Binaty Arithmetic and Binary coding schemes

Dr. Anita Goel
Assoc. Prof. (Computer Sci.)
Dyal Singh College, University of Delhi

 Introduction
 Binary Arithmetic
 Signed and Unsigned Numbers
 Binary Data Representation
 Binary Coding Schemes
2

 Uses digits 0-9
 Digits combined to form numbers like 104,
4561
 Decimal arithmetic operations
 Addition, subtraction, multiplication, division
 For e.g., a chocolate costs Rs. 5/-. Total
cost of 2 chocolates will be Rs. 10/- i.e.
(5*2) or (5+5)
3

 Used in computer systems
 Uses digits 0’s and 1’s only
 Digits combined to form numbers like
1001, 11000110
 A digit 0 or 1 is called a bit (binary digit)
 1001 is a 4-bit number.
 11000110 is an 8-bit number
4

 All data is represented internally in a
computer by a combination of bits.
 Each symbol is represented by a
combination of bits.
5

 Arithmetic operations performed on binary
numbers is called binary arithmetic.
 addition, subtraction, multiplication, division.
 Computer systems actually perform only
Binary Addition and Binary Subtraction.
 Binary Multiplication and Division is
performed using some simple operations
6

 Involves addition of two or more binary
numbers.
 Uses Binary addition rules
7

8
Input 1 Input 2 Sum Carry
0 0 0 No carry
0 1 1 No carry
1 0 1 No carry
1 1 0 1

9
Input 1 Input 2 Input 3 Sum Carry
1 1 1 1 1

1. Start by adding bits in unit column (rightmost
column)
2. Result of adding bits of a column is a sum with
or without a carry.
3. Write sum in result of that column.
4. If carry is present, carry is carried-over to
addition of the adjacent left column.
5. Then repeat the above steps, for each of the
columns, i.e., tens column, hundreds column
and so on.
10

Example 1. Add 10 and 01. Verify answer with the
help of decimal addition.
11
Binary Addition
1 0
+ 0 1
11
Decimal Addition
2
1+
3
112 = 310

12
Binary Addition
0 1
+ 1 1
00
Decimal Addition
1
3+
4
1
1
1002 = 410

13
Binary Addition
1 1
+ 1 1
01
Decimal Addition
3
3+
6
1
1
1102 = 610

Example 4. Add 1101 and 1111. Verify answer with
the help of decimal addition.
14
Binary Addition
1 1 0 1
+ 1 1 1 1
01
Decimal Addition
1 3
1 5+
2 8
1
1
11
01
111002 = 2810

Example 5. Add 10111, 11100 and 11. Verify answer
with the help of decimal addition.
15
Binary Addition
1 0 1 1 1
1 1 1 0 0
+ 1 1
01
Decimal Addition
2 3
2 8
3+
5 4
1
0
11
11
1
1
1101102 = 5410

 Uses Binary subtraction rules
16
Input 1 Input 2 Difference Borrow
0 0 0 No borrow
0 1 1 1
1 0 1 No borrow
1 1 0 No borrow
Binary Subtraction Rules

1. Start by subtracting bit in lower row from bit in
upper row, in unit column.
2. If bit in upper row is less than the bit in lower row,
borrow 1 from upper row of adjacent left column.
3. Result of subtracting two bits is the difference.
4. Write difference in result of that column.
5. Then repeat the above steps, for each of the
columns, i.e., tens column, hundreds column and
so on.
17

Example 1. Subtract 01 from 11. Verify answer
with the help of decimal subtraction.
18
Binary Subtraction
1 1
- 0 1
01
Decimal Subtraction
3
1-
2
102 = 210

19
Binary Subtraction
1 0
- 0 1
10
Decimal Subtraction
2
1-
1
10
012 = 110
0

20
Binary Subtraction
1 1 1 0
- 0 1 1 1
10
Decimal Subtraction
1 4
7-
7
10
10
11
10
01112 = 710
000

Example 4 Subtract 10010 from 10101. Verify
answer with the help of decimal
subtraction.
21
Binary Subtraction
1 0 1 0 1
- 1 0 0 1 0
10
Decimal Subtraction
2 1
1 8-
3
10
100
000112 = 310
0

Example 5. Subtract 101111 from 110001. Verify
answer with the help of decimal
subtraction.
22
Binary Subtraction
1 1 0 0 0 1
- 1 0 1 1 1 1
10
Decimal Subtraction
4 9
4 7-
2
101010
00 0 0
11
0000102 = 210
0

 A binary number may be positive or
negative.
 Generally, symbols “+” and “-” represent
positive and negative numbers,
respectively.
 In computer, sign of a binary number has
to be represented using 0 and 1.
23

 n-bit signed binary number consists of two parts
 Sign bit, and Magnitude.
 Left most bit is called Most Significant Bit (MSB).
 MSB is the sign bit.
 Remaining n-1 bits denote magnitude of number.
24
MSB
Sign bit Magnitude
1 bit n-1 bits

 Sign bit is 0 for a positive number and 1 for
a negative number.
 0 1100011 is a positive number. Sign bit is 0,
and,
 1 1001011 is a negative number. Sign bit is 1.
25
0 1100011
MSB
1 1001011
MSB
Positive Number Negative Number

 Data range for 8-bit signed number is:
 -128 to +127 (-27 to +27-1).
 Leftmost bit is sign bit.
 In n-bit unsigned binary number,
magnitude of number n is stored in n bits.
 Data range for 8-bit unsigned number is:
 0 to 255 (28= 256).
26

 Used in computer for simplification of
subtraction operation.
 Two types of complements
 1’s complement, and
 2’ s complement.
27

 Change bits 1 to 0 and bits 0 to 1.
 Some examples
Binary
Numbers
1’s
Complement
28
1 1 0
0 0 1
1 0 1 1
0 1 0 0 0 0 1 0 0 0 0
1 1 0 1 1 1 1

 Add 1 to the 1’s complement of the binary number.
 Some Examples:
Binary
Numbers
1’s
Complement
2’s
Complement
29
0 1 0 0 1 0 1 0 0 1 0 0 0 1
1 1 0 1 0 1 1 1 1 0 1 1 1 1
0 0 1 0 1 0 0 0 0 1 0 0 0 0

 Binary number can also have a binary
point, in addition to sign.
 Binary point used for representing
fractions, integers and integer-fraction
numbers.
 Registers are high-speed storage areas in
CPU of computer. All data is brought into a
register before it gets processed.
30

 Two ways of representing position of
binary point in a register
 Fixed Point Number Representation, and
 Floating Point Number Representation.
31

 Assumes binary point is fixed at one position.
 Represents +ve integer binary signed number as-
 Sign bit is 0. Magnitude is a positive binary number.
 Represents -ve integer binary signed number as-
 Sign bit is 1
 Magnitude is represented in any one of three ways-
 Signed Magnitude representation
 Signed 1’s complement representation
 Signed 2’s complement representation
32

 Signed Magnitude representation
 Magnitude is positive binary number itself.
 Signed 1’s complement representation
 Magnitude is 1’s complement of positive binary
number.
 Signed 2’s complement representation
 Magnitude is 2’s complement of positive binary
number.
33

+18
0 0 0 1 0 0 1 0
Sign bit is 0 Binary equivalent of +18
-18
Signed
magnitude
representation
1 0 0 1 0 0 1 0
Sign bit is 1 Binary equivalent of +18
Signed 1’s
complement
representation
1 1 1 0 1 1 0 1
Sign bit is 1 1’s complement of +18
Signed 2’s
complement
representation
1 1 1 0 1 1 1 0
Sign bit is 1 2’s complement of +18
Fixed Point Representation of
Signed Number 18
34

 Signed magnitude and signed 1’s
complement representation are rarely used
in computer arithmetic.
 Signed 2’s complement representation is
used to represent negative numbers.
35

 Represent positive number in binary form.
 For e.g., +5 is 0000 0101, +10 is 0000 1010
 Represent negative number in 2’s
complement form.
 For e.g., -5 is 111 1 1011, -10 is 1111 0110
 Add bits of the two signed binary numbers.
 Ignore any carry out from sign bit position.
36

 Negative output is automatically in 2’s
complement form.
 Get decimal equivalent of negative output
number
 Find its 2’s complement, and
 Attach a negative sign to the obtained result.
37

Example 1. Add binary +5 and +10. Verify answer
38
Binary Addition
0 0 0 0 0 1 0 1
+ 0 0 0 0 1 0 1 0
0 100 0
Decimal Addition
5
1 0+
1 5
+
+1 11
0000 11112 = +1510

Example 2. Add binary -5 and +10. Verify answer
39
Binary Addition
1 1 1 1 1 0 1 1
+ 0 0 0 0 1 0 1 0
0 000 0
Decimal Addition
5
1 0+
5
-
+0 11
111111
0000 01012 = +510

Example 3. Add binary +5 and -10. Verify answer
40
Binary Addition
0 0 0 0 0 1 0 1
+ 1 1 1 1 0 1 1 0
1 111 1
Decimal Addition
5
1 0-
5
+
-1 10
1
1111 10112 = -510

Example 4. Add binary -5 and -10. Verify answer
41
Binary Addition
1 1 1 1 1 0 1 1
+ 1 1 1 1 0 1 1 0
1 011 1
Decimal Addition
5
1 0-
1 5
-
-0 10
111 1 11 1
1111 00012 = -1510

 Changed to addition of two signed numbers.
 Sign of second number is changed before
performing the addition operation.
42

(+A) – (+B) = (+A) + (-B)
(+A) – (-B) = (+A) + (+B)
(-A) – (-B) = (-A) + (+B)
(-A) – (+B) = (-A) + (-B)
43

 Uses two registers
 1st register stores number without binary point.
 2nd register stores a number that indicates position
of binary point in first register.
 Consists of two parts:
 Mantissa, and Exponent.
 Mantissa is a signed fixed point number.
 Exponent shows position of binary point in
mantissa.
44

45
Represent binary number +11001.11 with
8-bit mantissa and 6-bit exponent
Floating point number is: Mantissa x 2exponent
+ (.1100111) x 2+5.
Mantissa Exponent
number is positive binary equivalent of +5
0 11 0 0 111 0 0 0 1 0 1
11 0 0 1 . 11 . 11 0 0 111 x 2 5

 Data - alphabetic, numeric, alphanumeric,
sound, video.
 Data represented as combination of bits in
computer.
 Bits are grouped in a fixed size.
 Code is made by combining bits of definite
size.
46

 Represents symbols in a standard code.
 Combination of bits represents a unique
symbol.
 Standard code enables programmers to
use same combination of bits to represent
a symbol in data.
47

 Commonly used binary coding schemes:
 ASCII,
 EBCDIC, and
 Unicode
48

 EBCDIC stands for Extended Binary Coded
Decimal Interchange Code
 8-bit code. 4 bits for zone; 4 bits for digit.
 Allows 28 = 256 combinations.
 Represents 256 unique symbols.
 Used mainly in mainframe computers.
49

 ASCII stands for American Standard Code
for Information Interchange
 Two types of ASCII codes
 ASCII-7 and
 ASCII-8.
50

 Standard ASCII code.
 7-bit code. 3 bits for zone; 4 bits for digits.
 ASCII-7 modified by IBM to ASCII-8.
51

 Extended version of ASCII-7.
 8-bit code. 4 bits for zone; 4 bits for digit.
 Widely used to represent data in computer.
52

 ASCII-8 code represents 256 symbols.
 0-31 for control characters.
 Non-printable. Carriage Return (CR), Bell
(BEL).
 48-57 for numeric 0-9.
 65-90 for uppercase letters A-Z.
 97-122 for lowercase letters a-z.
 128-255 are extended ASCII codes.
53

 Universal character encoding standard
 Represents text, symbols, characters in multi-
lingual environments.
 Uniquely represent a symbol in languages like
Chinese, Japanese etc.
 Represents mathematical and scientific
symbols.
 32 bit code.
 Allows 232 = approx. 4 billion combinations.
54

 Compatible with ASCII-8 codes.
 Unicode’s first 256 codes identical to ASCII-
8 codes.
 Implemented by character encodings.
 UTF-8 : A character encoding
 Most commonly used encoding scheme.
 Uses 8-32 bits per code.
55

Unicode character encoding
in MS-Word 2007
56

Binaty Arithmetic and Binary coding schemes

Binaty Arithmetic and Binary coding schemes

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