Repeating decimals.

We can use the fractions formula.

Like 0.xxxxxxx=0.x˙=x9 \displaystyle 0.xxxxxxx\cdots = 0.\dot x = \frac x9 , but if 0.99999999=0.9˙=99=1 \displaystyle 0.99999999\cdots = 0.\dot 9 = \frac 99 = 1 .

And there are too many ways to prove that 0.999999=0.9˙=1 \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 0.xxxxxxx\cdots .

It is almost for that a.bbbbbbbbbbb=a.b˙=aba9 \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.

0.abc˙=abcab900 \displaystyle 0.ab\dot c = \frac { abc - ab } { 900 } .

0.ab˙c˙=abca990 \displaystyle 0.a\dot b \dot c = \frac { abc - a } { 990 } .

0.a˙b˙c˙=abc999 \displaystyle 0.\dot a \dot b \dot c = \frac { abc } { 999 } .

a.b˙=aba9 \displaystyle a.\dot b = \frac { ab - a } { 9 } .

a.bc˙=abcab90 \displaystyle a. b\dot c = \frac { abc - ab } { 90 } .

a.b˙c˙=abca99 \displaystyle a.\dot b \dot c = \frac { abc - a } { 99 } .

\displaystyle \cdots .

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

Others are the same as abc abc .

. . - 1 month, 1 week ago

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

David Stiff - 1 month, 1 week ago

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

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

. . - 1 month, 1 week ago

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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=abc123 = \overline{abc}. That way, people know you're not talking about aa times bb times cc.

David Stiff - 1 month, 1 week ago

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@David Stiff Of course, anyone can use 0.4545454545454545=0.4˙5˙=0.45=4599=4559911=511 \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

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

Michael Huang - 1 month ago

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And \displaystyle command in LaTeX also helps.

. . - 1 month, 1 week ago

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