Art of Programming

Number Systems

Prof. Vishal Parikh

CSE Department

Institute of Technology

Nirma University

Number System

• 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

• A computer can understand positional number

system where there are only a few symbols called

digits and these symbols represent different values

depending on the position they occupy in the

number.

• A value of each digit in a number can be

determined using:

₋ The digit

₋ The position of the digit in the number

₋ The base of the number system

Radix or Base

• The number of digits require to represent all the

numbers in a particular number system is called

radix/base.

• For example, the radix of binary system is 2

because in binary number only 2 digits are used

i.e. 0 and 1

• To distinguish two numbers in different number

system we write it as (10)₂ or (10)₁₀

• Positional Notation

N = (an-1an-2 ... a1a0 . a-1a-2 ... a-m)r where

. = radix point

r = radix or base

n = number of integer digits to the left of the

radix point

m = number of fractional digits to the right of

the radix point

an-1 = most significant digit (MSD)

a-m = least significant digit (LSD)

• Interpreted as:

N = an-1rⁿ-1 + an-2r

n-2... a1r 1 + a0r

0 + a-1r -1 + a-2 r

-2…

a-m r -m

• The symbols used in above representation should

be one of the r symbols allowed in the system

i.e. 0 to r-1

Number Base Conversion

1. Other base system to Decimal System:-

• Step 1 - Determine the position value of each

digit

• Step 2 - Multiply the obtained column values (in

Step 1) by the digits in the corresponding

columns.

• Step 3 - Sum the products calculated in Step 2.

The total is the equivalent value in decimal.

For example:-

1) (11010.11)2 = 1 x 24 + 1 x 23 + 0 x 22 + 1 x 21 +

0 x 20 + 1 x 2-1 + 1 x 2-2

= (26.75)10

2) (11010.11)2 = 1 x 24 + 1 x 23 + 0 x 22 + 1 x 21 +

0 x 20 + 1 x 2-1 + 1 x 2-2

= (26.75)10

3) (B65F)16 = 11 x 163 + 6 x 162 + 5 x 161 +

15x160 = (46,687)10

2. Decimal to Other Base System:-

For integers:

• Step 1 - Divide the decimal number to be

converted by the value of the new base.

• Step 2 - Get the remainder from Step 1 as the

rightmost digit (least significant digit) of new

base number.

• Step 3 - Divide the quotient of the previous

divide by the new base.

• Step 4 - Record the remainder from Step 3 as the

next digit (to the left) of the new base number.

• Decimal Number: 2910

• Calculating Binary Equivalent:

Step Operation Result Remainder

Step 1 29/2 14 1

Step 2 14/2 7 0

Step 3 7/2 3 1

Step 4 3/2 1 1

Step 5 1/2 0 1

ANSWER is (11101)2

For fractions:

• Step 1- Identify fractional part and consider 0 in

integer part

• Step 2 - Multiply the decimal number to be

converted by the value of the new base.

• Step 3 - Record the integer part from Step 2 as

the next digit of the new base number.

• Step 4 -Repeat steps(1,2and 3)

• Decimal number (0.1245)10 to binary

• Calculating Binary Equivalent:

Fraction Integer Coefficient

0.1245*2=0.249

0.249 0 a-1

0.249*2=0.498

0.498 0 a-2

0.498*2=0.996

0.996 0 a-3

0.996*2=1.992

0.992 1 a-4

0.992*2=1.984

0.984 1 a-5

0.984*2=1.968

0.968 1 a-6

ANSWER is (0.000111)2

3. Shortcut method - Binary to Octal

• Step 1 - Divide the binary digits into groups of three (starting from the right).

• Step 2 - Convert each group of three binary digits to one octal digit.

• Binary number (10101)2

• Calculating Octal Equivalent:

Step Binary Number Octal Number

Step 1 101012 010 101

Step 2 101012 28 58

Step 3 101012 258

ANSWER is (25)8

4. Shortcut method - Octal to Binary

• Step 1 - Convert each octal digit to a 3-digit

binary number (the octal digits may be treated as

decimal for this conversion).

• Step 2 - Combine all the resulting binary groups

(of 3 digits each) into a single binary number.

• Octal Number: 258

• Calculating Binary Equivalent:

Step Octal Number Binary Number

Step 1 258 210 510

Step 2 258 0102 1012

Step 3 258 0101012

ANSWER is (10101)2

5. Shortcut method - Binary to Hexadecimal

• Step 1 - Divide the binary digits into groups of

four (starting from the right).

• Step 2 - Convert each group of four binary

digits to one hexadecimal symbol.

• Binary Number: 101012

• Calculating hexadecimal Equivalent:

Step Binary Number Hexadecimal Number

Step 1 101012 0001 0101

Step 2 101012 110 510

Step 3 101012 1516

ANSWER is (15)16 or (15)H

6. Shortcut method - Hexadecimal to Binary

• Step 1 - Convert each hexadecimal digit to a 4-

digit binary number (the hexadecimal digits may

be treated as decimal for this conversion).

• Step 2 - Combine all the resulting binary groups

(of 4 digits each) into a single binary number.

• Hexadecimal Number: 1516

• Calculating Binary Equivalent:

Step Hexadecimal Number

Binary Number

Step 1 1516 110 510

Step 2 1516 00012 01012

Step 3 1516 000101012

ANSWER is (10101)2

Exercises:

1. Convert the following to decimal: ANSWERS

(i) (10101)2 (iii) (28)16

(ii) (4307)8 (iv) (BC12)16

2. Convert into binary: ANSWERS

(i) (1998)10 (iii) (4E)16

(ii)(30)8 (iv) (4A01)16

Exercises:

3. Convert the following to Octal: ANSWER

(i) (23)10 (iii) (1A03)16

(ii) (11011)2

4. Convert into Hexadecimal: ANSWER

(i) (0100101000000001)2

(ii) (29)10 (iii) (37)8

Answers to que.1:

(i) (21)10

(ii) (2247)10 Back to Questions

(iii) (40) 10

(iv) (48146)10

Answers to que.2:

(i) (11111001110)2

(ii) (11000)2 Back to Questions

(iii) (01001110)2

(iv) (0100101000000001)2

Answers to que.3:

(i) (27)8

(ii) (33)8

(iii) (15003)8 Back to Questions

Answers to que.4:

(i) (4A01)16

(ii) (1D)16 Back to Questions

(iii) (1F)16

Representation of Characters in

Memory

1. BCD (Binary Coded Decimal)

2. ASCII (American Standard Code for

Information Interchange)

3. EBCDIC (Extended Binary Coded Decimal

Interchange Code)

BCD Code

• BCD Codes are used to send the decimal

information into digital system or to get decimal

information from digital system.

• Ex: 8421, 7421, 6311, 5311 etc.

• 8421 is most popular.

• In 8421 each decimal digit is expressed by its 4

bit binary equivalent.

(429)10 = (0100 0010 1001)BCD

Decimal BCD (8421) Code

5311 Code Binary

0 0000 0000 0000

1 0001 0001 0001

2 0010 0011 0010

9 1001 1001

10 0001 0000 1010

ASCII Code

• Most popular in Data Communication

• ASCII-7 (7 bits)

• ASCII-8 (8 bits)

• ASCII-7 represents 2^7 = 128 different

combinations

• ASCII – 8 represents 2^8 = 256 different

combinations

EBCDIC

• 8-bit character coding by IBM mainframes