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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.

bjcolby15
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Charlee
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    Welcome. I, for example, not have understood nothing. – Sebastiano Feb 13 '23 at 19:54
  • 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? – Arthur Feb 13 '23 at 19:58
  • Well, remember that, if $a-b=c$, then $a=b+c$. At least that lets you check possible answers quickly. – lulu Feb 13 '23 at 20:02
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    It equals neither of those. It is $0.5$. – eyeballfrog Feb 13 '23 at 20:05
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    Search for subtract repeating decimals . You will find several web sites, at least one video. – Ethan Bolker Feb 13 '23 at 20:05
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    Convert them both to fractions and subtract the fractions! – Amaan M Feb 13 '23 at 20:05
  • If I didn't know better I'd say you were a finitist trying to set a trap. The answer is that you can't because you cannot perform an infinite number of operations in finite time. In the case of repeating decimals, there are tricks which may be used, but this can get quite delicate when the lengths of the repetends differ. – Charles Hudgins Feb 13 '23 at 20:17
  • As an example of what @AmaanM and the existing answer suggest: Suppose you want to subtract $0.5 - 0.0\overline{5}$. You can convert both values into their respective fractions $1/2$ and $1/18$ (there are standard ways to do that), and then subtracting them yields $9/18 - 1/18 = 4/9 = 0.\overline{4}$. Notice that if you add these values decimally, you get $0.\overline{9} = 1$. – Brian Tung Feb 13 '23 at 20:21
  • If it's the "infinite" part that's causing problems, try taking smaller steps. What's $0.5-0.05$? $0.55-0.055$? $0.555-0.0555$? Do you see a pattern as you keep tacking on more digits? – user170231 Feb 13 '23 at 20:25
  • If you know what $\overline{05}$ is, you should find the correct result. If you don't know what $\overline{05}$ is, you should not use this notation. – Lourrran Feb 14 '23 at 13:59

1 Answers1

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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 $\dfrac {421}{990}$.

bjcolby15
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