### Introduction

• The number system is a form of expressing the different types of numbers used in an electronic or digital computer functioning.
• A Number System is a method to represent different types of numbers mathematically. It can use arithmetic operations to represent every number uniquely. To represent a number from a number system, it requires its base or radix.

### Definition

• Number system conversions are the study of changing the base of a number system from one system to another.

### Types of Number System

• There are a variety of number systems present in Mathematics that can be exercised or used in a computer system as needed. These are –
• #### Binary Number System

• The base value of a Binary Number System is 2.
• The total numbers in this system are 2 (hence the base is 2) i.e. 0 and 1 hence any binary value/number consists of a sequence of 0 and 1 one or multiple times with base value 2.
• This number system can be represented mathematically as ( … )2 such as (10 )2, ( 1011 )2,  (10110 )2, etc.
• #### Octal Number System

• The base value of an Octal Number System is 8.
• The total numbers in this system are 8 (hence the base is 8) i.e. 0,1,2,3,4,5,6, and 7 hence any octal value/number consists of a sequence of 0 to 7 one or multiple times with base value 8.
• This number system can be represented mathematically as ( … )8 such as (10)8, (75)8,  (32256)8, etc.
• #### Decimal Number System

• The base value of a Decimal Number System is 10.
• The total numbers in this system are 10 (hence the base is 10) i.e. 0,1,2,3,4,5,6,7,8 and 9 hence any decimal value/number consists of a sequence of 0 to 9 one or multiple times with base value 10.
• This number system can be represented mathematically as ( … )10 such as (10)10, (75)10,  (30299256)10, etc.
• #### Hexadecimal Number System

• The base value of a Hexadecimal Number System is 16.
• The total numbers in this system are 16 (hence the base is 16) i.e. 0,1,2,3,4,5,6,7,8,9, A(10), B(11), C(12), D(13), E(14), F(15) hence any hexadecimal value/number consists of a sequence of 0 – 9 and A-F one or multiple times with base value 16.
• This number system can be represented mathematically as ( … )16 such as (110)16, (75)16,  (302AC)16, etc.

### Methods of Number Conversion

• There are some common methods used in any type of Number conversion:-
• Direct Conversion:
• In this method of number conversions, each digit of the original number is converted to the corresponding digit in the target base.
• Divide and Conquer Conversion:
• In this method of number conversions, the original number is divided by the target base, and the remainder is the rightmost digit. This process is repeated until the quotient is zero or less than the dividend value.
• Place Value Conversion:
• In this method of number conversions, each place value of the original number is converted to the corresponding place value in the target base.
• Through Formulae/Shortcut Method Conversion:
• In this method of number conversions, some specific formulas or conventions are used to convert the number.
• This method may also contain some exceptions that may not apply to each case.

### Example Practices of Number System Conversion

• Number system conversions can be understood easily by categorized into two major groups –
• Non-Fractional Number Conversions (for Positive Integer values)
• This method of conversion of non-fractional numbers can be simplified as –
(1.) First of all we learn to convert Any other numbers to Decimal numbers (Multiplication Method)
(i) Binary Number to Decimal Number
(1101)2 to (..?..)10
= (1×23) + (1×22) + (0x21) + (1×20
= 8 + 4 + 0 + 1
= 13
Thus, (1101)2 is equivalent to  (13)10
(ii) Octal Number to Decimal Number
(46)8 to (..?..)10
= (4×81) + (6×80)
= 32 + 6
= 38
Thus, (46)8 is equivalent to  (38)10
(iii) Hexadecimal Number to Decimal Number
(4A6)16 to (..?..)10
= (4×162) + (Ax161)  + (6×160)
= (4×256) + (10×16) + (6×1)
= 1024 + 160 + 6
= 1190
Thus, (4A6)16 is equivalent to  (1190)10
(2.) Now we learn to convert Decimal numbers to Any other numbers (Division Method)
(a) Decimal Number to Binay Number
(184)10 to (..?..)2
Thus, the (184)10 is equivalent to ( 10111000 ) taking from MSB to LSB.
(b) Decimal Number to Octal Number
(484)10 to (..?..)8
Thus, the (484)10 is equivalent to (744) taking from MSB to LSB.
(c) Decimal Number to Hexadecimal Number
(523)10 to (..?..)16
Thus, the (523)10 is equivalent to (20B)16  taking from MSB to LSB.
(3.) Now we can learn to convert Any number system to Any other number system
(a) Octal Number to Binay Number
Step1:  First convert
Octal Number ⇒ Decimal Number (intermediate)
[using above 1 (ii) method]
Step2:  Finally convert
Decimal Number (intermediate) ⇒ Binary Number
[using above 2 (a) method]
(b) Binary Number to Hexadecimal Number
Step1:  First convert
Binary Number ⇒ Decimal Number (intermediate)
[using above 1 (i) method]
Step2:  Finally convert
Decimal Number (intermediate) ⇒ Hexadecimal Number
[using above 2 (c) method]
• Fractional Number Conversions (for Positive Float/Real/Values having decimal point)
• This method of conversion of fractional numbers can be simplified as –
(AA) First of all, we learn to convert Any other numbers to Decimal numbers
(i) Binary Number to Decimal Number
(1101.101)2 to (..?..)10
= (1×23) + (1×22) + (0x21) + (1×20) + (1×2-1) + (0 x2-2) + (1 x2-3)
= 8 + 4 + 0 + 1 + (1×1/2) + (0x1/4) + (1×1/8)
= 8 + 4 + 0 + 1 + 0.500 + 0.000 + 0.125
= 13.625
Thus, (1101.101)2 is equivalent to  (13.625 )10
(ii) Octal Number to Decimal Number
(46.35)8 to (..?..)10
= (4×81) + (6×80)  + (3×8-1) + (5×8-2)
= 32 + 6 + (3x 1/8) + (5×1/64)
= 32 + 6 + 0.375 + 0.078125
= 38.453125
Thus, (46.35)8 is equivalent to  (38.453125 )10
(iii) Hexadecimal Number to Decimal Number
(4A6.35)16 to (..?..)10
= (4×162) + (Ax161)  + (6×160) + (3×16-1) + (5×16-2)
= (4×256) + (10×16) + (6×1) + (3×1/16) + (5×1/256)
= 1024 + 160 + 6 + 0.1875 + 0.01953125
= 1190.20703125
Thus, (4A6.35)16 is equivalent to  (1190.20703125)10
(BB) Now we learn to convert Decimal numbers to Any other numbers
(a) Decimal Number to Binay Number
(184.625)10 to (..?..)2
Step 1: The integer part (here 184) is converted using the 2(a) method above = 10111000.
Step 2: To convert the fractional part (0.625) of a decimal number to binary:-
(i) Multiply the fractional part by 2.
(ii) Record or store the obtained integer part of the result (0 or 1).
(iii) Continue multiplying the remaining fractional part of the result by 2 until we reach a fractional part of 0 or we achieve the desired precision.
Convert the Fractional Part (0.625)

0.625 × 2 = 1.25 → record/strore  1

0.25 × 2 = 0.5 → record/strore  0

0.5 × 2 = 1.0 → record/strore   1

Thus, the fractional binary representation of 0.625 is : ($0.101_2$.

Step 3: Combine the Integer and fractional part result
Thus, the (184.625)10 is equivalent to (10111000.101)2.
(b) Decimal Number to Octal Number
(484.375)10 to (..?..)8
Step 1: The integer part (here 484) is converted using the 2(b) method above is = 744.
Step 2: To convert the fractional part (0.625) of a decimal number to octal:-
(i) Multiply the fractional part by 8.
(ii) Record or store the obtained integer part of the result (0 to 7).
(iii) Continue multiplying the remaining fractional part of the result by 8 until we reach a fractional part of 0 or we achieve the desired precision.

To Convert Fractional Part 0.375 to Octal

0.375 × 8 = 3.0 → record/store 3

.

Thus, the octal representation of 0.375 is = ($0.3_8$

Step 3: Now, Combine the Integer and fractional part of the result
Thus, the (484.375)10 is equivalent to (744.3)8.
(c) Decimal Number to Hexadecimal Number
(10.625)10 to (..?..)16
Step 1: The integer part (here 10) is converted using 2(c) method above is = 10(A).
Step 2: To convert the fractional part (0.625) of a decimal number to hexadecimal:-
(i) Multiply the fractional part by 16.
(ii) Record or store the obtained integer part of the result (0 to 9 and A to F for 10-15).
(iii) Continue multiplying the remaining fractional part of the result by 16 until we reach a fractional part of 0 or we achieve the desired precision.

To Convert Fractional Part 0.625 to Hexadecimal

0.625 × 16 = 10.0 → record/store 10(A)

Thus, the hexadecimal representation of 0.625 is = ($0.3_8$

Step 3: Now, Combine the Integer and fractional part of the result.
Thus, the (10.625)10 is equivalent to (A.A)16.

Categories: Computer Organization