The document discusses number systems and binary arithmetic operations. It provides information on r's and r-1's complements used for simplifying subtraction. It gives examples of binary, octal and decimal addition and subtraction. It also provides practice problems on converting between decimal, binary, octal and 2's complement representations and performing arithmetic operations in these number systems.

1. Number System

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33. 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’s
Here 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 COMPLEMENTS
 9’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

35. 10’s Complement Examples
35
-003
+1
↓↓↓
996
997
-214
+1
↓↓↓
785
786
Example #1
Example #2
Complement Digits
Add 1
Complement Digits
Add 1

36. 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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37. Example
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38. 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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44. BINARY ARITHMETIC OPERATION
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45. BINARY ADDITION EXAMPLE
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46. BINARY ARITHMETIC OPERATION (Cont…)
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We have already discussed these two.

47. BINARY SUBTRACTION EXAMPLE
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49. OCTAL ARITHMETIC OPERATION
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Flowchart for steps in C program Execution

59. Practice problems
1. Given the binary number 11011011, what is its decimal value if it
is a
a. unsigned binary number
b. Signed binary number
c. 2's complement number
2. Convert +24 into a 2's complement number
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Ans: 219
Ans: -37
Ans: -37
Ans: 011000 Note the leading zero!

60. 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…)

61. 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 = 10010111
vii. 255 = viii. -255 =
ix. -6 =
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00000001 11111111
01001000 01111111
Not Possible
11111010
Not Possible

62. Practice problems (Cont…)
Convert the following 2's complement numbers to decimal.
i. 1111 = ii. 11110000 =
iii. 001101 = iv. 11111111 = -1
v. 0000000 = 0 vi. 01 = 1
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-1 -16
13

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65. 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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67. #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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