s 7’s 15’sHere r is base or radix of number system

r’s Complement LEAVE ALL LSD ZEROS UNCHANGED SUBSTRACT FIRST NONZERO LSD FROM r SUBSTRACT ALL OTHER HIGHER DIGITS FROM r-1

r-1’s Complement SUBSTRACT EVERY DIGIT FROM r-1

Note: r’s Complement=r-1’s Complement +1

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RULES TO GET COMPLEMENTS9’S

SUBSTRACT EVERY DIGIT FROM 9 Example: 9’s complement of 134795 is 865204

10’s=(9’s complement +1)10’s complement can be Directly calculated by following steps:

LEAVE ALL LSD ZEROS UNCHANGED SUBSTRACT FIRST NONZERO LSD FROM 10 SUBSTRACT ALL OTHER HIGHER DIGITS FROM 9 Example:10’s complement of 134795 is 865205 Example: find the 9’s and 10’s complements of 314700.

2’S LEAVE ALL LS ZEROS AND FIRST NON ZERO DIGIT UNCHANGED REPLACE 1 BY 0 AND 0 BY 1 IN ALL HIGHER DIGITS

1’S 1 IS CHANGED TO 0 AND 0 IS CHANGED TO 1

Note: COMPLEMENT OF A COMPLEMENT RESTORES THE ORIGINAL VALUE OF THE NUMBER

10’s Complement Examples

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-003

+1

996

997

-214

+1

785

786

Example #1

Example #2

Complement Digits

Add 1

Complement Digits

Add 1

Subtraction using r’s complement:

To find M-N in base r, we add M + r’s complement of N

1) If M > N then result is M – N + c (c is an end around carry and

Just neglect it. 2) If M < N then result is rn –(N-M) which is the

r’s complement of the result. (to get actual number take r’s

complement again and put – sign before number)

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Example

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Subtraction using r-1’s complement

The same rules apply to subtraction using the (r-1)’s complements.

The only difference is that when an end carry is generated, it is not discarded but added to the least significant digit of the result.

Also, if no end carry is generated, then the answer is negative and in the (r-1)’s complement form.

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BINARY ARITHMETIC OPERATION

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BINARY ADDITION EXAMPLE

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BINARY ARITHMETIC OPERATION (Cont…)

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We have already discussed these two.

BINARY SUBTRACTION EXAMPLE

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OCTAL ARITHMETIC OPERATION

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Flowchart for steps in C program Execution

Practice problems 1. Given the binary number 11011011, what is its

decimal value if it is aa.unsigned binary numberb.Signed binary numberc.2's complement number

2. Convert +24 into a 2's complement number

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Ans: 219Ans: -37

Ans: -37

Ans: 011000 Note the leading zero!

3. Convert -24 into a 2's complement number

4. Convert -1 into a 2's complement number (assume a 4-bit result)

5. What is the minimum (maximum negative) number which can be

represented using a 6-bit 2's Complement representation? What is the maximum (positive) number?

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Ans: 1111

Ans: 101000

Ans: +31, -32

Practice problems (Cont…)

Practice problems (Cont…)

Convert, if possible, the following decimal numbers to 2's complement assuming an 8-bit binary representation for all.

i. 1 = ii. -1 =iii. 72 = iv. 127 =v. -127 = 10000001 vi. -105 = 10010111vii. 255 = viii. -255 = ix. -6 =

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00000001 1111111101001000 01111111

Not Possible 11111010

Not Possible

Practice problems (Cont…)

Convert the following 2's complement numbers to decimal.

i. 1111 = ii. 11110000 =iii. 001101 = iv. 11111111

= -1v. 0000000 = 0 vi. 01 = 1

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-1 -1613

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As a general rule, it is a good idea to avoid the use operators that cause side effects inside of compound expressions. This includes all assignment operators, plus the increment and decrement operators.

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Note: As a general rule, it is a good idea to avoid the use of operators that alter the value of same variable more than one in the same expression, because this causes side effects. This includes all assignment operators, plus the increment and decrement operators.For all side effects result will be compiler dependent.

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#include<stdio.h>void main() {

int i=3;printf("%d",++i + ++i); o/p 9 for turbo c

10 by codepad} The results are undefined. You're modifying a variable more

than once in an expression (or sequence point to be more accurate).

Modifying a variable more than once between sequence points is undefined, so don't do it.

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