0.999... equals 1

Example

The decimal expansion $0.999\dots$, with the digit $9$ at every place after the point, represents the real number $1$. By definition, $0.999\dots = \sum_{k=1}^{\infty} 9 \cdot 10^{-k}$. This is a geometric series with first term $9/10$ and ratio $1/10$, thus its sum is $\frac{9/10}{1 - 1/10} = 1$. Thus the number $1$ has two decimal expansions, $1.000\dots$ and $0.999\dots$.

Remarks

Another proof: the $n$th truncation is $1 - 10^{-n}$. The supremum of these numbers is $1$, by the Archimedean property. The same happens for each number with a finite expansion. For example, $0.25 = 0.24999\dots$.

Explanations

L1 human

Many people feel that $0.999\dots$ is a little less than $1$. But the difference $1 - 0.999\dots$ is smaller than $10^{-n}$ for every $n$. The only nonnegative real number with this property is $0$. Another view: $1/3 = 0.333\dots$, and three times this gives $1 = 0.999\dots$.