The explanation follows from simple arithmetic. We know that the fraction "one third" can be written as "0.333...." (the "..." means that the string of numbers goes on forever). So we have 0.333.... = 1/3. Multiply both sides by 3: 0.999... = 1. This is a proposition in pure numbers and it does not involve any approximation, nor any units of measurement. Mathematics is always interesting: mathematics with an infinite number of digits is particularly interesting because you can find truths that are surprising.

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