Write 5/7 as a decimal instantly

Question

Solve in your head, no long division. What is 	frac{5}{7} as a decimal (first six digits)? Aim: under 5 seconds.

Accepted Answer

Every seventh is a rotation of the one repeating block 142857. You only need to find where to start reading it. Estimate the first two digits with the rule k 	imes 14 \approx the start, since 	frac17 \approx 0.14: here 5 	imes 14 = 70, so 	frac57 \approx 0.71\ldots. Enter the 142857 wheel at the digit 7 (the one followed by 1): 	frac{5}{7} = 0.714285\,714285\ldots So the first six digits are 714285. As a sanity check, 	frac57 is just above 0.7, and 0.7142 fits. Takeaway. For any k/7, estimate the lead digits with k 	imes 14, then read the 142857 wheel from that point. Learn one number …

Write 5/7 as a decimal instantly

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