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 $\displaystyle 0.999999\cdots = 0.\dot 9 = 1$ .

Please write down your proofs in the comments freely.

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

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

#Calculus

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Comments

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 $abc$ .

. . - 3 months, 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.

David Stiff - 3 months, 1 week ago

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

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

. . - 3 months 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 $c$ .

David Stiff - 3 months ago

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

. . - 3 months 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 $c = 3$ .

Michael Huang - 3 months ago

And \displaystyle command in LaTeX also helps.

. . - 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$