Introduction to Number System

NUMBER SYSTEM 

ü It is a system for representing numeric values or quantities using different symbols (digits).

ü It deals with numbers and their representation in different systems.

ü Base of a system is equal to the number of symbols used in that system.

ü The base is also called the radix of the system.

 

Type of Number System

In Computer System there are 4 types of number system used. These are –

1.Decimal Number System

2.Binary Number System

3.Octal Number System

4.Hexa-Decimal Number System

 

1. Decimal Number System:

In real life we use Decimal Number System. Recently we used.

Base: 10 (Construct numbers combination of 10 digits)

Digits used: 0, 1, 2, 3, 4, 5 ,6, 7, 8, 9 All numbers can be represented by these digits.

Counting just like: 0,1,2, 3 …4,5… 10, 11 ……100, 101 … and so on.

Example: 1234, 4566, 10100, (1234)₁₀, (10100)₁₀

 

(a) NATURAL NUMBERS-

• In natural numbers are those used for counting and ordering.
• It denoted by N and defined as:

               N = { 1, 2, 3, 4 . . . . .  }

(b) WHOLE NUMBERS-

• If we include 0 with natural numbers then the set of natural numbers are called whole numbers.
• It denoted by W and defined as:

        W = {0, 1, 2, 3, 4 . . . . .  }

 

(c) INTEGERS –

• All natural numbers, 0 and negative of counting numbers together forms a set of integers.
• It is denoted by Z: 

              Z = {  . . . -3, -2, -1, 0,1, 2, 3 . . . . .  }

• The set of integers is also denoted by I

(d) RATIONAL NUMBERS-

• A number which can be expressed as a ratio of two integers is called a rational number.
• The set of rational number is denoted by Q.

           Examples of rational numbers are:

 

(e) IRRATIONAL NUMBER –

• System of the number which cannot be represented in the form of  p/q   is called irrational number system.
• Examples of irrational numbers are:

          √2, √3, pi(π), etc.

 

(f) REAL NUMBER-

• Real numbers include rational and irrational numbers.
• A real number is any number that can be placed on a number line or expressed as in infinite decimal expansion.
• They can be both positive or negative and are denoted by the symbol R.
• Examples of real numbers are:

           √2, √3, pi(π), 5, -3/5, 0 etc.

(f) COMPLEX NUMBERS

•   Complex Number is the most general set of numbers that include all types of numbers.
•   All numbers are Complex Numbers.
•   Complex Number having two parts (i) Real Part and  (ii) Imaginary Part
•   When the imaginary part is zero, we have only the real part.
•   Real number is a subset of complex number.
•     Complex number system is denoted by C and defined as:

          Example: 2+3i , 4.5x + i 6y, 9i etc.

(2) Binary Number System: Used in Computer Machine (Digital Circuits)

Base: 2 (Construct numbers combination of 2 digits)

Digits used: 0, 1 (only 2 digits not allowed other digits)

Example:(111.01)₂, (10100)₂, (.10100)_2, 1111₂

Counting just like:  0, 1, 10, 11, 100, 101, 110, 111, 1000 … and so on.

Invalid Binary Number: 1111  (Base not indicate therefore it is decimal)

 

(3) Octal Number System: Used in Data Representation

Base: 8 (Construct numbers combination of 8 digits)

Digits used: 0, 1, 2, 3, 4, 5, 6, 7 (only 8 digits not allowed other digits)

Counting just like:  0, 1, 2 … 6,7, 10, 11, 12… 16, 17, 20, 21… and so on.

Example:(111.01)₈, (15601)₈, (.151)_8, 1111₈

Invalid Octal Number: 1111  (Base not indicate therefore it is decimal)

                                      (1078)₈ (8 digit not allowed)

                                      (3479)₈ (9 not allowed)

 

(4) Hexa-Decimal Number System: Used in Data Representation

Base: 16 (Construct numbers combination of 16 digits)

Digits used:  0, 1, 2, 3, 4, 5, 6, 7, 8, 9  (10 digits) +

A, B, C, D, E, F (6 digits as character form)

(F is largest digit and 0 is smallest digit in Hexa-Decimal)

Here A is 10, B is 11, C is 12, D is 13, E is 14 and F is 15 in decimal number.  

Counting just like:  0, 1, .. 9, A, B, …. E, F, 10, 11 …. 30, 31, 20, 21… and so on.

Example:(111.01)₁₆, (F0A9E)₁₆, (.151)_16, 1111₁₆

 

Binary Representation for Signed Numbers

• Computers can handle both positive (unsigned) and negative (signed) numbers.
• The simplest method to represent negative binary numbers is called Signed Magnitude. 
• In signed magnitude method the left most bit is Most Significant Bit (MSB) is called parity bit or sign bit. 
• Bit means Binary Digit either 0 or 1 (single digit).
• The numbers are represented in computers in different ways:

  (1) Signed Magnitude representation:

•      The value of the whole numbers can be determined by the sign used before it.
•      If the number has no sign or ‘+’ sign it will be considered as positive.
•        If the number has ‘–’ sign it will be considered as negative.

Example:  +27 or 27 is a positive number

                  –27 is a negative number

• In signed binary representation, the left most bit is considered as sign bit.
• If this bit is 0, it is a positive number and if it 1, it is a negative number.
• Therefore, if binary number stored in 8 bits then 7 bits used for storing values (Magnitude) and 1 bit is used for sign.

 

(2)1’s Complement representation-

• This is an easier approach to representation to signed numbers.
• This is for negative numbers only i.e. the number whose MSB is 1.
• The following steps used to find 1’s complement of a number:

           Step 1: Convert given Decimal number into Binary

           Step 2: Check if the binary number contains 8 bits/16 bits, if less add 0 at the left

                        most bit, to make it as 8 bits/16bits.

           Step 3: Find 1’s Compliment (i.e. Change 1 as 0 and 0 as 1)

Example: Find 1’s complement for (–36)₁₀

                        (-36)₁₀    =  ( )₂

                                                   =   (1 0 0 1 0 0 )₂

                                                   =    0 0 1 0 0 1 0 0 (8-bit format)

                       1’s Compliment   =   1 1 0 1 1 0 1 1

        (subtract each digit from 1)

                  Therefore, (-36)₁₀ =  (1 1 0 1 1 0 1 1 )₂

 

(3)2’s Complement representation

• The 2’s-complement method for negative number is as follows:

          Step 1: Find 1’s Compliment (Subtract each digit from 1)

          Step 2:   Add 1 to the result to the Least Significant Bit (LSB).

Example  2’s Complement represent of (-31)₁₀

                    Binary equivalent of +31   = 1 1 1 1 1

                    8 bit format                        =  0 0 0 1 1 1 1 1

                    1’s Compliment                 =  1 1 1 0 0 0 0 0

                    Add 1 to LSB                    =  + 1

                    2’s Compliment 0f -31      =  1 1 1 0 0 0 0 1

 

University  Exam Solved Questions for Number Systems

BCA Exam 2019

BCA Exam 2018

BCA Exam 2017