What is the mistake in balancing this equation? It says $.99999 = 1$, in step 3 $x$ is subtracted from the left side and $.99999$ from the right side. Shouldn't that mean there is some division by $0$ stopping the equation from being right?
2026-04-02 11:15:14.1775128514
What is the mistake in balancing the equation in a demonstration that $0.999\ldots=1$?
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There is no mistake in these proofs. Both of them are completely valid and proof that $1 = 0.9999\dots$. This statement is, indeed, true. This can be sometimes confusing to people and it's completely normal that it feels weird at first.
If you think about it in the following way, it might make sense: What would $1-0.9999\dots$ be? The difference between these two numbers is smaller then any rational number. The only number that is smaller than any other is $0$. Thus: $1-0.9999\dots = 0$, which implies $1 = 0.9999\dots$.