Lecture-2(2): Number System & Conversion

Lecture 02
Number Systems &
Conversion
Mr. Mubashir Ali
Lecturer(Dept. of Computer
Science)
dr.mubashirali1@gmail.com

TOPICS
• Decimal Number Systems
• Binary number Systems
• Octal
• Hexadecimal
• Number conversion

Decimal Number System
• The word decimal is a derivative of decem,
which is the Latin word for ten.
• Numbers today are almost universally written
in a form of notation known as positional
number representation.
• A number is represented as a string of digits.

Decimal Number Systems
• In a decimal number there are ten such digits
that may be used, ranging in value from zero to
nine
• Decimal numbers are positional numbers that
have a base or radix of ten.
• Why ten ?

• Exactly ten distinct numerals in order to
represent all possible values for each position in
the number, and hence to enable us to represent
all possible integer numbers in decimal notation.
• 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 values
• For example, a number like "0123456789" has
ten positions and each position can contain the
digits 0-9.

• Each digit position has a weight associated
with it.
• Each digit corresponds to a power of 10 based
on its position in the number
• Number’s value = a weighted sum of the digits

• Number’s value = a weighted sum of the digits
• Number’ value = digit * 10x + digit * 10x where
x = (position number - 1).
It sounds complicated ?

• The powers of 10 increment from 0, 1, 2,
etc. as you move right to left
Hundreds Tens Units

• The BASE is 10 or Dec or simply D
• BASE = 10
10 3 Power or Exponent
Base 10
103=1000
102=100
101=10
100=1

• Example
123410 = 1x 103 + 2x 102 + 3x101 +4x100
= 1000 + 200 + 30 + 4
= 123410

• Example
9876D = 9x 103 + 8x 102 + 7x101 +6x100
= 9000 + 800 + 70 + 6
= 9876D

• Summary
• In the decimal system, there are 10 digits (0
through 9) which combine to form numbers as
follows:
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
19 20 21 22 ...

Binary Number Systems
• The term computer numbering formats refers to
the schemes implemented in digital computer and
calculator hardware and software to represent
numbers
• Digital Computers and Calculator use a binary
systems
• Why binary?

• Easy to represent binary values electrically
• Voltages and currents.
• Can be implemented using circuits
• Create the building blocks of modern computers

• Reliability
• With only 2 values, can be widely
separated, therefore clearly
differentiated
• Binary numbers are made of binary
digits

• Binary DigiTs (BITs) can be represented
electronically:
… 0 (no signal)
… 1 (signal)

Base is 2 or ‘b’ or ‘B’ or ‘Bin’
Two symbols: 0 and 1
Each place is weighted by the power of 2

• All the information in the digital computer is
represented as bit patterns
• What is a bit pattern?
01010101
This is called as the bit pattern
This is one bit

• Look at this bit pattern
0101 0101
• How many bits are present ?
• Count the number of ones and the zeros in the
above pattern
Answer = Total 8 bits

0101 0101
This pattern is represented as follows in the digital
computer
Bit7 Bit 6 Bit 5 Bit 4 Bit 3 Bit 2 Bit 1 Bit 0
0 1 0 1 0 1 0 1

• A single bit can represent two states:0 1
• Therefore, if you take two bits, you can use them
to represent four unique states:
00, 01, 10, & 11
• And, if you have three bits, then you can use them
to represent eight unique states:
000, 001, 010, 011, 100, 101, 110, & 111

•With every bit you add, you double the number of
states you can represent. Therefore, the expression
for the number of states with n bits is 2n. Most
computers operate on information in groups of 8
bits,

0 1 0 1 0 1 0 1
• There are 8 bits in the above table
• Group of 4 bits = 1 Nibble
• Group of 8 bits = 1 Byte
• Group of 16 bits = 1 Word 2 Bytes = 1 Word

0 1 0 1 0 1 0 1
• There are 8 bits in the above table
• Bit 0 is called the Least Significant Bit LSB
• Bit 1 is called the Most Significant Bit MSB

27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1
Bit positions and their values

Convert Decimal to Binary
For Each Digit Position
1.Divide decimal number by the base (e.g. 2)
2. The remainder is the lowest-order digit
3. Repeat first two steps until no divisor remains.

Convert Decimal to Binary
Example: Convert Decimal 13 (13 10) to Binary :
Repeated division by 2 (till quotient is zero)
Divide-by -2 Quotient Remainder Binary Bits
13/2
6 / 2
3 / 2
1 / 2
6
3
1
0
1
0
1
1
Bit 0 = 1
Bit 1 = 0
Bit 3 = 1
Bit 4 = 1
Answer = 11012

Convert Binary to Decimal
Example: Convert 11012
Multiply each 1 bit by the appropriate power of 2
and add them together.
1 0 1 1
Bit 1Bit 2Bit 3 Bit 0

Convert Binary to Decimal
Example: Convert 11012
27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1
1 1 0 1
Multiply with
these values
8 x 1 + 4 x 1 + 2 x 0 + 1x 1 = 8 + 4 + 0 + 1
= 13

Example:
10112 or 1011 B
= 1 x 23 + 0 x 22 + 1 x 21 + 1 x 20
= 8 + 0 + 2 +1
= 1110

Example:
101102 or 10110 B
1 x 24+0 x 23 + 1 x 22 + 1 x 21 + 0 x 20
= 16 + 0 + 4 +2+0
= 2210

Other Number Systems
• Octal and hex are a convenient way to represent
binary numbers, as used by computers.
• Computer mechanics often need to write out
binary quantities, but in practice writing out a
binary number such as

Other Number Systems
• 1001001101010001
is tedious, and prone to errors.
• Therefore, binary quantities are written in a
base-8 ("octal") or, much more commonly, a
base-16 ("hexadecimal" or "hex") number
format.

Octal Number Systems
• Base = 8 or ‘o’ or ‘Oct’
• 8 symbols: { 0, 1, 2, 3, 4, 5, 6, 7}
• Example 123, 567, 7654 etc
987 This is incorrect why?
• How to represent a Decimal Number using a
Octal Number System ?

• Repeated Division by 8
• Example
21310 = ( )8 ?
Divide-by -8 Quotient Remainder Octal digit
213 / 8
26 / 8
3 / 8
26
3
0
5
2
3
Lower digit = 5
Second digit =2
Third digit =3
Answer = 3258

• How to convert 3258 back to Decimal ?
• Use this table and multiply the digits with the position
values
Digit
8
Digit
7
Digit
6
Digit
5
Digit
4
Digit
3
Digit
2
Digit
1
87 86 85 84 83 82 81 80
…… …… 32768 4096 512 64 8 1

• How to convert 3258 back to Decimal ?
• Consider the above number
3 2 5 (8)
3 x 82 + 2 x 81 + 5 x 80 = 3 x 64 + 2 x 8 + 5 x 1
= 192 +16 + 5
= 213
Digit 1
Digit 2Digit 3

• Example Convert 6118
6 1 1 (8)
6 x 82 + 1 x 81 + 1 x 80 = 6 x 64 + 1 x 8 + 1 x 1
= 384 + 8 + 1
= 393
Digit 1
Digit 2Digit 3

• Convert 393 to octal
Divide-by -8 Quotient Remainder Octal digit
393 / 8
49 / 8
6 / 8
49
6
0
1
1
6
Lower digit = 1
Second digit =1
Third digit =6
Answer = 6118

Hexadecimal Number
Systems
• Base = 16 or ‘H’ or ‘Hex’
16 symbols: { 0, 1, 2, 3, 4, 5, 6, 7,8,9 }
{ 10=A, 11=B, 12=C, 13=D, 14=E, 15= F}

Hexadecimal Number
Systems
• {0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F} It uses 6
Letters !
• Example AB12, 876F, FFFF etc
• How to represent a Decimal Number using a
Hexadecimal Number System ?

Hex Number Systems
• Repeated Division by 16
• Example
21310 = ( )16 ?
Divide-by -16 Quotient Remainder Hex digit
213 / 16
13 / 16
13
0
5
13
Lower digit = 5
Second digit =D
Answer = D516

Hex Number Systems
• How to convert D516 back to Decimal ?
• Use this table and multiply the digits with the position
values
Digit
8
Digit
7
Digit
6
Digit
5
Digit
4
Digit
3
Digit
2
Digit
1
167 166 165 164 163 162 161 160
…… …… ….. …… 4096 256 16 1

Hex Number Systems
• How to convert D516 back to Decimal ?
D 5 (16)
D x 161 + 5 x 160 = 13 x 16 + 5 x 1
= 208 + 5
= 213
Digit 1
Digit 2

Reference & Readings
• Digital Logic & Design by Moris Mano
(Chapter 1) + Exercise

Lecture-2(2): Number System & Conversion

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