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My friend says that 1/3 is not 0.3333. She thinks that 1/3 is a rounded version of 0.3333... Is 1/3 the exact same as 0.3333333...?

Sungho Moon
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    Yes, $1/3$ is not $0.3333$. $1/3$ is the exact same as $0.3333333\ldots$. – peterwhy Jun 27 '22 at 12:32
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    To be clear, $\frac{1}{3}$ is not $0.3333$ where it has exactly four $3$'s following the decimal point. It is however equal to what we mean when we say $0.3333\cdots$ where the dots imply that we "have an infinite amount of $3$'s after the decimal" which by itself is not a formal description of what we mean. What we really mean by $0.3333\dots$ is that we mean $\lim\limits_{N\to\infty}\sum\limits_{n=1}^N 3\cdot 0.1^n$ which does indeed equal precisely $\frac{1}{3}$ – JMoravitz Jun 27 '22 at 12:33
  • This is closely related to the longstanding question of whether $0.9999... = 1$. I'm sure we have some of those on here explaining why that's true. Also, wouldn't $0.33333...$ be the rounded version of $1/3$, not the other way around? – eyeballfrog Jun 27 '22 at 12:33
  • To emphasize... if you use symbols and expect to be able to make any statements about the meanings of the symbols you have written, you must know the formal definition of what those symbols represent. This question stems from not knowing the formal definition of what is meant when we say $0.3333\dots$ – JMoravitz Jun 27 '22 at 12:47

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One way to think about the meaning of $0.333333...$ is to consider the following sequence of numbers: $$ \{0, 0.3, 0.33, 0.333, 0.3333, 0.33333, ...\}. $$ Clearly, each one is larger than the last, but also there are numbers greater than all of them (such as $0.34$). Out of all of those numbers, there is some number $r$ that is smaller than all the rest, but still greater than every element of the sequence. This is called the supremum of the sequence, abbreviated as $\sup$. Let's use that do define what $0.333333...$ actually means. That is, $$ 0.333333... = \sup\{0, 0.3, 0.33, 0.333, 0.3333, 0.33333, ...\} $$ Note that this works for other numbers, too $$ 0.142857... = \sup\{0, 0.1, 0.14, 0.142, 0.1428, 0.14285, ...\} $$ $$ 3.14159... = \sup\{3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ...\} $$

With that out of the way, let's see why $0.333333... = 1/3$. How do we do that? Well, one way is to show that if you multiply it by $3$, you get $1$. I think it's fairly logical that whatever $0.333333...$ is, we should have $$ 3\cdot0.333333... = \sup\{0, 0.9, 0.99, 0.999, 0.9999, 0.99999, ...\} $$ So what is the value of the right-hand side? Well, obviously $1$ is greater than every element of the sequence. Is there any number $r$ less than $1$ with this property? No, because we can write their difference in scientific notation: $1-r = a\cdot 10^{-n}$ for some $1 \le a < 10$. Then $0.99999...[n\;9\mathrm{s}]...9 > r$. So $\sup\{0, 0.9, 0.99, 0.999, 0.9999, 0.99999, ...\} = 1$. Thus we have $$ 3\cdot0.333333... = 1 \Longrightarrow 0.333333... = \frac{1}{3} $$

eyeballfrog
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$0.333...$ with an infinite number of $3$ is equal to

$$3 \sum_{i=1}^\infty \frac{1}{10^i} = 3\frac{\frac{1}{10}}{1 - \frac{1}{10}} = \frac{1}{3}$$ using the sum of a geometric sequence.

mathcounterexamples.net
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