Chapter 2.1 introduction to number system

Chapter 2: Number System
2.1 Binary, Octal & Hexadecimal Number Systems and their conversions
2.1.1 Representation of Signed Numbers, Floating Point Number
2.1.2 Binary Arithmetic
2.2 Representation of BCD, ASCII, Excess 3, Gray Code, Error Detection
and Correcting Codes.
1

 A number system defines how a number can
be represented using distinct symbols.
 A number can be represented differently in
different systems.
For example, the two numbers (2A)16 and (52)8
both refer to the same quantity, (42)10, but their
representations are different.
3

Number system can be categorized as
1. Decimal number system
2. Binary number system
3. Octal number system
4. Hexadecimal Number System
4

o Each number system is associated with a base or radix
o The decimal number system is said to be of base or
radix10
o A number in base r contains r digits 0,1,2,...,r-1
o Decimal (Base 10): 0,1,2,3,4,5,6,7,8,9
5

 The word decimal is derived from the Latin root
decem(ten). In this system the base b = 10 and
we use ten symbols.
S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
6

 The word binary is derived from the Latin root
bini (or two by two).
In this system the base b = 2 and we use
only two symbols,
S = {0, 1}
The symbols in this system are often referred
to as binary digits or bits.
7

 The word hexadecimal is derived from the Greek root
hex (six) and the Latin root decem (ten).
In this system the base b = 16 and we use sixteen
symbols to represent a number.
 The set of symbols is
S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E,F}
 The symbols A, B, C, D, E, F are equivalent to
10, 11, 12, 13, 14, and 15 respectively.
 The symbols in this system are often referred to as
hexadecimal digits.
8

The word octal is derived from the Latin root
octo (eight).
In this system the base b = 8 and we use
eight symbols to represent a number.
The set of symbols is:
S = {0, 1, 2, 3, 4, 5, 6, 7}
9

Common Number Systems
10
System Base Symbols Used by
Humans?
Used in
Computers?
Decimal 10 0, 1, … 9 Yes No
Binary 2 0, 1 No Yes
Octal 8 0, 1, … 7 No No
Hexa-
decimal
16 0, 1, … 9,
A, B, … F
No No

Quantities / Counting
11
Decimal Binary Octal
Hexa-
decimal
0 0 0 0
1 1 1 1
2 10 2 2
3 11 3 3
4 100 4 4
5 101 5 5
6 110 6 6
7 111 7 7
Decimal Binary Octal
Hexa-
decimal
8 1000 10 8
9 1001 11 9
10 1010 12 A
11 1011 13 B
12 1100 14 C
13 1101 15 D
14 1110 16 E
15 1111 17 F

Conversion among Bases
• Possibilities
• Example
12
Hexadecimal
Decimal Octal
Binary
2510 = 110012 = 318 = 1916 Base

Decimal to Binary
• Technique
– Divide by two, keep track of the remainder
– First remainder is bit 0 (LSB, least-significant bit)
– Second remainder is bit 1 and so on
13
Decimal Binary

Example (Decimal to Binary)
14
12510 = ?2 2 125 1
2 62 0
2 31 1
2 15 1
2 7 1
2 3 1
2 1 1
0
12510 = 11111012

Example (Decimal to Binary)
15
0.687510 = ?2
0.6875 x 2 = 1.3750 1 0.3750
+
0.3750 x 2 = 0.7500 0 0.7500
+
0.7500 x 2 = 1.5000 1 0.5000
+
0.5000 x 2 = 1.0000 1 0.0000
+
0.687510 = 0.10112
integer fraction

Binary to Decimal
• Technique
– Multiply each bit by 2n, where n is the “weight” of the
bit
– The weight is the position of the bit, starting from 0
on the right
– Add the results
16
Binary Decimal

Example (Binary to Decimal)
17
1 0 1 0 1 1
1 x 20
1 x 21
0 x 22
1 x 23
0 x 24
1 x 25 +
+
+
+
+
1
2
0
1010112 =
0
32 +
+
+
+
+
4310
4310
8

Example (Binary to Decimal)
18
1 1 . 1 1
1 x 20
1 x 21 +
1
2
11.112 =
+
3.7510
3.7510
1 x 2-2
1 x 2-1 +
0.25
0.5 +
+
+

Decimal to Octal
• Technique
– Divide by eight
– Keep track of the remainder
19
Decimal Octal

Example (Decimal to Octal)
20
12510 = ?8 8 125 5
8 15 7
8 1 1
0
12510 = 1758

Example (Decimal to Octal)
21
0.687510 = ?8
0.6875 x 8 = 5.5000 5 0.5000
+
0.5000 x 8 = 4.0000 4 0.0000
+
0.687510 = 0.548
integer fraction

Octal to Decimal
• Technique
– Multiply each bit by 8n, where n is the “weight” of the
bit
on the right
– Add the results
22
Octal Decimal

Example (Octal to Decimal)
23
7 2 4
4 x 80
2 x 81
7 x 82 +
+
7248 =
46810
46810
4
16
448 +
+

Example (Octal to Decimal)
24
4 3 . 2 5
3 x 80
4 x 81 +
3
32
43.258 =
+
35.328110
35.328110
5 x 8-2
2 x 8-1 +
0.0781
0.25 +
+
+

Decimal to Hexa-Decimal
• Technique
– Divide by 16
– Keep track of the remainder
25
Decimal Hexa-Decimal

Example (Decimal to HexaDecimal)
26
123410 = ?16 16 1234 2
123410 = 4D216
16 77 13=D
16 4 4
0

Hexa-Decimal to Decimal
• Technique
– Multiply each bit by 16n, where n is the “weight” of
the bit
on the right
– Add the results
27
Hexa-Decimal Decimal

Example (HexaDecimal to Decimal)
28
A B C
C x 160
B x 161
A x 162 +
+
ABC16 =
274810
274810
12 x 160
11 x 161
10 x 162
+
+
12
176
2560 +
+

Octal to Binary
• Technique
– Convert each octal digit to a 3-bit equivalent binary
representation
29
Octal Binary

Octal - Binary Table
Octal Binary
0 000
1 001
2 010
3 011
4 100
5 101
6 110
7 111
30

Example (Octal to Binary)
31
7058 = ?2
7 0 5
101
000
111
7058 = 1110001012

Binary to Octal
• Technique
– Group bits in threes, starting on right
– Convert to octal digits
32
Binary Octal

Example (Binary to Octal)
33
10110101112 = ?8
1
011
001
10110101112 = 13278
111
010
3 2 7

Hexa-Decimal to Binary
• Technique
– Convert each hexadecimal digit to a 4-bit equivalent
binary representation
34
Hexa-Decimal Binary

Hexa-Decimal to Binary
Hexa-
Decimal
Binary Hexa-
Decimal
Binary
0 0000 8 1000
1 0001 9 1001
2 0010 A 1010
3 0011 B 1011
4 0100 C 1100
5 0101 D 1101
6 0110 E 1110
7 0111 F 1111
35

Example (Hexa-Decimal to Binary)
36
10AF16 = ?2
1 0 A F
1111
1010
0000
10AF16 = 10000101011112
0001

Binary to Hexa-Decimal
• Technique
– Group bits in fours, starting on right
– Convert to hexadecimal digits
37
Binary Hexa-Decimal

Example (Binary to Hexa-Decimal)
38
10110101112 = ?16
0010
10110101112 = 2D716
0111
1101
2 D 7

Octal to Hexa-Decimal
• Technique
– Convert Octal to Binary
– Regroup bits in fours from right
– Convert Binary to Hexa-Decimal
39
Octal Hexa-Decimal

Example (Octal to Hexa-Decimal)
40
10768 = ?16
1 0 7 6
110
111
000
10768 =23E16
001
1110
0011
0010
E
3
2

Hexa-Decimal to Octal
• Technique
– Convert Hexa-Decimal to Binary
– Regroup bits in three from right
– Convert Binary to Octal
41
Hexa-Decimal Octal

Example (Hexa-Decimal to Octal)
42
1F0C16 = ?8
1 F 0 C
1100
0000
1111
1F0C16 = 174148
0001
100
001
100
4
1
4
111
7
001
1
000
0

Positional Number System
• A positional number system is also known as
weighted number system.
• As the name implies there will be a weight
associated with each digit.
• Few examples of positional number system
are decimal number system, Binary number
system, octal number system ,etc.
43

Non-Positional Number System
• Non-positional number system is also known
as non-weighted number system.
• Digit value is independent of its position. Non-
positional number system is used for shift
position encodes and error detecting purpose.
• Few examples of non-weighted number
system are gray code, roman code, excess-3
code, etc.
44

Signed numbers
• Digital systems such as computer must be able to
handle both positive and negative numbers. A signed
binary number consists of both signed and magnitude
information.
• Eg: Signed number : +7
7 = Binary Number 111
+7 = 0 111
• Sign indicates whether a number is positive or negative
and magnitude is the value of the number. Left most
bit is the sign bit.
• ‘0’ sign bit indicates a positive number and ‘1’ sign bit
indicates a negative number.
45

Signed numbers
There are 3 forms in which sign integer numbers
can be represented in binary.
1. Signed magnitude
2. 1’s Complement
3. 2’s Complement
46

Signed Magnitude form
• When a signed binary number is represented in a sign
magnitude form, left most bit is the signed bit and
remaining bits are magnitude bit.
• The magnitude bits are in a true binary for both positive
and negative numbers.
• Eg : Represent decimal number +25 on 8-bit signed
binary number.
25  (11001)2
+25  0 0011001
+ Sign Magnitude

Example
Q. Similarly -75 is represented (8-bit) as ???
Ans: 1 1001011
75 = 1001011
-75 = 1 1001011
48

1’s Complement
49
1 1 1 1
1 1 0 1
0 0 1 0
-
(1’s complement of 1101)
1 1 1 .
1 0 1 .
0 1 0 .
-
(1’s complement of 101.01)
1 1
0 1
1 0
 1’s complement of a binary number is obtained by subtracting
each digit of that binary number from 1.
 Example

1’s Complement Form
• Positive Numbers are simply represented as it is.
• If the number is negative, First represent the number
with positive sign and then take 1’s complement of that
number.
• Ex: Using 8-bit, find 1’s complement of decimal number
-25.
Step-1 : +25 = 00011001
Step-2 : 1’s complement : 11100110
Q. Find 1’s complement of -75 using 8-bit
Step-1 : +75 = 0 1001011
Step-2 : 1’s complement : 10110100

2’s Complement
• 2’s complement of a binary number is obtained by
adding 1 to its 1’s complement.
51
1 1 1 1
1 1 0 0
0 0 1 1
-
0 1 0 .
-
1 1 1 . 1 1
1 0 1 . 0 1
1 0
1
0 1 0 0
1
+ +
0 1 0 1 1
.
 Example

2’s Complement form
• Positive Numbers are simply represented as it is.
• If the number is negative, First represent the number
with positive sign and then take 2’s complement of that
number.
E.g. : Find 2’s complement of -25 using 8-bit
+25 = 00011001
1’s complement = 11100110
+ 1
11100111
Q. Find 2’s complement of -75
Ans: (10110101)

Representation of negative number in 2’s
complement form
• Express -65.5 in 12 bit 2’s complement form.
53
2 65 1
2 32 0
2 16 0
2 8 0
2 4 0
2 2 0
2 1 1
0
0.5 x 2 = 1.0
65.510 = 01000001.10002
So, result in 12-bit binary is as follows:
For negative number, we have to
convert this into 2’s complement form
-65.510 =10111110.10002

Assignment
Questions:
1. Convert (1101010)2 to octal, decimal and hexadecimal.
2. Convert (1011.01)2 to octal, decimal and hexadecimal.
3. Convert (35.43)10 to binary, octal and hexadecimal.
4. Convert (635.74)8 to binary, decimal and hexadecimal.
5. Convert (D6A)16 to binary, octal & decimal.
6. Express the decimal number -39 as an 8-bit number in
sign magnitude, 1’s complement and 2’s complement
form.

9’s Complement
• 9’s complement of a decimal number is obtained by
subtracting each digit of that decimal number from 9.
55
9 9 9 9
3 4 6 5
6 5 3 4
-
9 9 9 .
7 8 2 .
2 1 7 .
-
9 9
5 4
4 5
 Example

10’s Complement
• 10’s complement of a decimal number is obtained by
adding 1 to its 9’s complement.
56
9 9 9 9
3 4 6 5
6 5 3 4
-
2 1 7 .
-
9 9 9 . 9 9
7 8 2 . 5 4
4 5
1
6 5 3 5
1
+ +
2 1 7 4 6
.
 Example

Subtraction using 9’s complement
• Obtain 9’s complement of subtrahend
• Add the result to minuend and call it
intermediate result
• If carry is generated then answer is positive
and add the carry to Least Significant Digit
(LSD)
• If there is no carry then answer is negative and
take 9’s complement of intermediate result
and place negative sign to the result.
57

Example
58
7 4 5 .
4 3 6 .
-
8 1
6 2
1) 745.81 – 436.62
7 4 5 . 8 1
5 6 3 .
+ 3 7
3 0 9 .
1 1 8
+ 1
3 0 9 . 1 9
3 0 9 . 1 9
9’s complement

Example
59
7 4 5 .
4 3 6 .
- 8 1
6 2
2) 436.62 - 745.81
2 5 4 .
+ 1 8
6 9 0 . 8 0
3 0 9 . 1 9
3 0 9 . 1 9
-
4 3 6 . 6 2
-
9’s complement
9’s complement
As carry is not generated, so take 9’s
complement of the intermediate
result and add ‘ – ‘ sign to the result

Subtraction using 10’s complement
• Add the result to minuend
• If carry is generated then ignore it and result
itself is answer
take 10’s complement of result and place
negative sign to the result.
60

Example
61
7 4 5 .
4 3 6 .
-
8 1
6 2
1) 745.81 – 436.62
5 6 3 .
+
7 4 5 . 8 1
3 8
3 0 9 .
1 1 9
3 0 9 . 1 9
Ignore the carry
10’s complement

Example
62
7 4 5 .
4 3 6 .
- 8 1
6 2
2) 436.62 - 745.81
2 5 4 .
+ 1 9
6 9 0 . 8 1
3 0 9 . 1 9
3 0 9 . 1 9
-
4 3 6 . 6 2
-
10’s complement
10’s complement
As carry is not generated, so take
10’s complement of the intermediate

Binary Addition
• Rules for binary addition
63
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10 i.e.
0 with a carry
of 1
1 1 0 1
0 1 1 1
0 1 0 1
1 1 1 1
1
+
. 1 0 1
. 0 1 1
. 0 0 0
1 1

Binary Subtraction
• Rules for binary subtraction
64
0 – 0 = 0
1 – 1 = 0
1 – 0 = 1
0 – 1 = 1, with
a borrow 1
1 0 1 0
0 1 1 1
0 0 1 0
-
. 0 1 0
. 1 1 1
. 0 1 1
1
0
1
1
1
0
1
1
0

Binary Multiplication
• Four basic Rules:
A B Output
0 0 0
0 1 0
1 0 0
1 1 1

Eg: 1 1 1
x 1 0 1
1 1 1
0 0 0 X
+ 1 1 1 X X
1 0 0 0 1 1

Eg: 1 1 .0 1
x 1 0 .1 1
1 1 0 1
1 1 0 1 X
0 0 0 0 X X
+ 1 1 0 1 X X X
1 0 0 0. 1 1 1 1

Binary Multiplication
68
1 0 1 1 1
1 0 0 1 1
x
1
1
1
0
1
1
1
1
0
1
0
0
0
0
0
0
0
0
0
0
1
1
1
0
1
1
0
1
0
1
1
0
1
1

• Solve the following binary multiplication
2. (101.01)2 and (100.1)2
Ans: 101111.01

Binary Division
A B Output
0 0 0
0 1 0
1 1 1
1 0 Infinity

Binary Division
74
1 0 1 1 0 1
110 0
0 0 0
1 0 11
1 1 0
1 0 1 0
1 1 0
1 0 0 1
1 1 0
1 1
111.1
0
1 1 0
0 0 0

• Solve the following division
1.(11001) by (101)
Ans:101
2. (111)2 by (11)2
Ans: 10 remainder 1
3. (110000) by (100)
Ans:

Subtraction using 1’s Complement
• Add the result to minuend and call it
intermediate result
• If carry is generated then answer is positive
and add the carry to Least Significant Digit
(LSD)
take 1’s complement of intermediate result
and place negative sign to the result.
76

Example
77
68.
27.
-
75
50
1) 68.75 – 27.50
+ 41.25
1’s complement
000 . 1 1
100
01 0 0
+ 100 .
100
11 0 11 1
+1
00 1.
1 00
01 0
0
1 11
01 0 . 0 1
00 1 01 0 0

Example
78
2) 43.25 - 89.75
43.
89.
-
25
75
- 46.50
1’s complement
101 . 0 1
011
00 0 0
+ 100 .
110
10 0 01 1
00 1.
0 01
10 1
1 11
11 0.
1 10
01 0
0 00
1’s complement

Subtraction using 2’s Complement
• Add the result to minuend
• If carry is generated then ignore it and result
itself is answer
take 2’s complement of result and place
negative sign to the result.
79

Example
80
68.
27.
-
75
50
1) 68.75 – 27.50
+ 41.25
2’s complement
000 . 1 1
100
01 0 0
+ 100 .
100
11 1 00 0
00 1.
1 01
01 0
0
1 00
00 1.
1 01
01 0
0 00
Ignore Carry bit

Example
81
2) 43.25 - 89.75
43.
89.
-
25
75
- 46.50
2’s complement
101 . 0 1
011
00 0 0
+ 100 .
110
10 0 10 0
00 1.
0 10
10 1
1 00
11 0.
1 10
01 0
0 00
2’s complement

Signed Binary Numbers
• To represent negative integers, we need a notation for negative values.
• It is customary to represent the sign with a bit placed in the leftmost position
of the number since binary digits.
• The convention is to make the sign bit 0 for positive and 1 for negative.
• Different methods of representations (example):
• Table 1.3 lists all possible four-bit signed binary numbers in the three
representations.

• Arithmetic addition
– The addition of two numbers in the signed-magnitude system follows the rules of
ordinary arithmetic. If the signs are the same, we add the two magnitudes and give the
sum the common sign. If the signs are different, we subtract the smaller magnitude
from the larger and give the difference the sign if the larger magnitude.
– The addition of two signed binary numbers with negative numbers represented in
signed-2's-complement form is obtained from the addition of the two numbers,
including their sign bits.
– A carry out of the sign-bit position is discarded.
• Example:

• Arithmetic Subtraction
– In 2’s-complement form:
• Example:
1. Take the 2’s complement of the subtrahend (including the sign bit)
and add it to the minuend (including sign bit).
2. A carry out of sign-bit position is discarded.
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
A B A B
A B A B
      
      
( 6)  ( 13) (11111010  11110011)
(11111010 + 00001101)
00000111 (+ 7)

BINARY REPRESENTATION OF FLOATING POINT
NUMBERS
Converting decimal fractions into binary
representation.
Consider a decimal fraction of the form: 0.d1d2...dn
We want to convert this to a binary fraction of the
form:
0.b1b2...bn (using binary digits instead of decimal
digits)

Algorithm for conversion
Let X be a decimal fraction: 0.d1d2..dn
i = 1
Repeat until X = 0 or i = required no. of binary fractional
digits {
Y = X * 2
X = fractional part of Y
Bi = integer part of Y
i = i + 1
}

EXAMPLE 1
Convert 0.75 to binary
X = 0.75 (initial value)
X* 2 = 1.50. Set b1 = 1, X = 0.5
X* 2 = 1.0. Set b2 = 1, X = 0.0
The binary representation for 0.75 is thus
0.b1b2 = 0.11b

Let's consider what that means...
In the binary representation
0.b1b2...bm
b1 represents 2-1 (i.e., 1/2)
b2 represents 2-2 (i.e., 1/4)
...
bm represents 2-m (1/(2m))
So, 0.11 binary represents
2-1 + 2-2 = 1/2 + 1/4 = 3/4 = 0.75

EXAMPLE 2
Convert the decimal value 4.9 into binary
Part 1: convert the integer part into binary: 4 = 100 b

Part 2.
Convert the fractional part into binary using multiplication
by 2:
X = .9*2 = 1.8. Set b1 = 1, X = 0.8
X*2 = 1.6. Set b2 = 1, X = 0.6
X*2 = 1.2. Set b3 = 1, X = 0.2
X*2 = 0.4. Set b4 = 0, X = 0.4
X*2 = 0.8. Set b5 = 0, X = 0.8,
which repeats from the second line above.

Since X is now repeating the value 0.8,
we know the representation will repeat.
The binary representation of 4.9 is thus:
100.1110011001100...

COMPUTER REPRESENTATION OF FLOATING
POINT NUMBERS
In the CPU, a 32-bit floating point number is
represented using IEEE standard format as follows:
SIGN | EXPONENT | MANTISSA
where SIGN is one bit, the EXPONENT is 8 bits, and
the MANTISSA is 23 bits.

• The mantissa represents the leading significant bits in
the number.
• The exponent is used to adjust the position of the
binary point (as opposed to a "decimal" point)
• The mantissa is said to be normalized when it is
expressed as a value between 1 and 2. I.e., the mantissa
would be in the form 1.xxxx.

• The leading integer of the binary representation is
not stored. Since it is always a 1, it can be easily
restored.
• The "SIGN" bit is used as a sign bit and indicates
whether the value represented is positive or negative
(0 for positive, 1 for negative)
• If a number is smaller than 1, normalizing the
mantissa will produce a negative exponent.
• But 127 is added to all exponents in the floating
point representation, allowing all exponents to be
represented by a positive number.

Example 1. Represent the decimal value 2.5 in 32-
bit floating point format.
2.5 = 10.1
In normalized form, this is: 1.01 * 21
The mantissa: M = 01000000000000000000000
(23 bits without the leading 1)
The exponent: E = 1 + 127 = 128 = 10000000
The sign: S = 0 (the value stored is positive)
So, 2.5 = 0 1000000001 000000000000000000000

Example 2: Represent the number -0.00010011 in
floating point form.
0.00010011 = 1.0011 * 2-4
Mantissa: M = 00110000000000000000000 (23 bits
with the integral 1 not represented)
Exponent: E = -4 + 127 = 01111011
S = 1 (as the number is negative)
Result: 1 01111011 00110000000000000000000

Exercise 1: represent -0.75 in floating point format.
Exercise 2: represent 4.9 in floating point format.

Chapter 2.1 introduction to number system

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Chapter 2.1 introduction to number system