Ninths are the easiest decimals

Question

Solve in your head, no long division. What is 	frac{7}{9} as a decimal? And what is the rule for any 	frac{k}{9}? Aim: under 5 seconds.

Accepted Answer

For 	frac{k}{9} with k a single digit, the decimal is simply that digit repeating forever: 	frac{7}{9} = 0.7777\ldots = 0.\overline{7}. The reason is that 	frac19 = 0.1111\ldots, so multiplying by k just repeats the digit k: 	frac29 = 0.\overline{2}, 	frac49 = 0.\overline{4}, and so on up to 	frac89 = 0.\overline{8}. (The pattern stops at 	frac99 = 1, not 0.\overline{9}, though those two are in fact equal.) Takeaway. A single digit over 9 is that digit repeating: 	frac{k}{9} = 0.\overline{k}. This is the base case of a whole family, over 99 you get a repeating two-digit block, over 99…

Ninths are the easiest decimals

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