This document discusses different number systems including binary, decimal, octal, and hexadecimal. It provides details on each system such as their number bases and allowed digits. The document also describes how to convert between these different number systems using methods like dividing numbers by the target base or grouping binary digits into sets of four for hexadecimal conversion. The goal is to understand representation of numbers in computing systems which commonly use binary and hexadecimal formats.

1. NUMBER
SYSTEM
-By
Snehamay Roy .
Roll No. : 34900321029 .
Presentation submitted for CA1 examination of the
Paper : Digital System design (EC302)
Department of ECE

2. Content
 Introduction
 Types of Number Systems
 Binary Number System
 Octal Number System
 Decimal Number System
 Hexadecimal Number System
 Conversion
 Conclusion
 References

3. Introduction
 A number system is defined as the representation of numbers by using digits or other
symbols in a consistent manner.
 The value of any digit in a number can be determined by a digit, its position in the number,
and the base of the number system.
 The numbers are represented in a unique manner and allow us to operate arithmetic
operations like addition, subtraction, and division.

4. Types of Number System
 Based on the base value and the number of allowed digits, number systems are of many
types. The four common types of Number System are:
1.Decimal Number system.
2.Binary Number system.
3.Octal Number system.
4.Hexadecimal Number system.

5. 1. Decimal Number system
 Decimal number system is the number system we use every day and uses digits from 0 - 9 i.e.
0, 1, 2, 3, 4, 5, 6, 7, 8, & 9.
 The base number of the decimal number system is 10 as the total number available in
this number system is 10.
 If any number is represented without a base, it means that its base is 10.
 For example: 7310,13210,5267107310,13210,526710 are some examples of numbers in the
decimal number system.
 Let us look at an example for a better understanding, (134)10(134)10 = 1 × 102 + 3 × 101 + 4 ×
100, (78)10(78)10 = 7 × 101 + 8 × 100. A number with a decimal point in the decimal number
system is expressed in the decreasing power of 10 after the decimal point. For
example, (24.5)10(24.5)10 = 2 × 101 + 4 × 100 + 5 × 10-1

6. 2. Binary Number system.
 Binary number system is used to define a number in binary system.
 The binary number system uses only two digits: 0 and 1. The numbers in this system have
a base of 2. Digits 0 and 1 are called bits and 8 bits together make a byte.
 The data in computers is stored in terms of bits and bytes. The binary number system
does not deal with other numbers such as 2,3,4,5 and so on.
 For example: 100012, 1111012, 10101012 are some examples of numbers in the binary number
system.
 The binary number system is used commonly by computer languages like Java, C++. As
the computer only understands binary language that is 0 or 1, all inputs given to a
computer are decoded by it into series of 0's or 1's to process it further.

7. 3. Octal Number system
 Octal Number System has a base of eight and uses the numbers from 0 to 7.
 The octal numbers, in the number system, are usually represented by binary numbers when they
are grouped in pairs of three.
 For example, an octal number 128 is expressed as 0010102 in the binary system, where 1 is
equivalent to 001 and 2 is equivalent to 010.
 If we solve an octal number, each place is a power of eight.
1248 = 1 × 82 + 2 × 81 + 4 × 80
.

8. 4.Hexadecimal Number system
 The hexadecimal number system is a type of number system, that has a base value equal
to 16. It is also pronounced sometimes as ‘hex’. Hexadecimal numbers are represented by
only 16 symbols. These symbols or values are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F.
Each digit represents a decimal value. For example, D is equal to base-10 13.
 Examples: (255)10 can be written as (FF)16,(1096)10 can be written as (448)16
 Hexadecimal number systems can be converted to other number systems such as binary
number (base-2), octal number (base-8) and decimal number systems (base-10).
 The list of 16 hexadecimal digits with their equivalent decimal, octal and binary
representation is given here in the form of a table, which will help in number system
conversion. This list can be used as a translator or converter also.

9. Conversion
 Conversion between numbers systems is quite an easy task. Any number from any number system can be
converted to other number systems with the help of certain methods that will be discussed below:
1. Conversion from Decimal Number System to Other Number Systems
A. Decimal to Binary Conversion:
 Step 1: Identify the base of the required number. Since we have to convert the given number into the octal
system, the base of the required number is 8.
 Step 2: Divide the given number by the base of the required number and note down the quotient and the
remainder in the quotient-remainder form. Repeat this process (dividing the quotient again by the base)
until we get the quotient less than the base.
 Step 3: The given number in the octal number system is obtained just by reading all the remainders and the
last quotient from bottom to top.

10. A. Decimal to Octal Conversion:
 Step 1: Divide the Decimal Number with the base of the number system to be converted to.
Here the conversion is to octal, hence the divisor will be 8.
 Step 2:The remainder obtained from the division will become the least significant digit of the
new number.
 Step 3:The quotient obtained from the division will become the next dividend and will be
divided by base i.e. 8.
 Step 4: The remainder obtained will become the second least significant digit i.e. it will be
added in the left of the previously obtained digit.

11. A. Decimal to Hexadecimal Conversion:
 Step 1: Divide the Decimal Number with the base of the number system to be converted to.
Here the conversion is to Hex hence the divisor will be 16.
 Step 2:The remainder obtained from the division will become the least significant digit of
the new number.
 Step 3:The quotient obtained from the division will become the next dividend and will be
divided by base i.e. 16.
 Step 4:The remainder obtained will become the second least significant digit i.e. it will be
added in the left of the previously obtained digit.

12. 2. Conversion from Binary Number System to Other Number
Systems
A. Binary to Decimal Conversion:-
 Step 1: Multiply each digit of the Binary number with the place value of that digit, starting
from right to left i.e. from LSB to MSB.
 Step 2: Add the result of this multiplication and the decimal number will be formed.
.

13. C. Binary to Hexadecimal Conversion:
 Step 1: Divide the binary number into groups of four digits starting from right to left i.e.
from LSB to MSB
 Step 2: Convert these groups into equivalent hex digits.

14. Conclusion
 The most commonly used Number system is the decimal positional number system, the
decimal referring to the use of 10 numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 to construct all the
required numbers.
 This discovery was made by the Indians. There are other two common number systems
which are used in computers and computing science.
 They are the binary system, and these are denoted by 0's and 1's, and the hexadecimal
system, which has 16 symbols (We can understand by the term Hex which is 16) 0, 1, 2, 3, 4,
5, 6, 7, 8, 9, A, B, C, D, E, F.

15. References :
 1. Book “ Digital logic and computer design “-R.P
. Jain.
 2. Internet: Google.
Thank You,