Dividing 52/59

What would be the strategy to divide 52/59? Can be solved through logarithms, and if yes, how could it be?

I like divisions. I calculated 52/59 to 59 decimal places.
(0,88135593220338983050847457627118644067796610169491525423728)

What I just do is divide 52 by 5,9 and then multiply the remainder by 10 and then divide that by 5,9 again and continue like this.

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Ok. But dividing by 5.9 is the same as dividing by 59…

Another strategy perhaps would be more useful? Using your method I got the answer till 5 decimal places.

I take a similar route as Johnny.
Instead of dividing by 59 I divide by 60 and make a correction to the remainder.

I always make the analogy with putting apples in baskets.
If I have 120 appels and divide them over 60 baskets, it is easy to see that I have 2 appels in each basket.
120 / 60 = 2, remaining: 0.

However; if I divide by 59 instead of 60, then it is also easy to see that if I do the above division with 60 baskets and after that remove one basket that I have a division over 59 baskets.

So imagine you put 2 apples into 60 baskets and then remove the last basket (so I have 59 baskets), keeping the 2 apples from the removed basket.

In other words:
120 / 59 = 2, remaining: 2.

Back to the calculation:
520 / 60 = 8 R 40 (8 * 60 = 480, removed from 520 gives 40 remaining)

520 / 59 = 8 R 48 (8 apples into 60 baskets give 40 remaining. I take the 8 apples from the last basket. Now I have 48 apples remaining)

The digit that we find in each step is the correction to the remainder. In the previous step we found the digit 8 with a remainder of 40. Add the digit 8 to the remainder of 40 to get the new remainder of 48.

= 88 R 8 (480 apples / 60 = 8 R 0. So 480 / 59 = 8 R 8. 8R0 => 8R8)
= 881 R 21 (80 / 60 = 1 R 20. 80 / 59 = 1 R 21)
= 8813 R 33 (210 / 60 = 3. 210 - 180 = 30. 3R30 => 3R33)
= 88135 R 35 (330 / 60 = 5R30 => 5R35)
= 881355 R 55 (350 / 60 = 5R50 => 5R55)
= 8813559 R 19 (550 / 60 = 9R10 =>9R19)
= 88135593 R 13
= 881355932 R 12
= 8813559322 R 2
= 88135593220 R 20
= 881355932203 R 23
= 8813559322033 R 53 etc.

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I prefer multiplication but I’m pretty much doing the same thing that @Kinma is doing backwards. Also, since 59 is a prime, it only has 58 recurring decimals, so @albinoblanke could have just continued writing down that number because the last 8 is the same as the first 8.

…also note that if I cut his sequence in half so it’s easier to so, the digits of the top and bottom row always add to 9. So you only have to do half the work (the bottom row) and then add the difference from 9 on the top one… but let me start with the base case 1/59

01694915254237288135593220338
98305084745762711864406779661   x6

First, put the 1 (numerator) at the bottom right, then instead of 59 (denominator) use 60 or rather 6 and start multiplying:

  • 1 x 6 = 6 …so put it left of 1 for 61
  • 6 x 6 = 36 …so put the unit digit left of 61 for 661 and carry the 3
  • 6 x 6 + 3 = 39 …so put 9 for 9661 and carry the 3
  • 9 x 6 + 3 = 57 …so put 7 to the left and carry 5
  • etc.

Once your result equals numerator - denominator = 59 - 1 = 58, you can stop and fill the top row with the differences to 9. That’s it… next step find your question 52/59 instead of 1/59

01694915254237288135593220338
98305084745762711864406779661

I’ve highlighted all the 2s for you… those are: 2/59, 12/59, 22/59, 32/59, 42/59 , 52/59. In order to find 52 you just need to divide 52 by 6 which is 8 and some remainder we don’t care about. Now Find the 2 that is followed by an 8 and there you go… place you decimal point between the two digits and you can read off the answer from there… I’ve highlighted the first 5 digits:

016949152542372.88135593220338
98305084745762711864406779661

once you get to …38 at the end of the row just continue on the bottom row …98 and once you get to …61 at the end of the bottom just continue at the top …01 until you get to the decimal point you put there.

tl;dr

  1. multiply by 6 for the bottom row
  2. add the difference from 9 for the top row
  3. highlights unit digits of numerator to find
  4. divide number by 6 and find that one
  5. place a decimal point between the two

This works the same way for 19, 29, 89, etc… maybe try 19 to familiarize yourself with the algorithm.

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That is why I only went to 59 decimal places, I noticed that the sequence was going to repeat. If it was a different division then I could’ve gone to a hundred decimal places :slight_smile: