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