Other Base To Other Base

Other Base To Other Base 

In other base to other base number system, there are three types of conversion method which is as follows:

1. Octal To Binary & Binary To Octal

2. Hexadecimal to Binary & Binary To Octal 

3. Octal To Hexadecimal & Hexadecimal to Octal 

1. Octal To Binary & Binary To Octal

In the octal number system the each digit equivalent to 3 bits to binary number so octal to binary and binary to octal we used three bit number.

Octal To Binary:

The octal to binary is simple and understand below example.

Example:

Binary To Octal: 

To convert binary to octal use the following rules:

First, make 3 bits pair of digits.

If a binary number is an integer, then it starts from right to left side.

If fractional than make 3 bit  pair from left to right side.

Check the last pair of both integer and fractional if less than 3 bits then add a zero. 

Example:

      2.Hexadecimal To Binary & Binary To Hexadecimal 

Hexadecimal To Binary Number:

To convert from hexadecimal to binary there is the same method like used in octal to binary but there is little change that each digit of hexadecimal represent in the binary equivalent of 4 bits. To understand how to convert hexadecimal to binary with the help of following below example.

Example:

Binary To Hexadecimal Number System:

To convert from binary to hexadecimal using following rules:

First, make 4 bits pair of digits.

If a binary number is an integer, then it starts from right to left side.

If fractional than make 3 bit  pair from left to right side.

Check the last pair of both integer and fractional if less than 4 bits then add a zero.

Example:

Octal To Hexadecimal & Hexadecimal To Octal

 Octal To Hexadecimal:

To convert from octal to hexadecimal first convert octal to binary, then make 4 bits pair of binary digit if fractional, then left to right and if integer, then right to the left side and check the last pair of  binary digit for both integer and fractional if last pair incomplete than add zero make 4 bits than it convert into hexadecimal.

Example:

Hexadecimal To Octal:

To convert to hexadecimal first step we convert hexadecimal to binary than binary digits make 3 pair bits and convert it into octal.

Example:

Other Base To Decimal Number System

   Other Base To Decimal Number System

This is the second type of the conversion method in number system. The other base to decimal number system are as follows.

1. Binary to Decimal Number System
2. Octal  to Decimal Number System
3. Hexadecimal to Decimal Number System

Rules For Conversion:

There are the same rules used from another base to decimal number system in case of integer and fraction value there is just only change there base power. 

Integer:
In case of an integer value the base power started zero and increases from the right side toward the left digits and so on. The base power of an integer is positive.

Fractional:
In case of a fractional value the base power started -1 from the left side to the right side and so on. The base power of an integer is negative.

                 1.Binary To Deicmal Number System

To convert the binary number into the decimal number system is explained in following example mentioned below.

Example: Convert the following binary number into the decimal number system.

Solution:

(1101.11)2 

Step1:
The above binary number (1101)2 converts into decimal number so in this step  all bases multiply by digits than this  write as below:

1x2+1x2+0x2+1x2

Step2:
In this step we give power to their bases. explains as below: 
In this binary number the integer number has 4 digits and fractional has 2 so 
The 1st binary number in right side, the base 2 power is 0and written as 2⁰.
The 2nd binary number 0 and its base 2 power are 1 and written as 2¹.
The 3rd binary number 1 and its base 2 power are 2 and written as 2².
The 4th binary number 1 and its base 2 power are 3 and written as 2³.
So we write the above base's power as below:

1x2³+1x2²+0x2¹+1x2⁰

Step3: 
This is the most important step because in which we explain base's power multiplication. In case of an integer value the base power increase from right to left side so we start it from right side then.

The 1st digit in the right side is 1 and its base 2 power is 0 equal to 1.
The 2nd digit is 0 and its base 2 power is 2 written as 2x1=0.
The 3rd digit is 1 and its base 2 power is 2 written as 2x2=4.
The 4th digit is 1 and its base 2 power is 3 written as 2x3=8.
And the above bases, power answer is written as.

1x8+1x4+0+1

Step4:
In the above step we  multiply bases answer with digits than addition and get our final answer in decimal.
=8+4+0+1
=13


The answer is  (1101)₂ = (13)₁₀


If in the above example, only for integer value if  the binary number system consists of both value combined decimal and fractional which is known in numeral system is float numbers.

Example:
For example  a binary number 

(110.11)2 which we want to convert into decimal than we used same step which is used for previous example.

Step1: Multiply all digits with base 2 so we have both values integer and fractional than for easy understand the integer number digit, write in blue and fractional write in red font color.


1x2+1x2+0x2+1x2+1x2

Step2:
 In the step first we start fractional digits, which are shown in red color and we know that the fractional value written from left side to right in negative power than in this example we have two digits for fractional value so 

The first digit 1  base 2 power is -1 which is the minimum value and written as 2⁻¹. 
The second digit is also 1 and its base 2 power is -2 and written as 2⁻².

In the previous example, we already know about that integer power start from the right side to the left side so here we start from the right side, Then we write it from blue digits which show in this example as an integer.

Than 
The  first digit is 0 and we know the first number base 2 power is 0 and written as 2⁰.
The  second number 1 and its base 2 power 1 and written as 2¹.
The third number 1 and its base 2 power 2 and written as 2².
So the integer and fraction power written as below:

1x2²+1x2¹+0x2⁰+1x2⁻¹+1x2⁻²

Step3:

In this step for negative base value first we take their inverse then multiply their bases which explain below.

For 2⁻¹ take inverse 1/2¹ than base 2x1 =2 and write as 1/2
For 2⁻² take inverse 1/2² than base 2x2=4 and write as 1/4

And for integer base power value use the same rule as we use in the previous example.

For integer 0 bases  2⁰  = 2x1 =2
For integer 1 bases  2¹  = 2x1 =2
For integer 1 bases  2²  = 2x2 =4

Then we write it as mentioned below:

1x4+1x2+0x1+(1/2)+(1/4)

Step4:

In this step integer value base multiplies with a digit and fraction value divided than this all value added and our  answer come in decimal.

4+2+0+0.50 +0.45 =6.45

So the binary number (110.11) answer in decimal number (6.45)

Octal To Decimal Number System 

In octal, decimal number system we used the same rule to convert in in decimal as used in binary to decimal number system.

Example:

Convert the following octal number into the decimal number system.

(776.253)₈
Step1:
 In the first step we multiply digit with their base's 8  like binary to decimal number system. In this octal number system the integer value written in blue color and fractional value with red color as mentioned below.

7x8+7x8+6x8+2x8+5x8+3x8

Step2:
In the second step we write base's power  according to their position same give in the previous example of binary to octal number system both integer and fractional.


7x8²+7x8¹+6x8⁰+2x8⁻¹+5x8⁻²+3x8⁻³

Step3:
In the third multiply the base's power like same method which is used in binary to decimal number system so it mentioned below.

7x64+7x8+6x1+ (2/8) + (5/64) + (3/512)


Step 4:  In this step we used the same binary method for integer value base  answer multiplies with a digit and fraction value divided than this all value added and our answer come in decimal.

448+54+6+.25+.0781+.0058
508+.3339 =508.3339

This is our final answer for the decimal number system (508.3339)₁₀.

3. Hexadecimal To Decimal 

To convert the hexadecimal into decimal using same method like binary and octal number convert into

decimal number system  for  both integer or fraction digits.

Example:

(2A9.78)

Step 1: In this step first multiplies all digits with base 16 like same method used binary and octal.

Here important note that we write 10 instead of A because hexadecimal include 6 letters which is starting from 10 to 15 respectively A to F, So A=10 than we write 10 here:

2x16+10x16+9x16+7x16+8x16

Step 2:

In this step write base power for integer right to left side in a blue color and fraction left to right side 

in red color like same method adopt in binary and octal example.

2x16²+10x16¹+9x16⁰+7x16⁻¹+8x16⁻²

Step3:

In this step, multiply all base's power like to adopt in binary and octal for the integer and fractional value than it write as below mentioned:

2x256+10x16+9x1+ (7/16) + (8/256) 

Step4: 

In this step,  like in  binary and octal performs arithmetic calculations then

683+0.0329 = (683.0329)10

Number System Conversion Method

Number System Conversion Method

There are three types of the conversion method for number system. 

1. Decimal To Other Base Number System

2. Other Base Number To Decimal Number System

3. Other Base to Other base Number System

          1. Decimal To Other Base Number System:

The decimal to other base are as follow:

1. Decimal To Binary Number

2. Decimal To Octal Number

3. Decimal To Hexadecimal Number 

1. Decimal To Binary Number

In, Decimal to binary number, If the decimal number is an integer, then divided it by base 2 and if the decimal number in fraction than it multiply by 2.  

For Integer Example:

Example For Fractional Value:

2. Decimal To Octal

In, Decimal to binary number, If the decimal number is an integer, then divided it by base 8 and if the decimal number in fraction than it multiply by 8.  

For Integer Example:

Example For Fractional Value:

3. Decimal To Hexadecmial:

In, Decimal to Hexadecimal number, If the decimal number is an integer, then divided it by base 16 and if the decimal number in fraction than it multiply by 16.  

For Integer Example:

Example For Fractional Value:

Base For Number System

Base For Number System

Base:

The base defined as:

                       

                         "The number that is going to be raised the power is called a base.

Base Formula For Power Or Position Of Number:

                    aXb³+a Xb²+aX b¹+aXb⁰+ a  X b(n-₁)+a X b(n-₂)+a X b(n-₃)-------n

There is above formula for the base

Where  a= digit of numbers

            b= base 

            n0,n1,n2,n3= power increase for integer value right side to left 

            And 
            n-1, n-2,n-3 show the power decrease for fraction value from left side to right side .

The base another name is the radix.

Characters Of Base In Number System:

1.The power of numbers started for  integers from right side and increase to left side.

2.The power of numbers started for fraction value at negative -1 and decrease from left side     to right side.

3.The base is used when another base converts  into decimal numbers.

4.The base is used when subtraction is used in different type of number system.

5.The base power of integer and fraction value is decreasing from the left side to right side.

Example:

Ex1:    1289 is the decimal integer number  when gives power to its digits  than its write as follows.

(1289)₁₀ = 1x10³+2x10²+8x10¹+9x10⁰

In this example the base of digit increase from the right side of numbers to the left side.

It started from the  right side so 

The 1st digit is 9 and their 10 base power is 0.

The 2nd digit is 8 with their 10 base power is 1.

The 3rd digit is 2 with their 10 base power is 2.

The 4th digit is 2 with their 10 base power is 3.

This base power shows the position of digits or weighted of the digits in the number.

Ex2:  0.166 is the decimal fraction number when gives power to this number than  its Write as follows:

(0.166)₁₀ = 1x10ᐨ¹+6x10ᐨ²+6x10ᐨ³

This example  fraction value we start to give base power from the left side to right side

so the first digit is 1 and its base power is -1 then

The 2nd number is 6 and its base power is -2 then 

The 3rd and last number is 6 with their base power -3.

In above two examples noted the base power is decreasing from left to right side and increase right to left side.

Number System And Types Of Number System

Number System

Definition:

The number system defined as:

             "Number System is an organized and systematic way to representation of number."

Explanation:

The number system is used in computer architecture. The data which we enter in the computer  to show on monitor screen or to take print from the computer can be saved in computer memory in the form of the number system.

                            Type Of Number System

There are  main four types of number systems.
1. Binary number system
2. Octal number system
3. Decimal number system
4. Hexadecimal number system



1.Binary Number System:

The binary number system:

Digit:
Include two digit numbers which is 0 and 1 (0,1)
Base-2 System:
It is also called base-2 system because only two digits exist.

Integer Value Base Power:
Its base power value started zero from right side to the left side of the digit and it is written as 2⁰.

Fraction Value Base Power:
In fraction value its base power started negative -1 from left side to right side written as 2ᐨ¹.


2.Octal Number System:

The octal number system:

Digit:
Include eight digit numbers which start from 0 to  end 8.

Bit Represent In Binary:
They each digit of octal represent 3 bits in binary numbers

Base-8 System:
It is also called base-8 system because  8 digits exist.

Integer Value Base Power:
Its base power value started zero from right side to the left side of the digit and it is written as 8⁰.

Fraction Value Base Power:
In fraction value its base power started negative -1 from left side to right side written as 8ᐨ¹.

3.Decimal Number System:

The Decimal number system:

Digit:
Include ten digit numbers which start from 0 to  end 9.

Base- System:
It is also called base-8 system because  8 digits exist.

Integer Value Base Power:
Its base power value started zero from right side to the left side of the digit and it is written as 10⁰.

Fraction Value Base Power:
In fraction value its base power started negative -1 from left side to right side written as 10ᐨ¹.

4.Hexadecimal Number System:

The Hexadecimal number system:

Digit:
Include 10 digit numbers (0 to 9) and 6 alphabetic letters (A to F).

Bit Represent In Binary:
They each digit of hexa represent 4 bits in binary numbers

Base-16 System:
It is also called base-16 system because  16 digits exist.

Integer Value Base Power:
Its base power value started zero from right side to the left side of the digit and it is written as 16⁰.

Fraction Value Base Power:
In fraction value its base power started negative -1 from left side to right side written as 16ᐨ¹.