Minimum Number of Bits Required for Conversion The Easy Way || Lesson 25 || Digital Electronics ||

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Here we will try to understand Minimum Number of Bits Required for Conversion The Easy Way.

The discussion on how to convert a number in some base to base 10 and a number in base 10 other bases.

Minimum Number of Bits Required for Conversion The Easy Way

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The minimum number of bits required means, to convert a number in a given number system to another what is the minimum number of bits required in the other number system.

To understand this concept we will consider the following examples.

1. ( 4786 ) 10 to ( ) 2

2. ( 4786 ) 10 to ( ) 8

3. ( 677 )8 to ( ) 10

Example 1:

( 4786 ) 10 to ( ) 2

How to find the minimum number of bits required?

The procedure is to convert the Base 10 number to Base 2 and count the number of bits.

To do that we have to divide the number 4786 by 2 until the quotient is zero.

It is a lengthy procedure and needs a lot of time to solve.

We will try to understand the logic behind the solution and try to get a generalized solution.

In a two-digit decimal number, what is the maximum number possible. It is 99.

Similarly, what is the maximum number possible with a three-digit number? It is 999.

Which means the maximum number possible with the given number of decimal digits is (10^n) – 1.

n is the number of digits.

The logic is applied for binary or any number system.

For example, to get the maximum number in a 3-bit binary number is 111.

In decimal conversion it 7.

Which means (2^n) -1.

n is the number of bits.

(2^3) – 1 = 7.

To generalize this it is (X^n) – 1.

X is the base of the number system.

In our example, we have a four-digit number 4786.

What is the maximum number possible with a four-digit number? It is 9999.

Now we can say that given any four-digit number it is always 9999 great= four-digit number ( 4786 ) great 0.

Now from the above equation ( 9999 great= four-digit number ( 4786 ) great 0 ) we can say that 4786 less= ( x^n ) -1.

4787 less= ( x^n ).

Apply log base x on both sides.

n = log 4787 base x.

So to find the minimum number of bits n required for conversion is log 4787 base 2.

Which is equal to 12.224 which means the minimum number of bits required is 13 bits to represent 4787 in binary.

The base 2 equivalent of ( 4786 )10 is 1001010110010.

Example 2:

( 4786 ) 10 to ( ) 8

Similar to the above example in order to find the minimum number of bits required to represent (4786)10 in ( )8 we have to find log4786 base 8.

Log 4786 base 8 is 4.07.

So the minimum number of bits required is 5.

The base 8 equivalent of ( 4786 )10 is 11262.

Example 3:

( 677 )8 to ( ) 10

To find the minimum number of bits required to represent (677)8 in ( )10 we have to find log677 base 10.

Log 677 base 10 is 2.85.

So the minimum number of bits required is 3.

The base 10 equivalent of ( 677 )8 is 447.

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