# Why Does 0.999… = 1? (Index)

I’m using the Twelve Days of Christmas (and perhaps a few extra days besides) to do something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on different techniques that I’ll use to try to convince students that $0.999\dots = 1$.

Part 1: Converting the decimal expansion to a fraction, with algebra.

Part 2: Rewriting both sides of the equation $1 = 3 \times \displaystyle \frac{1}{3}$.

Part 3: Converting the decimal expansion to a fraction, using infinite series.

Part 4: A proof by contradiction: what number can possibly be between $0.999\dots$ and $1$?

Part 5: Same as Part 4, except by direct reasoning.

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