Bizarre

Today I had need for a non-round number, so that when I resized the spacing of a grid, I wouldn't get any coincident grid points. So I chose 1.23456789, which might seem natural enough. But when I took the old spacing of 1, and divided by my factor, I got 0.810000007371. That seemed oddly round. Was it really supposed to be 0.81 exactly (and some computational round-off error?). I tried again with more accuracy, and sure enough,

0.810000007371000067076100610392515554571…

there it is with 6 zeroes right near the front. And then 4 digits, and then 2 fewer (4) zeroes, and then 2 more (6) digits, and then 2 fewer (2) zeroes, and then 2 more (8) digits, and then 2 fewer (0) zeroes, and then a movie phone number (555-4571). And beyond that, a suspicious number of triply-repeated digits.

So then, of course, I tried it the other way around. What is 1/0.81? This answer also looks quite non-arbitrary:

1.234567901234567901234567901

So then I was thinking, what's so special about 0.81 that makes it have a reciprocal with such a strange pattern; so close to counting, but without the 8's? And then it hit me…

Bonus points for anybody (without a PhD) who can figure out what's so special about 0.81?

In the meantime, for your enjoyment, here are the first 1000 digits of 1/1.234567890:

.8100000073710000670761006103925155545718

91546604213074098338974294884666083450461

35939919837053270517184761706381331528070

11690543806383948638093932606654786720558

55915708288832945428379803398256210924131

51940959682662733112230871321300929238384

54116929932464062385422967707349006136875

95584557119819469790357175092250293339477

66938924679144214580212352679932409387384

92542520282136934567446104563795515302119

18924928462216849006173325956177266201213

12243103941412245866851437388348080233967

53012910452417485116999114564691942538696

67710213976162947183082813660536562310882

71702903272496419779717419995428521958399

54982143590337506672071310715848927514225

24037944968745299215582222861798228042363

87518551126418815250112187787420908865530

27067632546315456171470651160382925559484

62259131006558092159678638653075611742988

06686119140843684181677526053265487084715

93247091498548526367916469948039876527162

87639718217521435779445065592950096895845

88175219752394499746789947695788524031675

568688247675…

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5 Responses

  1. I shot an 81 at St. Andrews in 2001. Is that it???

  2. I can’t believe you got it so quickly! I bet after your 81 you forgot to call 555-4571 afterwards; Lee Trevino was waiting by the phone to congratulate you.

  3. Does it have anything to do with the fact that 3^2=9, and 9^2=81? (In other words, 81 is a square squared?)

  4. It's not to do with squareness (as far as I can determine), but it has everything to do with 9, and its behavior as the largest digit in our base ten system. 0.81 is 81/100 = (9/10)*(9/10), so 1/0.81 = (10/9)*(10/9) = 100*(1/9)*(1/9) =100*(.111…)*(.111…). So discarding the decimals, and the 100, we can see the effect by multiplying 1*1, 11*11, 111*111, … You can best see what happens if you manually square 111111111 (9 ones), and then 1111111111 (10 ones). In the 9-case, you'll get 9 shifted copies of itself, added together, with the deepest stack being 9 ones, so the answer will be 12345678987654321. But in the 10-case, you get maximum stacks that are ten-deep, so you have to carry, and that turns a 9-stack also into 10, and that carry is what turns the 8 into 9 — hence the missing 8. It's easier to see if you work it yourself on paper.

    But anyways, for your further enjoyment (if you're not the Forester, who can just enjoy his bonus points), here are the first 25 products in the 1*1, 11*11, 111*111, … series:

    +1
    +121
    +12321
    +1234321
    +123454321
    +12345654321
    +1234567654321
    +123456787654321
    +12345678987654321
    +1234567900987654321
    +123456790120987654321
    +12345679012320987654321
    +1234567901234320987654321
    +123456790123454320987654321
    +12345679012345654320987654321
    +1234567901234567654320987654321
    +123456790123456787654320987654321
    +12345679012345678987654320987654321
    +1234567901234567900987654320987654321
    +123456790123456790120987654320987654321
    +12345679012345679012320987654320987654321
    +1234567901234567901234320987654320987654321
    +123456790123456790123454320987654320987654321
    +12345679012345679012345654320987654320987654321
    +1234567901234567901234567654320987654320987654321

  5. That was my next guess.

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