Is 1 divided by 3 equal to 0.333...? [duplicate]
Here is a simple reasoning that $1/3=0.3333...$.
Lets denote $0.333333......$ by $x$. Then
$$x=0.33333.....$$ $$10x=3.33333...$$
Subtracting we get $9x=3$. Thus $x=\frac{3}{9}=\frac{1}{3}$.
Since $x$ was chosen as $0.3333....$ it means that $0.3333...=\frac{1}{3}$.
You can find the sum of $\frac{3}{10} + \frac{3}{100} + \frac{3}{1000} + \cdots$ using the formula of sum of infinite geometric progression.
$$a_1 = \frac{3}{10}$$
$$r=\frac{1}{10}$$
$$\sum =\frac{a_1}{1-r}=\frac{3}{10}\times\frac{10}{9} =\frac{1}{3}$$
The problematic part of the question is "no matter how many ones you add, 0.111... will never equal precisely 1/9."
In this (imprecise) context $0.111\ldots$ is an infinite sequence of ones; the sequence of ones does not terminate, so there is no place at which to add another one; each one is already followed by another one. Thus, $10\times0.111\ldots=1.111\ldots$ is precise. Therefore, $9\times0.111\ldots=1.000\ldots=1$ is precise, and $0.111\ldots=1/9$.
I say "imprecise" because we also say $\pi=3.14159\dots$ where ... there means an unspecified sequence of digits following. A more precise way of writing what, in the context of this question, we mean by $0.111\dots$ is $0.\overline{1}$ where the group of digits under the bar is to be repeated without end.
In this question, $0.333\ldots=0.\overline{3}$, and just as above, $10\times0.\overline{3}=3.\overline{3}$, and therefore, $9\times0.\overline{3}=3.\overline{0}=3$, which means $0.\overline{3}=3/9=1/3$.
You didn't follow the thread here Is it true that $0.999999999\ldots = 1$?.
Well, $\frac{1}{3}=0.33333\ldots$
You can use $1$.Geometric Progression. Or $2$. The one N.S suggested.
This question is similar to show that $0.999\ldots=1$.
I give here a proof and tou can see Does .99999... = 1? for another proofs. We have $$0.999\ldots\leq 1\leq 0.999\ldots+\frac{1}{10^n},\, \forall n\in\mathbb{N},$$ so by passing at limit ($n\to\infty$) we find $$0.999\ldots\leq1\leq 0.999\ldots$$ which allows us to conclude.