Number system and conversions Digitalelectronics.ppt

Types Of Numbers
 Natural Numbers
 The number 0 and any number obtained by
repeatedly adding a count of 1 to 0
 Negative Numbers
 A value less than 0
 Integer
 A natural number, the negative of a natural number,
and 0.
 So an integer number system is a system for
‘counting’ things in a simple systematic way

Exponent Review
 An exponent (power) tells you how many
times to multiply the base by itself:
 21
= 2
 22
= 2 x 2 =4
 23
= 2 x 2 x 2 = 8
 20
= 1 (ANY number raised to power 0 is 1)
 1 / x2
= x -2

Decimal Numbering
System
 How is a positive integer represented in decimal?
 Let’s analyze the decimal number 375:
375 = (3 x 100) + (7 x 10) + (5 x 1)
= (3 x 102
) + (7 x 101
) + (5 x 100
)
3
3 7
7 5
5
10
100
0
10
101
1
10
102
2
Position weights
Position weights
Number digits
digits
5 x10
5 x100
0
= 5
= 5
7 x10
7 x101
1
= 70
= 70
3 x 10
3 x 102
2
= 300
= 300
+
+
375
375

Decimal System Principles
 A decimal number is a sequence of digits
 Decimal digits must be in the set: {0, 1,
2, 3, 4, 5, 6, 7, 8, 9} (Base 10)
 Each digit contributes to the value the number
represents
 The value contributed by a digit equals the product of
the digit times the weight of the position of the digit in
the number

Decimal System Principles
 Position weights are powers of 10
 The weight of the rightmost (least significant digit)
is 100
(i.e.1)

The weight of any position is 10x
, where x is the
number of positions to the right of the least
significant digit
10
100
0
10
101
1
10
102
2
Position weights
Position weights
digits
digits
10
103
3
10
104
4
5
7
3

Bits
 In a computer, information is stored using digital
signals that translate to binary numbers
 A single binary digit (0 or 1) is called a bit
 A single bit can represent two possible states,
on (1) or off (0)
 Combinations of bits are used to store values

Data Representation
 Data representation means encoding data into bits
 Typically, multiple bits are used to represent the
‘code’ of each value being represented
 Values being represented may be characters, numbers,
images, audio signals, and video signals.
 Although a different scheme is used to encode each type
of data, in the end the code is always a string of zeros
and ones

Decimal to Binary
 So in a computer, the only possible digits we can use to
encode data are {0,1}

The numbering system that uses this set of digits is
the base 2 system (also called the Binary Numbering
System)
 We can apply all the principles of the base 10 system to
the base 2 system
2
20
0
2
21
1
2
22
2
Position weights
Position weights
digits
digits
2
23
3
2
24
4
1
1
0
1

Binary Numbering System
 How is a positive integer represented in binary?
 Let’s analyze the binary number 110:
110 = (1 x 22
) + (1 x 21
) + (0 x 20
)
= (1 x 4) + (1 x 2) + (0 x 1)
1
1 1
1 0
0
2
20
0
2
21
1
2
22
2
Position weights
Position weights
Number digits
digits
0 x2
0 x20
0
= 0
= 0
1 x2
1 x21
1
= 2
= 2
1 x 2
1 x 22
2
= 4
= 4
+
+
6
6
 So a count of SIX is represented in binary as 110

Binary to Decimal
Conversion
 To convert a base 2 (binary) number to
base 10 (decimal):
 Add all the values (positional weights)
where a one digit occurs
 Positions where a zero digit occurs do
NOT add to the value, and can be
ignored

Binary to Decimal
Conversion
Example: Convert binary 100101 to decimal
(written 1 0 0 1 0 12 ) =
1*20
+
0*21
+
1*22
+
0*23
+
0*24
+
1*25
3710
1 +
4 +

Binary to Decimal
Conversion
positional powers of 2: 24
23
22
21
20
decimal positional value: 16 8 4 2 1

Example #2: 101112
binary number: 1 0 1 1 1
16 + 4 + 2 + 1 = 2310

Binary to Decimal
Conversion
24
23
22
21
20
decimal positional value: 32 16 8 4 2 1
binary number: 1 1 0 0 1 0
32 + 16 + 2 = 5010

Example #3: 1100102

Decimal to Binary
Conversion
The Division Method:
1) Start with your number (call it N) in base 10
2) Divide N by 2 and record the remainder
3) If (quotient = 0) then stop
else make the quotient your new N, and go back to step 2
The remainders comprise your answer, starting with the last
remainder as your first (leftmost) digit.
In other words, divide the decimal number by 2 until you reach
zero, and then collect the remainders in reverse.

Decimal to Binary
Conversion
Example 1: 2210
=
2 ) 22 Rem:
2 ) 11 0
2 ) 5 1
2 ) 2 1
2 ) 1 0
0 1
101102
Using the Division Method:
Divide decimal number by 2 until you reach zero, and then
collect the remainders in reverse.

Decimal to Binary
Conversion
Using the Division Method
Example 2: 5610
=
2 ) 56 Rem:
2 ) 28 0
2 ) 14 0
2 ) 7 0
2 ) 3 1
2 ) 1 1
0 1
1110002

Decimal to Binary
Conversion
The Subtraction Method:
 Subtract out largest power of 2 possible
(without going below zero), repeating until you
reach 0.
 Place a 1 in each position where you
COULD subtract the value
 Place a 0 in each position that you could
NOT subtract out the value without going
below zero.

0 1
- 1
- 4
1
Decimal to Binary
Conversion
Example 1: 2110
21 26
25
24
23
22
21
20
64 32 16 8 4 2 1
Answer: 2110
= 101012
- 16
5 1 0 1

Decimal to Binary
Conversion
Example 2: 5610
56 26
| 25
24
23
22
21
20
- 32 64| 32 16 8 4 2 1
24 | 1 1 1 0 0 0
- 16
8
- 8 Answer: 5610
= 1110002
0

Octal Numbering System
 Base: 8
 Digits: 0, 1, 2, 3, 4, 5, 6, 7

Octal number: 3578
= (3 x 82
) + (5 x 81
) + (7 x 80
)
 To convert to base 10, beginning with the
rightmost digit, multiply each nth digit by 8(n-
1)
, and add all of the results together.

Octal to Decimal Conversion

Example 1: 3578
81
80
decimal positional value: 64 8 1
Octal number: 3 5 7
(3 x 64) + (5 x 8) + (7 x 1)
= 192 + 40 + 7 = 23910

Octal to Decimal Conversion

Example 2: 12468
82
81
80
decimal positional value: 512 64 8 1
Octal number: 1 2 4 6
(1 x 512) + (2 x 64) + (4 x 8) + (6 x 1)
= 512 + 128 + 32 + 6 = 67810

Decimal to Octal Conversion
In other words, divide the decimal number by 8 until you reach
zero, and then collect the remainders in reverse.

Using the Division Method:
Example 1: 21410
=
8 ) 214 Rem:
8 ) 26 6
8 ) 3 2
0 3
3268

Example 2: 433010
=
8 ) 4330 Rem:
8 ) 541 2
8 ) 67 5
8 ) 8 3
8 ) 1 0
0 1
103528

 Subtract out multiples of the largest power
of 8 possible (without going below zero) each
time until you reach 0.
 Place the multiple value in each position
where you COULD subtract the value.
below zero.

Example 1: 31510
82
81
80
64 8 1
315
- 256 (4 x 64)
59
- 56 (7 x 8)
3
- 3 (3 x 1)
0
Answer: 31510
= 4738
7
4 3

Example 2: 201810
2018 84
83
82
81
80
-1536 (3 x 512) 4096 512 64 8 1
482 3 7 4 2
- 448 (7 x 64)
34
- 32 (4 x 8)
2
- 2 (2 x 1) Answer: 201810
= 37428
0

Hexadecimal (Hex)
Numbering System
 Base: 16
 Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
 Hexadecimal number: 1F416
= (1 x 162
) + (F x 161
) + (4 x 160
)

Hexadecimal (Hex)
Extra Digits
Decimal Value Hexadecimal Digit
10 A
11 B
12 C
13 D
14 E
15 F

Hex to Decimal Conversion
 To convert to base 10:
 Begin with the rightmost digit
 Multiply each nth digit by 16(n-1)
 Add all of the results together


Example 1: 1F416
162
161
160
Hexadecimal number: 1 F 4
(1 x 256) + (F x 16) + (4 x 1)
= (1 x 256) + (15 x 16) + (4 x 1)
= 256 + 240 + 4 = 50010


Example 2: 25AC16
162
161
160
Hexadecimal number: 2 5 A C
(2 x 4096) + (5 x 256) + (A x 16) + (C x 1)
= (2 x 4096) + (5 x 256) + (10 x 16) + (12 x 1)
= 8192 + 1280 + 160 + 12 = 964410

Decimal to Hex Conversion
In other words, divide the decimal number by 16 until you
reach zero, and then collect the remainders in reverse.

Using The Division Method:
Example 1: 12610
=
16) 126 Rem:
16) 7 14=E
0 7
7E16

Example 2: 60310
=
16) 603 Rem:
16) 37 11=B
16) 2 5
0 2
25B16

 Subtract out multiples of the largest power
of 16 possible (without going below zero)
each time until you reach 0.
 Place the multiple value in each position
where you COULD to subtract the value.
below zero.

Example 1: 81010
162
161
160
256 16 1
810
- 768 (3 x 256)
42
- 32 (2 x 16)
10
- 10 (10 x 1)
Answer: 81010
= 32A16
2
3 A

Example 2: 15610
162
161
160
256 16 1
156
- 144 (9 x 16)
12
- 12 (12 x 1)
0
Answer: 15610
= 9C16
9 C

Binary to Octal Conversion
The maximum value represented in 3 bit is:
23
– 1 = 7
So using 3 bits we can represent values from
0 to 7
which are the digits of the Octal
numbering system.
Thus, three binary digits can be converted to
one octal digit.

Three-bit Group Decimal Digit Octal Digit
000 0 0
001 1 1
010 2 2
011 3 3
100 4 4
101 5 5
110 6 6
111 7 7

Octal to Binary Conversion
111 100 010
7428 = 1111000102
Ex : Convert 7428 to binary
Convert each octal digit to 3 bits:
7 = 111
4 = 100
2 = 010

Ex : Convert 101001102 to octal
Starting at the right end, split into groups of 3:
10 100 110 
110 = 6
100 = 4
010 = 2 (pad empty digits with 0)
101001102 = 2468

Binary to Hex Conversion
The maximum value represented in 4 bit is:
24
– 1 = 15
So using 4 bits we can represent values from
0 to 15
which are the digits of the Hexadecimal
numbering system.
Thus, four binary digits can be converted to one
hexadecimal digit.

Binary to Hex
Conversion
Four-bit Group Decimal Digit Hexadecimal Digit
0000 0 0
0001 1 1
0010 2 2
0011 3 3
0100 4 4
0101 5 5
0110 6 6
0111 7 7
1000 8 8
1001 9 9
1010 10 A
1011 11 B
1100 12 C
1101 13 D
1110 14 E
1111 15 F

Binary to Hex Conversion
Ex : Convert 1101001102 to hex
Starting at the right end, split into groups of 4:
1 1010 0110 
0110 = 6
1010 = A
0001 = 1 (pad empty digits with 0)
1101001102 = 1A616

Hex to Binary Conversion
Ex : Convert 3D916 to binary
Convert each hex digit to 4 bits:
3 = 0011
D = 1101
9 = 1001
0011 1101 1001 
3D916 = 11110110012 (can remove leading zeros)

Conversion between Binary
and Hex - Try It Yourself

Convert the following numbers:
 10101111012 to Hex
 82F16 to Binary
 (Answers on NEXT slide)

Answers
 10101111012  10 1011 1101
= 2BD16
 82F16 = 0100 0010 1111 
100001011112

Octal to Hex Conversion
 To convert between the Octal and
Hexadecimal numbering systems
 Convert from one system to binary first
 Then convert from binary to the new
numbering system

Hex to Octal Conversion
Ex : Convert E8A16 to octal
First convert the hex to binary:
First convert the hex to binary:
1110 1000 10102
111 010 001 010 and re-group by 3
bits
(starting on the
right)
Then convert the binary to octal:
7 2 1 2
So E8A16 = 72128

Octal to Hex Conversion
Ex : Convert 7528 to hex
First convert the octal to binary:
111 101 0102
re-group by 4 bits
0001 1110 1010 (add leading zeros)
Then convert the binary to hex:
1 E A
So 7528 = 1EA16

Number system and conversions Digitalelectronics.ppt

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