Is 0.999... really equal to 1?

Question

Many people insist that 0.999..., with infinitely many 9s, is "just less than 1".

Show that it is exactly equal to 1, in at least three different ways. Then explain what the number would have to be if it were less than 1, and why no such number exists.

Accepted Answer

0.999... = 1 exactly. Argument 1: algebra. Let x = 0.999.... Then 10x = 9.999.... Subtracting, 10x minus x = 9.999... minus 0.999... = 9, so 9x = 9 and x = 1. Argument 2: thirds. Everyone accepts that 1/3 = 0.333.... Multiply both sides by 3: 1 = 0.999.... Argument 3: the series. 0.999... means 9/10 + 9/100 + 9/1000 + ..., a geometric series with first term 9/10 and ratio 1/10. Its sum is (9/10) / (1 minus 1/10) = (9/10) / (9/10) = 1. Argument 4: the gap. If 0.999... were less than 1, the difference 1 minus 0.999... would be a positive number. But it is smaller than 0.1, smaller than 0.01, sm…

Is 0.999... really equal to 1?

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