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Why Does 0.999… = 1?

In math, we often deal with numbers that have an infinite amount of decimal places. For example, $\pi=3.14159265\ldots$ going on forever. Similarly, $e=2.178282\ldots$ going on forever. These are special in that they are irrational numbers: they never repeat and never terminate. In another vein are repeating decimals, like $0.\overline{45}$, which means $0.454545\ldots$

A number that gets infinitely closer and closer to another value, like 0.99999… is considered to be equal to that value. Namely, $$ 0.99999\ldots = 1 $$

This often throws people for a loop. Shouldn’t there be “some number” between the infinitely-repeating decimal and 1? Shouldn’t you always be able to add another 9 at the end?

Well, there’s a simple way to explain why this isn’t the case:

$$ \begin{aligned} 3 \times \frac{1}{3} &= 1 \\ 3 \times 0.\overline{333} &= 1 \\ 0.\overline{999} &= 1 \end{aligned} $$