Wikipedia:Reference desk/Archives/Mathematics/2019 October 31

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October 31

Lengths of inverse of some decimal number.

So we know the inverse of 2.5 is relatively simple. 1 / 2.5 = simply .4 But what about the inverse of 2.54? 2.54 is special because 1 inch = 2.54 centimeters. Well, years ago, in a IRC chatroom I asked a math guru to calculate the inverse of that (1 / 2.54). He had a bot calculate it for him. 1st, he said "more than 10,000 digits." Then he said "more than 30,000 digits." So now I'm scratching my head, why is that? So does 1 / 2.54 ever stop? Can anyone calculate how many decimal digits for 1 / 2.51, 2.52 ... to 2.59? 1/2.50 is exactly .4, so is it actually infinite decimal digits? Thanks. 67.175.224.138 (talk) 04:31, 31 October 2019 (UTC).

Well I solved 1 / 2.56 is exactly 6 decimal digits, so why is 1 / 2.54 more than 30,000 decimal digits. 67.175.224.138 (talk) 04:35, 31 October 2019 (UTC).

The number 256 is the product of a prime number that is also found in the number 100. (256 is equal to 2⁸). However, 254 is different because it is the product of prime numbers 2 and 127. The number 2 is found in 1000, but 127 is not. The reciprocal of 127 (or 1.27) is an endless rational number (like 1/3 = 0.333333333...) so does not terminate after 10,000, or 30,000 or even more digits. I suspect the reciprocals of 2.51, 2.52, 2.53, 2.55, 2.57 etc. will also be non-terminating. Dolphin (t) 05:37, 31 October 2019 (UTC)

I see now that every number that is 2^x can be simple reciprocals, but what other numbers after 2^x? Apparently not 5^x. Apparently 1/5 and 1/25 work, but not 1/15 or 1/75, which are multiples of 5, but not 5^x. 67.175.224.138 (talk) 06:30, 31 October 2019 (UTC).

To explain this observation, note that the prime factors of 10 are 2 and 5, thus any number consisting of only those primes will have a reasonably short inverse, in base 10. However, if we used other bases, then this would change. For example, in base 3, the inverse of 3 (written as "10" in base 3) would be 0.1. SinisterLefty (talk) 17:36, 31 October 2019 (UTC)

The only denominators yielding terminating decimals for all numerators are products of a power of 2 and a power of 5. 15 and 75 also have a factor of 3 and therefore do not work. You have more flexibility when the base numeral system is a highly composite number base.--Jasper Deng (talk) 06:40, 31 October 2019 (UTC)

FWIW, 1/2.54 starts .393700787401574803149606299212598425196850 and then repeats these 42 digits ad infinitum. --RDBury (talk) 13:19, 31 October 2019 (UTC)

Go tell that to someone who wants to know which of 2 statements about inches and centimeters is true (is one inch exactly 2.54 centimeters or is one centimeter exactly .3937 inch??) Georgia guy (talk) 22:33, 31 October 2019 (UTC)

Oh yeah I remember that, it repeat the 42-digits infinitely. Anything significant about the number 42? Do most repeating digits are around 42? 67.175.224.138 (talk) 04:15, 1 November 2019 (UTC).

No, and many repeat just 1 digit, like 1/9, 2/9, 1/3, 4/9, 5/9, 2/3, 7/9, 8/9. I always found 1/7 to be interesting. It repeats 6 digits (0.142857), but there is a secondary pattern even more fascinating. This is the repeat of 2 digits, except that they double each time, although eventually that makes them overlap with the neighboring digits. Here's the first 9 reps:

  0.14
  0.0028
  0.000056
  0.00000112
  0.0000000224
  0.000000000448
  0.00000000000896
  0.0000000000001792
+ 0.000000000000003584
  ====================
  0.142857142857142...
  

And if you look at 1/14, we have the same pattern, although it starts at a different point in the 6 repeating digits: 0.07142857142857... And all the inverses of multiples of 7 seem to have a 6 digit repeating decimal. So, we have patterns within patterns within patterns, almost fractal-like. SinisterLefty (talk) 05:38, 1 November 2019 (UTC)

@SinisterLefty: I think you'll find that this pattern has something to do with this observation:

${\frac {1}{7}}={\frac {0.14}{0.98}}={\frac {0.14}{1-0.02}}=0.14\left(1+(0.02)+(0.02)^{2}+\ldots \right)$

Hope that doesn't spoil the magic :-) --Trovatore (talk) 00:45, 2 November 2019 (UTC)

To add to all the above, see also various later sections in the article Repeating decimal, which has some other results on the lengths of reciprocals; it turns out to be a tricky question. –Deacon Vorbis (carbon • videos) 13:34, 31 October 2019 (UTC)

Yes, specifically Repeating decimal#Totient rule, which speaks to the OP's question, "Anything significant about the number 42?" It tells us that 42, the repetend length, divides the totient function of the denominator 127, $\phi (127)$. Because 127 is prime, $\phi (127)=127-1=126$, and sure enough $126=3\cdot 42$.

The questioner may also be interested in the weaker rule which states that the decimal expansion of $1/n$ either terminates or repeats with a repetend length strictly less than $n$. This follows immediately from the totient rule since $\phi (n)<n$, but it is also intuitively obvious if you consider the repeated steps in long division by $n$ where the remainder is from $0,1,2,...,n-1$. A remainder of 0 means the decimal expansion terminates, so there are only $n-1$ possible remainders for a non-terminating decimal. Since the remainder wholly determines the next digit of the quotient (as the next step involves dividing into $10\cdot r$), then the first repeat of a remainder starts the repetition in the quotient.

More generally, for a reduced fraction $a/b$, you immediately know that the decimal expansion either terminates or repeats with a repetend length of at most $b-1$. If you further factor from $b$ all factors of 2 and 5 so that $b=2^{x}5^{y}n$ where neither 2 nor 5 are factors of $n$, then you know that the decimal expansion terminates if $n=1$ and repeats with a repetend length of at most $n-1$ otherwise.

For your example of 1/2.54 = 100/254 = 50/127, the repetend length must be strictly less than 127, which 42 satisfies. Likewise, if you were expanding 1/254 = 1/(2·127), you get the same repetend length limit result. -- ToE 23:00, 2 November 2019 (UTC)

Okay so you got 127 from the lowest denominator? Can we try 258 and 257?

258

100/258

50/129

129 - 1 = 128, 128 = 64 * 2 or 32 * 4 etc... = so what?

257

100/257 = can't be reduced, 257 - 1 = 256, 256 = a whole load of things, so I just know the repetend is less than 256? 67.175.224.138 (talk) 14:25, 3 November 2019 (UTC).

Two problems here. First, the simpler rule tells you that the length of the repetend for $1/n$ is strictly less than $n$ (that is, at most or less than or equal to $n-1$). So it tells you that the length of the repetend of 1/257 is less than or equal to 256, not "less than 256".

You may not think that rule gives you much, but it is a lot more than you came in with.

Second, if you are considering the totient rule, remember that $\phi (n)=n-1$ if and only if $n$ is prime. So yes, when considering 1/257, $\phi (257)=256$ because 257 is prime, so the length of the repetend divides 256 and is thus 1, 2, 4, 8, 16, 32, 64, 128, or 256.

But for 1/129, 129 is not prime, so you need to properly compute its totient. $\phi (129)=84$, so the length of the repetend is 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, or 84.

You may be interested in (sequence A051626 in the OEIS) and (sequence A000010 in the OEIS). -- ToE 16:09, 3 November 2019 (UTC)

So 1/257 how many digits does it go? And for 1/129, how did you get 84? I can't make sense of that pattern 1 2 3 4 6 7 12. 67.175.224.138 (talk) 17:58, 3 November 2019 (UTC).

They're the factors of 84. -- Jack of Oz ^{[pleasantries]} 18:16, 3 November 2019 (UTC)

1/129 seems to repeat every 21 digits, including a pair of leading zeros: 0.007751937984496124007751937984496124... SinisterLefty (talk) 18:22, 3 November 2019 (UTC)

Unfortunately (sequence A051626 in the OEIS) only lists through $n=90$. Perhaps someone here can link a longer list. Here is a program (which can be run live on that site) for calculating such values. You can edit main() to getPeriod() the value you want (as given, it tests 3 and 7), or even add a loop to print out as large a table as you wish. I have not checked out code for errors or limitations, but did test it with 129 and 257, getting $\lambda (129)=21$, agreeing with SinisterLefty above, and $\lambda (257)=256$, which per Repeating decimal#Totient rule means that 10 is a primitive root modulo 257. -- ToE 20:23, 3 November 2019 (UTC)

Here is a table of repetend length $\lambda (n)$ of $1/n$ for n=1..10,000. -- ToE 05:57, 4 November 2019 (UTC)

And while (sequence A051626 in the OEIS) only lists through $n=90$, follow their graph link to see a pin plot through $n=200$. -- ToE 22:20, 3 November 2019 (UTC)

Length Pairs

1/13 and 1/14, have the same repeat length as do 1/77 and 1/78 and 1/397 and 1/398. Is there any to predict there pairs?Naraht (talk) 18:00, 4 November 2019 (UTC)

Here are others (the smaller denominator): 13, 77, 104, 158, 259, 350, 397, 1095, 1213, 1453, 1687, 1924... I doubt that there is a way to predict such pairs. Bubba73 ^{You talkin' to me?} 03:44, 5 November 2019 (UTC)

Trying to show that 0.999... does not equal 1

Is it very common for many people to try to show that 0.999... does not equal 1?? (Uncyclopedia, which loves to make un-funny jokes, definitely tries so hard.) Georgia guy (talk) 17:50, 31 October 2019 (UTC)

Have a look at talk:0.999.../Arguments, which has 11 archive pages. After you read them all, we'll hear back from you :-) --Trovatore (talk) 18:49, 31 October 2019 (UTC)

I have seen many attempts, all by lay people who don't appear to know or understand the definition as a limit. It is not very intuitive if you don't know the concept of limits so I can see why they try. I have a harder time accepting that some of them think they know better than professional mathematicians. Talk:0.999... is listed at Wikipedia:Database reports/Talk pages by size at 4.9 MB. Many amateur proofs that it is 1 are also false or incomplete because they don't use a proper definition. PrimeHunter (talk) 19:02, 31 October 2019 (UTC)

Off topic, but I was trying to figure out why Talk:9 is even bigger. Turns out we have a bunch of articles starting with "9/11" and those all get treated as subpages of 9. --RDBury (talk) 19:19, 31 October 2019 (UTC)

To be precise, subpages are disabled in mainspace but enabled in Talk so Talk:9/11 is a subpage of Talk:9, but 9/11 is not a subpage of 9. PrimeHunter (talk) 22:25, 31 October 2019 (UTC)

• Have you seen 0.999...? Bubba73 ^{You talkin' to me?} 17:36, 2 November 2019 (UTC)

I was a younger man, just starting out graduate school, when the 0.999... page became featured. This led to a great influx of new readers to the article. Among them were also, unfortunately, misguided trolls who tried to argue that 0.999... does not equal 1. I spent quite a bit of time debating them on the matter, and you can see my name pop up in the archives of Talk:0.999... and Talk:0.999.../Arguments (a section which was only created some time later, after so many arguments in the main talk page that it became unusable for anything else).

I believe the answer to your question is - no, not many people. A few very loud and very persistent people. -- Meni Rosenfeld (talk) 21:48, 2 November 2019 (UTC)