I am currently in $8$th grade and I need to know if it’s like normal subtraction or it’s different because the number is infinite. For example: $$. \overline {5} - . \overline {05}$$ Would that equal $.45$ or $.\overline {45}$? I need to know, because I've searched everywhere for an answer and I cannot find anything on at at all.
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- 1$\begingroup$ Welcome. I, for example, not have understood nothing. $\endgroup$Sebastiano– Sebastiano2023-02-13 19:54:12 +00:00Commented Feb 13, 2023 at 19:54
- $\begingroup$ Are you asking about how to do subtraction the way you learn in school, starting from the right, when there is no right-side end to the numbers? $\endgroup$Arthur– Arthur2023-02-13 19:58:41 +00:00Commented Feb 13, 2023 at 19:58
- 1$\begingroup$ It equals neither of those. It is $0.5$. $\endgroup$eyeballfrog– eyeballfrog2023-02-13 20:05:05 +00:00Commented Feb 13, 2023 at 20:05
- 1$\begingroup$ Search for subtract repeating decimals . You will find several web sites, at least one video. $\endgroup$Ethan Bolker– Ethan Bolker2023-02-13 20:05:06 +00:00Commented Feb 13, 2023 at 20:05
- 5$\begingroup$ Convert them both to fractions and subtract the fractions! $\endgroup$Amaan M– Amaan M2023-02-13 20:05:13 +00:00Commented Feb 13, 2023 at 20:05
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1 Answer
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If you try it with decimals directly,
$0. \overline{5} - 0.\overline{05} = .55555555.... - .05050505....=.50505050.... \rightarrow 0.\overline {50}$
However, if you recognize that $0. \overline{5}$ is $\dfrac {5}{9}$ and $0. \overline{05} = \dfrac {5}{99}$, you can subtract the fractions to get $$\dfrac {5}{9} - \dfrac {5}{99} \rightarrow \dfrac {55}{99} - \dfrac {5}{99} \rightarrow \dfrac {50}{99} = 0.\overline {50}$$
To wit: anything that repeats in a single digit will have a denominator of $9$; anything that repeats with two digits will end in $99$, and anything repeating with $n$ digits will have a denominator of $10^{n}-1$.
ETA: There is also a trick for determining repeating decimals from Rapid Calculations (from Google Books...)
For the denominator take as many nines as there are recurring figures in the decimal, and as many zeroes as there are non-recurring figures. For the numerator take the entire number and deduct all the non-recurring figures.
So for $0. \overline{5}$, the denominator would be $9$ (only one recurring digit) and the numerator would be $5$, so we get $\dfrac {5}{9}$. For $0. \overline{50}$, the denominator is $99$ (two recurring digits) and the numerator is $50$, giving us $\dfrac {50}{99}$. For $0.4 \overline{25}$, the denominator will be $990$ (two recurring digits, one non-recurring digit) and the numerator will be $421 (425-4)$ to give us $\dfrac {421}{990}$.