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Thus the repéating decimal fraction 0.... is equivalent to the repéating 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 repéating binary fraction 0.00011... .  It may come as a surprise that terminating decimal fractions can have repéating expansions in binary. It is for this réason 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 repéating fractions, but otherwise the method is to shift the fraction to an integer, convert it as above, and then divide by the appropriate power of two in the decimal base. For example: