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Thus the repeating decimal fraction 0.... is equivalent to the repeating binary fraction 0.... .

Or for example, 0.110, in binary, is:

|0.1 × 2 = 0.2 < 1    || 0.0

|0.2 × 2 = 0.4 < 1    || 0.00

|0.4 × 2 = 0.8 < 1    || 0.000

|0.8 × 2 = 1.6 ≥ 1 || 0.0001

|0.6 × 2 = 1.2 ≥ 1 || 0.00011

|0.2 × 2 = 0.4 < 1    || 0.000110

|0.4 × 2 = 0.8 < 1    || 0.0001100

|0.8 × 2 = 1.6 ≥ 1 || 0.00011001

|0.6 × 2 = 1.2 ≥ 1 || 0.000110011

|0.2 × 2 = 0.4 < 1    || 0.0001100110

This is also a repeating binary fraction 0.0... .  It may come as a surprise that terminating decimal fractions can have repeating expansions in binary. It is for this reason that many are surprised to discover that 0.1 + ... + 0.1, (10 additions) differs from 1 in floating point arithmetic. In fact, the only binary fractions with terminating expansions are of the form of an integer divided by a power of 2, which 1/10 is not.

The final conversion is from binary to decimal fractions. The only difficulty arises with repeating fractions, but otherwise the method is to shift the fraction to an integer, convert it as above, dan then divide by the appropriate power of two in the decimal base. For example: