Math puzzle: Which fraction equals this repeating decimal?

A mathematical puzzle delves into the nature of numbers, specifically questioning whether a given real number with an infinite decimal expansion is rational. The number in focus is 0.123456789123456789..., which features the sequence '123456789' repeating infinitely after the decimal point. The core question is whether this number can be expressed as a fraction 'a/b', where 'a' and 'b' are integers.

The article confirms that the number is indeed rational. It demonstrates how to represent this repeating decimal as a fraction, initially arriving at 123,456,789/999,999,999. This fraction can be simplified by dividing both the numerator and denominator by 9, resulting in 13,717,421/111,111,111.

The underlying principle explained is that any repeating decimal can be converted into a fraction. The method involves setting the number equal to 'x', multiplying by a power of 10 to shift the repeating part, and then solving for 'x'. If a repeating decimal 'x' has a period of 'n' digits, represented by the integer 'p', the formula x = p/(10^n - 1) is derived. The denominator, 10^n - 1, precisely corresponds to a number composed of 'n' nines, thus proving the technique.