Just as an integer in base *b* is represented in terms of powers of *b*,
a fraction *N* in base *b* is represented in terms of the negative powers of *b*:

*
N = c _{1}b^{-1} + c_{2}b^{-2} + c_{3}b^{-3 + … + }c_{n}b^{-n} + …* ........ Equation (1)

where

So, converting a fraction given in any base to a corresponding fraction in base 10 is easy:

0.011

This is often called the

There is another way also to do this. That is to write the binary fraction as a ratio. For instance 0.011

**Multiplication Method**

How do we convert 0.375_{10} to base 2 or any other base?

To convert, we need to pick up the coefficients *c _{1}, c_{2}, c_{3},…*

Consider Equation (1). Multiply by

The integer part of the result is the coeffcient

If the remaining part is zero, we are finished.

If the remaining part is not zero, multiply that by

The integer part of the result is

Let us see why this works. Consider Equation (1) again.

*bN = c _{1} + q_{1}*, where

… …

where we continue until

Example:

Convert 0.375_{10} in to base 2 and base 5.

2*N* = 2×0.375 = 0.75 = 0 + 0.75 →→ *c _{1}* = 0 and

When a

So, 0.375

Similarly for base 5, we start with

5

Since

0.375

Having understood the principle behind the algorithm, we can now simply compute the binary version of the fraction 0.375_{10} as follows:

There is another way to convert a decimal fraction to binary. That is to first express the fraction as a ratio. Thus 0.375_{10} = (375/1000)_{10} = (111111111/11111011000)_{2} and then do the division in binary. As you can see, the above Multiplication method is easier, although it initially took a bit longer to understand.

Now that we know how to convert a fraction from DECIMAL to any other base, how about converting a fraction from any integer base to any other integer base? The case we presented is a special case, namely, from decimal fraction to fraction in any base.

** Converting a fraction between bases**

Let us consider converting 0.011_{2} to the corresponding fraction in base 5. From the discussion above, we know how to do this by first converting 0.011_{2} to decimal and then converting the decimal to base 5. But this goes via the decimal. A direct conversion is to multiply 0.011_{2} with 5 expressed in base 2. We get 0.011_{2} ×101_{2} = 1.111_{2}. The integer part is 1. This is the coefficient *c _{1}* for our fraction in base 5.

CONTINUE WITH THE NON-INTEGER PART AND MULTIPLY IT BY 5 EXPRESSED IN BASE 2: 0.111