UNIT - I

Number systems are the technique to represent numbers in the computer system architecture,
every value that you are saving or getting into/from computer memory has a defined number
system.
Computer architecture supports following number systems.
 Binary number system
 Octal number system
 Decimal number system
 Hexadecimal (hex) number system
1) Binary Number System
A Binary number system has only two digits that are 0 and 1. Every number (value) represents
with 0 and 1 in this number system. The base of binary number system is 2, because it has only
two digits.
2) Octal number system
Octal number system has only eight (8) digits from 0 to 7. Every number (value) represents
with 0,1,2,3,4,5,6 and 7 in this number system. The base of octal number system is 8, because
it has only 8 digits.
3) Decimal number system
Decimal number system has only ten (10) digits from 0 to 9. Every number (value) represents
with 0,1,2,3,4,5,6, 7,8 and 9 in this number system. The base of decimal number system is 10,
because it has only 10 digits.
4) Hexadecimal number system
A Hexadecimal number system has sixteen (16) alphanumeric values from 0 to 9 and A to F.
Every number (value) represents with 0,1,2,3,4,5,6, 7,8,9,A,B,C,D,E and F in this number
system. The base of hexadecimal number system is 16, because it has 16 alphanumeric values.
Here A is 10, B is 11, C is 12, D is 13, E is 14 and F is 15.
 Table of the Numbers Systems with Base, Used Digits, Representation, C
language representation:
Number system Base Used digits Example C Language assignment
Binary 2 0,1 (11110000)2 int val=0b11110000;
Octal 8 0,1,2,3,4,5,6,7 (360)8 int val=0360;
Decimal 10 0,1,2,3,4,5,6,7,8,9 (240)10 int val=240;
Hexadecimal 16 0,1,2,3,4,5,6,7,8,9, (F0)16 int val=0xF0;
A,B,C,D,E,F
Number System Conversions
There are three types of conversion:
 Decimal Number System to Other Base
[for example: Decimal Number System to Binary Number System]
 Other Base to Decimal Number System
[for example: Binary Number System to Decimal Number System]
 Other Base to Other Base
[for example: Binary Number System to Hexadecimal Number System]
Decimal Number System to Other Base
To convert Number system from Decimal Number System to Any Other Base is quite easy;
you have to follow just two steps:
A) Divide the Number (Decimal Number) by the base of target base system (in which you
want to convert the number: Binary (2), octal (8) and Hexadecimal (16)).
B) Write the remainder from step 1 as a Least Signification Bit (LSB) to Step last as a Most
Significant Bit (MSB).

Decimal to Octal Conversion Result

Decimal to Binary Conversion Result
Decimal Number is : (12345)10 Octal Number is
(30071)8
Decimal Number is : (12345)10 Binary Number is
(11000000111001)2

Decimal to Hexadecimal Conversion Result

Example 1 Hexadecimal Number is

Decimal Number is : (12345)10 (3039)16

Example 2 Hexadecimal Number is

Decimal Number is : (725)10 (2D5)16
Convert
10, 11, 12, 13, 14, 15
to its equivalent...
A, B, C, D, E, F
 Other Base System to Decimal Number Base
To convert Number System from Any Other Base System to Decimal Number System, you
have to follow just three steps:
A) Determine the base value of source Number System (that you want to convert), and also
determine the position of digits from LSB (first digit’s position – 0, second digit’s position –
1 and so on).
B) Multiply each digit with its corresponding multiplication of position value and Base of
Source Number System’s Base.
C) Add the resulted value in step-B.

Explanation regarding examples:

Below given exams contains the following rows:
A) Row 1 contains the DIGITs of number (that is going to be converted).
B) Row 2 contains the POSITION of each digit in the number system.
C) Row 3 contains the multiplication: DIGIT* BASE^POSITION.
D) Row 4 contains the calculated result of step C.
E) And then add each value of step D, resulted value is the Decimal Number.

Binary to Decimal Conversion

Binary Number is : (11000000111001)2

Octal to Decimal Conversion

Octal Number is : (30071)8

Hexadecimal to Decimal Conversion Result

Hexadecimal Number is : (2D5)16 =512+208+5
=725
Decimal Number is: (725)10

Binary Arithmetic

Binary arithmetic is essential part of all the digital computers and many other digital system.

Binary Addition
It is a key for binary subtraction, multiplication, division. There are four rules of binary
addition.

In fourth case, a binary addition is creating a sum of (1 + 1 = 10) i.e. 0 is written in the given
column and a carry of 1 over to the next column.

Example − Addition

Binary Subtraction
Subtraction and Borrow, these two words will be used very frequently for the binary
subtraction. There are four rules of binary subtraction.

Example − Subtraction

Binary Multiplication
Binary multiplication is similar to decimal multiplication. It is simpler than decimal
multiplication because only 0s and 1s are involved. There are four rules of binary
multiplication.

Example − Multiplication

Binary Division
Binary division is similar to decimal division. It is called as the long division procedure.
 Example − Division

What are Binary Codes?

By encoding, an explicit group of symbols are used for representing a number or
word or letter. Code implies to the explicit group of symbols. Binary code represents
the stored and transmitted digital data. Numbers and alphanumeric letters are used
for representing the Binary codes.
What are the advantages of Binary Code?
Some of the advantages offered by the binary code are as follows:
 Computer applications mostly use Binary codes.
 Digital communications mostly use Binary codes.
 Digital circuits can be analyzed and designed by using the binary codes.
 Implementation of binary codes is easy as only 0 and 1 are used.

How Binary codes are classified?

Binary codes are categorized into -
 Weighted Codes
 Non-Weighted Codes
 Binary Coded Decimal Code
 Alphanumeric Codes
 Error Detecting Codes
 Error Correcting Codes

What are Weighted Binary Codes?