For example, here is the decimal value '55' and we like to find the equivalent base 2 value.

Then, this is how we can do the conversion by hand:

55 / 2 = 27 (the quotient) + 1 (remainder)

27 / 2 = 13 (the quotient) + 1 (remainder)

13 / 2 = 6 (the quotient) + 1 (remainder)

6 / 2 = 3 (the quotient) + 0 (remainder)

3 / 2 = 1 (the quotient) + 1 (remainder)

Keep going until the quotient becomes smaller than the divisor, which is the base.

Now, put the very last quotient and all remainders (from the last to the first) of 55 together.

Here it becomes '110111'.

This process works for all base numbers. Let's try it out for the base 3:

55 / 3 = 18 (the quotient) + 1 (remainder)

18 / 3 = 6 (the quotient) + 0 (remainder)

6 / 3 = 2 (the quotient) + 0 (remainder)

Since the last quotient 2 is smaller than the divisor 3, we stop here.

Now, the decimal value 55 has been converted to the base 3, and that is '2001'.

How about for the base 8:

55 / 8 = 6 (quotient) + 7 (remainder)

Since the last quotient 6 is smaller than the divisor 8, we stop here.

Now, the decimal value 55 has been converted to the base 8, and that is '67'.

In general,  <1.>   Find the length/size(n) of the given base value.  <2.>   Compute in the following way:    base^(n-1)*(n^th base value)    +    base^(n-2)*((n-1)^th base value)    +   ...   +    base⁰*(1st base value).

Example 1)   base 2 value:   1 1 0 1    ( size: 4)

2³*1 + 2²*1 + 2¹*0 + 2⁰*1   =  13   in the Decimal.

Example 2)   base 8 value:   107    ( size: 3)

8²*1 + 8¹*0 + 8⁰*7   =   71   in the Decimal.