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Many years ago, I noticed that $987654321/123456789 = 8.0000000729\ldots$. I sent it in to Martin Gardner at Scientific American and he published it in his column!!! My life has gone downhill since then:)

My questions are:

  1. Why is this so?

  2. What happens beyond the "$729$"?

  3. What happens in bases other than $10$?

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9  
8.000000072900000663390006036849054935326399911470239194379176... –  copper.hat May 19 '13 at 7:10
87  
I don't see it, why is this number interesting? –  Matsemann May 19 '13 at 9:43
15  
8.000000072900000663390006036849054935326399911470239194379176668850507686539619‌​947510541522345927853347943465466285535743198375263105214894257455537742845393459‌​893080485027032413745994965088554182305843058983171836746863714396459801007784189‌​170836121454608705236939217656146880670936614105523188360261014078375228113214575‌​830252640055299024503221122979312219111741193916844864643288268253923241110701494‌​107383596377190727032435615995164105555993360559539581091810187935472710212801662‌​936495132722105707771161940717573660529920310822274828482700939192578546652464774‌​5374294482906079794445326129452467 –  Double AA May 19 '13 at 10:29
14  
Unrelated, but ever notice that Sqrt(9.87654321) approximates pi? The number system is full of this kind of stuff. –  cobaltduck May 19 '13 at 11:44
14  
@cobaltduck: It is a terrible approximation given how many digits you're putting in. It has more than $10$ times the error of $\sqrt{9.87}$ –  Jonas Meyer May 19 '13 at 18:28

5 Answers 5

up vote 80 down vote accepted

In base $n$ the numerator is $$p = n^{n-1} - \frac{n^{n-1}-1}{(n-1)^2}$$ and the denominator is $$q = \frac{n(n^{n-1}-1)}{(n-1)^2}-1.$$

Note that $p = (n-2)q + n-1$ and for the quotient we get

$$ \frac{p}{q} = n-2 + \frac{(n-1)^3}{n^n} \frac{1}{1 - \frac{n^2-n+1}{n^n}} = n-2 + \frac{(n-1)^3}{n^n} \sum_{k=0}^{\infty} \left(\frac{n^2-n+1}{n^n}\right)^k. $$

Indeed for $n=10$ this is

$$\frac{987654321}{123456789} = 8 + \frac{729}{10^{10}}\sum_{k=0}^{\infty}\left(\frac{91}{10^{10}}\right)^k $$

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22  
You and Frederica would make some scary babies... –  Sh3ljohn May 19 '13 at 8:44
30  
@Sh3ljohn So you know how I look... :-( –  WimC May 19 '13 at 8:45
1  
so methodical. Neat. –  oldrinb May 19 '13 at 13:52
    
@Sh3ljohn It's "Federica", there is no $R$ after $F$. –  Vÿska Sep 23 at 23:34

$$729=9^3$$ $$66339=9^3\cdot 91$$ $$6036849=9^3\cdot 91^2$$ $$...$$ $$987654321/123456789=8+9^3\cdot 10^{-10}\cdot\displaystyle\sum_{n=0}^{\infty}(91\cdot 10^{-10})^n$$

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Could this be generalized to the case where you append $ 10 $ and so forth. I think you would have to go to a higher base for it to be mathematically correct but there would still be a recognizable pattern if you converted the quotient back to decimal it would seem. –  Jon Claus May 19 '13 at 7:39
7  
@FedericaMaggioni Please tell me you didn't just "find" that...! –  Sh3ljohn May 19 '13 at 7:40
35  
Could you add a bit more detail? –  nbubis May 19 '13 at 8:08
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Is it just me who is too stupid to understand what the pattern is? –  Parth Kohli May 20 '13 at 10:11
    
@ΠάρτηΚοχλί look at Double AA's expansion in the comments above.. –  Federica Maggioni May 20 '13 at 10:26

Let $$S_n(a)=1 +2a+\ldots +na^{n-1}=\frac{na^{n+1}-(n+1)a^n+1}{(a-1)^2},$$ $$T_n(a)=a^{n-1}+2a^{n-2}+\ldots +n=a^{n-1}S_n(a^{-1}).$$

Then $$ \frac{S_n(a)}{T_n(a)}=\frac{na^{n+1}-(n+1)a^n+1}{a^{n+1}-(n+1)a+n}.$$ For $a=10,n=9$ we have $$ \frac{S_n(a)}{T_n(a)}\approx\frac{8\cdot 10^{10}+1}{10^{10}}. $$

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I really like this answer! Thank you for this! :) –  Sh3ljohn May 19 '13 at 7:53
    
@Sh3ljohn Thank you. –  Boris Novikov May 19 '13 at 7:55
    
$\large{+1}$ for my dear friend, Boris. Thanks again for my question in MathOverflow. –  Babak S. May 19 '13 at 10:41
    
@Babak S.:Dear Mohammad Reza, thank you. –  Boris Novikov May 19 '13 at 13:49

Just to add to the excellent answers above, some examples:

${987654321\,/\,123456789}\approx 8.00000007290000066339$

${{87654321}_9\,/\,{12345678}_9}\approx {7.000000628000056238}_9$

${{7654321}_8\,/\,{1234567}_8}\approx {6.0000052700046137}_8$

${{654321}_7\,/\,{123456}_7}\approx {5.00004260036036}_7$

${{\mathrm{fedcba987654321}}_{16}\,/\,{\mathrm{123456789abcdef}}_{16}}\approx {\mathrm{e.0000000000000d2f00000000000c693f}}_{16}$

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And what do these numbers show?? –  Matsemann May 21 '13 at 19:04
4  
They show empirically that the behavior seen in base 10 is present for all bases. –  Mark Adler May 21 '13 at 20:13
1  
And what behavior is that? I asked in a comment to the question what's so interesting about this, no one has answered. –  Matsemann Jun 13 '13 at 8:59
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The behavior is that the result is extremely close to the base minus two. Like a joke, if it has to be explained to you, it won't be funny. If you weren't intrigued by this when you first saw it, then you probably never will be. –  Mark Adler Jun 13 '13 at 15:00

$98765432 / 12345679 = 8$, exactly. You can see how the pattern works by multiplying out $12345679 * 8$ starting at either end.

This explains why your fraction is close to an integer. If you think the $729$ is interesting (I don't), it can be explained by some of the other answers here.

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