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I just started typing some numbers in my calculator and accidentally realized that $\frac{123456789}{987654321}=1/8$ and vice versa $\frac{987654321}{123456789}=8.000000072900001$, so very close to $8$.

Is this just a coincidence or is there a pattern behind this or another explanation? I tried it with smaller subsets of the numbers but I never got any similar pattern.

Bastian
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    No, it is an unexplainable behavior at least according to my models. Maybe there is a psychology stack exchange? – mathreadler May 06 '17 at 18:12
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    :) Well, thanks for that. I was just interested – Bastian May 06 '17 at 18:14
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    Is it correct your second fraction? Did you want to write $987654321/123456789$? In such a case it may be a rounding error of your calculator as the inverse of $1/8$ is exactly 8 – Ender Wiggins May 06 '17 at 18:14
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    @EnderWiggins Actually the $\frac18$ isn't exact, the actual answer is something like $0.1249999$ – John Doe May 06 '17 at 18:15
  • Yes right! Thank you very much. – Bastian May 06 '17 at 18:16
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    It is exactly $\frac{1}{8}$ if you just switch two digits around: $$\frac{123456789}{9876543\color{red}{12}}=\frac{1}{8}$$ – projectilemotion May 06 '17 at 18:20
  • @projectilemotion I tried inputting the your modified fraction into a calculator, which provides the exact result $\frac{1}{8}$ and doing the reciprocal gives $0.7999999994$ which is an inconsistency - whereas for the fraction provided by the OP, it did not provide an output of $\frac{1}{8}$ – mrnovice May 06 '17 at 18:41
  • @mrnovice If you look at my answer, you'll see that the difference is $9$ in the given equation, and from the earlier comment this is $12+9=21$. So your calculator just has issues with precision, or a too limited representation of numbers ("floating point inaccuracy"). – Yeti May 06 '17 at 18:51
  • Just a coincidence? LOL! In base b, form two numbers, one with digits from b-1 down to 1, the other with digits from 1 up to b-1, and divide the bigger one by the smaller one: the result will be close to b-2 (slightly bigger, though). –  May 06 '17 at 18:57
  • @ProfessorVector "the result will be close to", why? (rethorical) Of course it's bound to happen that if you try $\frac{321}{123}$ -> $\frac{4321}{1234}$ -> $\frac{54321}{12345}$ and you continue, you may end up with a number that has something special (in this case "nearly a whole number"). But that is still just a coincidence, and any explanation will likely be more difficult than the fraction itself. – Yeti May 06 '17 at 19:23

4 Answers4

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Let's define $d(b)=\sum^{b-1}_{k=1}k\,b^{k-1}$ and $u(b)=\sum^{b-1}_{k=1}(b-k)\,b^{k-1}$ (ours would be the special case $b=10$). Now from the well-known series $$\frac{1}{(1-x)^2}=1+2x+3x^2+4x^3+\cdots,$$ we have $$d(b)\approx\frac{b^{b-2}}{\left(1-\frac{1}{b}\right)^2}$$ and $$u(b)+d(b)=b\,\frac{b^{b-1}-1}{b-1}\approx\frac{b^{b-1}}{1-\frac{1}{b}}.$$ So $$\frac{u(b)}{d(b)}\approx b\,\left(1-\frac{1}{b}\right)-1=b-2.$$ I won't make the $\approx$ more precise, it's not hard, but tedious. Here are some numerical values for that ratio for various bases:
10: 8.00000007290000
11: 9.00000000350494
12: 10.00000000014928
13: 11.00000000000571
14: 12.00000000000020
15: 13.00000000000001
16: 14.00000000000000
17: 15.00000000000000
18: 16.00000000000000
19: 17.00000000000000
20: 18.00000000000000

  • In Dutch the word 'Mathematics' is translated as 'Wiskunde', which literally translated back to English is: Wis=Remove, Kunde=Art, thus, the "art of removing". What I am trying to say is... to me mathematics is a way of using symbols to transform complicated things like $1+1+1+1$ into simple things like $4$. As long as we know the number $4$, then it will be easier to use $4$ instead of $1+1+1+1$. To me you are doing the opposite of mathematics here, no offense ;) I think anyone who understands numbers and fractions, can just answer this question with: "it's just like that", just like $4/2=2$. – Yeti May 06 '17 at 21:22
  • Thank you very much for your answer! – Bastian May 07 '17 at 13:22
  • @Yeti The word "wis" in wiskunde has no relation to "wissen" or "removing" - it more accurately translates to something like "certain". And "kunde" does not translate to "art" either. Other than that you made a good point ;) – TMM Jun 02 '17 at 01:14
  • @TMM Thank you, in that regard the word seems more Frisian than Dutch, as the word 'wis' is Frisian for 'certain'/'sure' (but 'to remove' seemed to make more sense to me). This is getting a bit off-topic, but how else would you translate kunde/kundigheid? (note that I am not talking about art as in artistic) Maybe 'ability', 'skill', or 'expertise'? – Yeti Jun 03 '17 at 09:43
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Start with $\dfrac 1{9^2}=\dfrac 1{81}=0.012345679\underline{012345679}\;$ (this is well known see for example this thread) then $\;1-\dfrac 1{81}=\dfrac {80}{81}=0.987654320\underline{987654320}\;$ and conclude that : $$\dfrac {0.\underline{123456790}}{0.\underline{987654320}}=\dfrac {10}{81}\dfrac {81}{80}=\dfrac 18$$

Raymond Manzoni
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  • What do the lines underneath mean? Repeating decimals? I was taught that line was over the numbers; does this mean something else, or is it just equivalent notation? –  May 06 '17 at 21:45
  • @QPaysTaxes: Yes it is simply repeating decimal (as I learned it in France but it seems it should indeed be overlined...). – Raymond Manzoni May 06 '17 at 21:51
  • Well, if you were taught that way, both are probably correct. I was just taught the other way. –  May 06 '17 at 21:55
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Following Raymond Manzoni´s explanation, we can extend to:

$\dfrac {998877665544332211}{112233445566778899}=8,90000000000007000 $

$\dfrac {999888777666555444333222111}{111222333444555666777888999}=8,99000000000005000 $

$\dfrac {999988887777666655554444333322221111}{111122223333444455556666777788889999}=8,99900000000003000 $

....

cgiovanardi
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There does not seem to be an explanation that makes it easy to explain why $\frac{987654321}{123456789}$ is so close to $8$ other than a Mathematical coincidence.

However, since it is not exactly $8$, this does not count as a mathematical coincidence. Maybe the best way to rewrite it that shows why it is close to $8$ is:

$8\cdot123456789 + 9 = 987654321$

But the thing that is not hard to explain is the basic algebra that given:

$$\frac{123456789}{987654321} \approx \frac{1}{8}$$

Taking the inverse results in:

$$\frac{987654321}{123456789} \approx \frac{8}{1} (= 8)$$

Yeti
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  • I'm downvoting this answer because this is certainly not a mere coincidence. Raymond Manzoni and Professor Victor's answers both provide compelling explanations for why the answer is almost exactly 8. – Tanner Swett May 06 '17 at 21:30
  • @TannerSwett In my answer is a link, in which this coincidence is actually listed on Wikipedia as a 'Mathematical coincidence' under the header 'Decimal coincidences'. Of course, a mathematical coincidence is not necessarily a "coincidence", it is just a term for any peculiarity with "special" patterns in mathematics (maybe the only real coincidence is with constants). But about the answers you refer to, I think it is bad practice to explain the inner workings of a mathematics with mathematics, because I feel it becomes a tautology. – Yeti May 06 '17 at 22:29
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    Sure, I'm not going to argue about the meaning of the word "coincidence". But I still don't think that your statement that "there does not seem to be an explanation ... other than a mathematical coincidence" is accurate. Can you give an example of anything in mathematics which you think does have a satisfactory explanation? – Tanner Swett May 06 '17 at 22:38
  • I'd beg to differ between isolated facts and members of a series, like $10\cdot\frac{54321}{12345}=5\cdot 9-1+\frac{5\cdot 6}{12345}\approx 44$, $10\cdot\frac{654321}{123456}=6\cdot 9-1+\frac{6\cdot 7}{123456}\approx 53$, $10\cdot\frac{7654321}{1234567}=7\cdot 9-1+\frac{7\cdot 8}{1234567}\approx 62$, $10\cdot\frac{87654321}{12345678}=8\cdot 9-1+\frac{8\cdot 9}{12345678}\approx 71$, $10\cdot\frac{987654321}{123456789}=9\cdot 9-1+\frac{9\cdot 10}{123456789}\approx 80$. –  May 07 '17 at 06:18