# Repeating decimals.

We can use the fractions formula.

Like $\displaystyle 0.xxxxxxx\cdots = 0.\dot x = \frac x9$, but if $\displaystyle 0.99999999\cdots = 0.\dot 9 = \frac 99 = 1$.

And there are too many ways to prove that $\displaystyle 0.999999\cdots = 0.\dot 9 = 1$.

And it is not only for $0.xxxxxxx\cdots$.

It is almost for that $\displaystyle a.bbbbbbbbbbb\cdots = a.\dot b = \frac { ab - a } { 9 }$.

Note by . .
1 month, 1 week ago

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And fractions formulas are too many.

$\displaystyle 0.ab\dot c = \frac { abc - ab } { 900 }$.

$\displaystyle 0.a\dot b \dot c = \frac { abc - a } { 990 }$.

$\displaystyle 0.\dot a \dot b \dot c = \frac { abc } { 999 }$.

$\displaystyle a.\dot b = \frac { ab - a } { 9 }$.

$\displaystyle a. b\dot c = \frac { abc - ab } { 90 }$.

$\displaystyle a.\dot b \dot c = \frac { abc - a } { 99 }$.

$\displaystyle \cdots$.

And $abc$ means not $a \times b \times c$.

Others are the same as $abc$.

- 1 month, 1 week ago

It might make it clearer to use \overline{abc} (which looks like $\overline{abc}$) for instance to indicate that $abc$ is to be taken as one number.

- 1 month, 1 week ago

Maybe most people use 0.\dot a, a.\dot b, etc.

It is shown like $0.\dot a$, and $a.\dot b$.

- 1 month, 1 week ago

Oh, you mean for repeating decimals? Yes, you could use dots as well. I was referring to when you talk about a number but assign each digit a letter, such as $123 = \overline{abc}$. That way, people know you're not talking about $a$ times $b$ times $c$.

- 1 month, 1 week ago

Of course, anyone can use $\displaystyle 0.4545454545454545\cdots = 0.\dot4\dot5 = 0.\overline{45} = \frac { 45 } { 99 } = \frac { \cancel { 45 } ^ { 5 } } { \cancel { 99 } _ { 11 } } = \frac 5 { 11 }$.

- 1 month, 1 week ago

@. . He is taking about overline not used as a repeating decimal, but as letters. For instance, $\overline{abc} = 123$, where $a = 1$, $b = 2$ and $c = 3$.

- 1 month ago

And \displaystyle command in LaTeX also helps.

- 1 month, 1 week ago