CHAPTER NO.

1 : NUMBER SYSTEMS
 Binary Numbers
 Decimal
 Octal
 Hexadecimal numbers
 BCD
 ASCII Code
 Conversion from one number system to another
 Binary Addition
 Subtraction by One’s and Two’s complement
 Binary multiplication and Division.
1.1 Introduction to Number Systems
What is a Number?
A number is a mathematical value used for counting or measuring
or labelling objects. Numbers are used to performing arithmetic
calculations. Examples of numbers are natural numbers, whole
numbers, rational and irrational numbers, etc. 0 is also a number
that represents a null value.

In a number system, these numbers are used as digits. 0 and 1 are

the most common digits in the number system, that are used to
represent binary numbers. On the other hand, 0 to 9 digits are also
used for other number systems. Let us learn here the types of
number systems.

WHAT IS A NUMBER SYSTEM?

A number system is a method to represent (write) numbers.
It provides a unique representation of every number and
represents the arithmetic and algebraic structure of the figures. It
also allows us to operate arithmetic operations like addition,
subtraction, multiplication and division.
A number system is a system for representing numbers in a
consistent manner using digits or other symbols. The most
common system is the decimal system, but in computing, we also
use binary, octal, and hexadecimal systems.
Every number system has two things:
1. Base (or Radix) → the number of digits used.

2. Digits → the actual symbols used.

👉 Example:
 Decimal (Base-10) uses 10 digits → 0,1,2,3,4,5,6,7,8,9.
 Binary (Base-2) uses 2 digits → 0,1.
1.2 Types of Number Systems
1) Decimal System (Base-10):
The decimal number system is used in our day-to-day life. It is the
most used and popular number system. It is known as base-10
system since 10 digits (0 to 9) are used.
The decimal system is the most familiar system, consisting of 10
digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
It is positional, meaning the value of each digit depends on its
position in the number (right to left, increasing powers of 10).

2) Binary System (Base-2):

The binary system uses only two digits: 0 and 1.
Computers and electronic devices use the binary system because
they work with two states: ON (1) and OFF (0).
It is the foundation of all digital computers and systems because
computers represent data in binary.
The numbers in the binary system are represented to the base 2
and the positional multipliers are the powers of 2.

The left most bit in the binary number is called as the Most
Significant Bit (MSB) and it has the largest positional weight. The
right most bit is the Least Significant Bit (LSB) and has the
smallest positional weight.

3) Octal Number System (Base-8):

The octal number system uses eight digits: 0–7.

Each position in an octal number represents a power of 8.
In the decimal system (Base-10), we use 10 digits (0-9), but in the
octal system, only 8 digits are used. Therefore, each position in an
octal number represents a power of 8.
- Place Values in the Octal System
Just like the decimal system has place values based on powers of
10 (units, tens, hundreds, etc.), the octal system has place values
based on powers of 8.
These powers of 8 increase from right to left.
For example:
  80=18*0 = 180=1 (the rightmost place value is ones)
 81=88*1 = 881=8 (the next place value is eights)
 82=648*2 = 6482=64 (the next place value is sixty-fours)
 83=5128*3 = 51283=512 (the next place value is five hundred
twelves)
 And so on...
Octal is frequently used in computing as a shorthand for binary, as
each octal digit corresponds to exactly three binary digits (bits).
Why Do We Use Octal?
The octal system is commonly used in computing because it
provides a more compact and manageable way to represent binary
numbers. Each octal digit corresponds directly to three binary
digits (bits), making it simpler for humans to read and work with
binary data.
Example:
The Octal sequence (547)8 has the decimal equivalent:

4) Hexadecimal Number System (Base-16) :

This system is used to represent data in a more compact manner.

Since 16 symbols are used, 0 to F, the notation is called
hexadecimal.
The first 10 symbols are the same as in the decimal system, 0 to 9
and the remaining 6 symbols are taken from the first 6 letters of
the alphabet sequence, A to F .

The Hexadecimal number system is a base-16 system, meaning it

uses 16 distinct symbols:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.
In hexadecimal, the digits A to F represent the decimal values 10
to 15, respectively.
 A = 10

 B = 11

 C = 12

 D = 13

 E = 14

 F = 15

Each position in a hexadecimal number represents a power of 16.

For example:
 The rightmost position represents 160=116x0 = 1160=1,
 The next position represents 161=1616x1 = 16161=16,
 The next represents 162=25616x2 = 256162=256, and so on.
Hexadecimal is commonly used in programming and computer
science to represent binary numbers in a more compact form. Each
hexadecimal digit represents exactly four binary digits (bits).

Example
The hexadecimal sequence (25)16 has the decimal equivalent:

1.3 BCD

BCD stands for Binary Coded Decimal.

BCD is a way to store decimal digits (0–9) in binary, where

each decimal digit gets its own 4 bits.
 So instead of converting the whole number to one big binary
value, you encode each digit separately.
  Example:
Decimal 59 → digits 5 and 9
5 → 0100? (No!) 5 → 0101
9 → 1001
BCD for 59 = 0101 1001.
Think of BCD as “binary for each decimal digit.”

This code is useful for the conversion of numbers into binary form.
BCD numbers are binary-coded decimal numbers.

In BCD, each decimal digit is represented by its equivalent four-bit

binary number. The circuits of pocket calculators can process BCD
numbers because you enter decimal numbers through the
keyboard and see decimal answers on an LED.

Digital clocks, digital voltmeters, and electronic counters work

with BCD numbers.

Advantages of BCD
1.The length of the number is short.
2.Easy to convert to and from BCD.
3.Suitable for hexadecimal addition.

1.4 ASCII
ASCII (American Standard Code for Information
Interchange)
This is the most popular encoding system recognized by United
States.
Most of the computers use this system. Remember this encoding
system can handle English characters only.
To make computers understand letters (A, B, C…), digits (0–9),
punctuation marks (!, ?, #…), and special symbols (@, $, %…), a
standard code was created.
 This standard is called ASCII.
👉 ASCII assigns a unique number (in binary form) to each
character (letter, digit, or symbol).
ASCII Code Size
 Originally, ASCII used 7 bits → it could represent 128
characters (from 0 to 127).
 Later, it was extended to 8 bits → 256 characters (from 0 to
255).
👉 Extended ASCII added extra symbols, graphics, and special
characters.
The ASCII code equivalent to the uppercase letter ‘A’ is 65.

1.5 Conversion from one number system to another:

1.5.1 Conversion from Decimal to other Number Systems
To convert a decimal number to any other number system (binary,
octal or hexadecimal), use the steps given below.
Step 1: Divide the given number by the base value (b) of the
number system in which it is to be converted
Step 2: Note the remainder
Step 3: Keep on dividing the quotient by the base value and note
the remainder till the quotient is zero
Step 4: Write the noted remainders in the reverse order (from
bottom to top)
A)Decimal to Binary Conversion:
Since the base value of binary system is 2, the decimal number is
repeatedly divided by 2 following the steps given in above till the
quotient is 0.
Record the remainder after each division and finally write the
remainders in reverse order in which they are computed.
EXAMPLE:
1.CONVERT 122 INTO ITS BINARY
2.Convert (65)10 into its equivalent binary number

B)Decimal to Octal Conversion

Since the base value of octal is 8, the decimal number is repeatedly
divided by 8 to obtain its equivalent octal number.
EXAMPLE:
1)Convert (122)10 to octal number.

2)Convert (65)10 into its equivalent Octal number

C)Decimal to Hexadecimal Conversion
Since the base value of hexadecimal is 16, the decimal number is
repeatedly divided by 16 to obtain its equivalent hexadecimal
number.
EXAMPLE: Convert 65 decimal number into its hexadecimal
equivalent.

Therefore, 65 decimal in hexadecimal number is (41)16

1.5.2Conversion from Binary to other Number Systems