We had some fun analyzing Rüdiger Gamms performance in his division by 109.
Now let’s see how one can easily do the division by 167 in this video (starts at 0:00):
If you did not watch the video, he calculates: 62/167.
I have an idea how to do this calculation with ease, but first I would like to hear from you how you would do this calculation.
There are a few similarities to 109.
Starting with 1/167th, it’s a repeating decimal with a period of 166 digits (we’ve already talked about the special qualities of numbers where 1/n repeats after n-1 digits). Any digit in the nth place after the decimal point, where n>83, can be found by subtracting the digit at n-83 from 9:
DIGITS 001-042: 005988023952095808383233532934131736526946
DIGITS 043-083: 10778443113772455089820359281437125748502
DIGITS 084-125: 994011976047904191616766467065868263473053
DIGITS 126-166: 89221556886227544910179640718562874251497
Any quantity x/167, where x < 167, can be found by shifting the starting point of the number a proper number of places. 62/167, for example, is the same number as above, but starting at the 72nd digit: 0.37125748502994011… etc. The question, then, would be how to determine the proper starting point for a given number.
Excellent post, Greymatters!
This is not so difficult in this case. If we multiply both numbers by 6, we get 62/167 = 372/1002.
So the sequence needs to start at 371…
That’s so simple, Kinma, I can’t believe I didn’t see it!
There is one added step I’d take into consideration, however.
For any x from 0-83 in x/167, you want to multiply x by 6 and subtract 1 to get the starting digits. For any x from 84 to 166 in x/167, you’ll want to multiply x by 6 and subtract 2 instead.
For 109/167, for example, you’d multiply 109 by 6 to get 654, and then subtract 2 to get 652 as your starting digits.
Before we go too much farther with discussions like this, there are many numbers like this, where the decimal expansion of 1/n repeats after n-1 digits. The link below lists all such numbers from 1 to 1000:
http://oeis.org/A001913
All the prime numbers that qualify can also be calculated in Wolfram|Alpha. At the link below, the first 200 prime numbers are whittled down to the primes in which 1/n has a period of n-1:
Hi GreyMatters,
The way of finding the repeating sequence might become boring quickly in threads like these.
We can mathematically prove that any number that can be written as x/y (and x & y being an integer) either stops after a number of decimals, or repeats in a sequence. And if it ends in a repeating sequence, we can talk about quickly finding the start of the sequence and after, if we have memorized the sequence, we can just rattle the numbers ad infinitum.
We can talk about how to do the mental calculation in cases where we don’t have the sequence memorized.
We can talk about shortcuts of how to use ways that keep the amount of numbers that one needs to keep online in your brain as low as possible.
Let me know what interests you the most, please.
Kinma, this happens for these cyclic numbers (=reptend prime /long period prime) 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, 223, 229, 233, 257, 263, 269, 313, 337, 367, 379, 383, 389, 419, 433, 461, 487, 491, 499, 503, 509, 541, 571, 577, 593, 619, 647, 659, 701, 709, 727, 743, 811, 821, 823, 857, 863, 887, 937, 941, 953, 971, 977, 983, 1019, etc.
That’s why Corney or Gamm were able to divide so fast in documentaries and TV shows, where in fact it was memory recital of the cyclic and not mental calculation.
The easiest trick is to divide any number by the first cyclic prime (7) so you always get the decimal expansion 142857.Then you just need to find the starting point. Also, 142+857=999, so actually only 142 need to be remembered. Also 1+4+2=7… there lots of ‘magic’ properties for 7.
But talking about mental calculation, there was once a division suprise task in Magdeburg MCWC 2010: 46*67/(46+67) or 3082 / 113
So 30 competitors had to face such a task. I smiled when I saw that one. I was able to extract 21 decimal digits in 10’ and be placed 5th in that task. The winner Ali from Iran found 35 decimals in 10’.
Even if Gamm had come to Magdeburg with us 5 years ago, and let’s say he had memorised the all the 56 decimals needed for dividing by that 113 prime, he would still need around 1 minute to write all the decimals down. My point is that memory recital is actually much more comfortable than mental calculation, since you retrieve from memory and don’t process new information.
Indeed. And even if somebody memorizes these numbers, I find it much more impressive if you can actually calculate them quickly.
Nodas; when did you realize you could calculate much quicker than other people and when did you realize you could calculate much more complicated calculations compared to other people?
(It’s off topic. We could move the question to another thread if we like.)
.@Kimna,
This is a bit personal, but my mother (ex-kindergarten teacher)
told me that I was already doing 2-digit additions mentally at age 4 (back in 1988) before even learning to write anything. I have never used abacus.
Some adults in my family gatherings used to tell me their ages (e.g. 38+41), and I was keen on adding them. Of course I made mistakes sometimes, but they used to explain and correct me.
.
But below age 4 we hardly remember anything, so I can’t delve into this in further detail, I only rely on anecdotal evidence from my older relatives’ tales.
Nodas
Excellent! Great to hear. I had a similar experience. I was 5 or maybe 6 or so and my father explained the square root to me.
I understood it and I was able to extract the square root of 4, 9 & 16. I had no idea what the square root of 5 was, but I reckoned (pun intended) it had to be between 2 and 3 and closer to 2.
I always liked to calculate mentally.