After dividing 1 by 999-quattuordecillion (a number that’s 48 integers long), you get the Fibonacci sequence presented in neat, 24-digit strings. Here’s why that happens.

As a quick refresher, the Fibonacci sequence is the series of numbers, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, etc., in which each subsequent number is determined by adding the two numbers before it.

Now, the division problem that produces the Fibonacci sequence is actually pretty straightforward: It’s just one integer divided by another, after all, and 1 is pretty unassuming, as dividends go. What makes the equation uncanny is the strange choice of divisor (999-quattuordecillion), and the appearance of such a specific series of numbers in the quotient (and what’s with the 24-digit packets?):

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So how did the Fibonacci sequence find its way into this division problem? First, let’s look at the gigantic number by which we are dividing. In point of fact, the divisor is not exactly 999-quattuordecillion. The precise number is:

999,999,999,999,999,999,999,998,999,999,999,999,999,999,999,999

Notice that 998 in the middle? That’s actually a big clue as to why this equation works the way it does. It’s not God or aliens talking to us in code, but rather something a bit simpler. To help explain what’s going on, I recruited Spencer Greenberg from Ask a Mathematician.

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Greenberg tells io9 there’s nothing mysterious about the Fibonacci sequence appearing in 24-digit packets. For instance, he says, we can divide 1 by 99989999 and get 0.000000010001000200030005000800130021003400550089014402330. Again, we see the Fibonocci sequence emerge from the equation, but now the sequence is packaged in 4-digit sequences, not 24. Consequently, the quotient quickly runs out of room for numbers in the sequence with more than four digits.

By now you’ve probably noticed that 99989999 looks a lot like 999,999,999,999,999,999,999,998,999,999,999,999,999,999,999,999. Understanding how they differ, Greenberg says, can help you see how to produce a Fibonacci quotient with other divisors. For instance, you can make the Fibonacci sequence appear every d digits by choosing the number:

Q = 1/(100^d - 10^d - 1)

for d=4 we get:

Q = 1/99989999

= 0.00000001000100020003000500080013002100340055008901440233037706100987159725844181676609477713866163755037141

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for d=8 we get:

Q = 1/9999999899999999

= 0.0000000000000001000000010000000200000003000000050000000800000013000000210000003400000055000000890000014400000233000

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for d=16 we get:

Q = 1/99999999999999989999999999999999

= 0.00000000000000000000000000000001000000000000000100000000000000020000000000000003000000000000000500000000000000080000000000000013000

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“Though once the number of digits (in the next value of the Fibonacci sequence) exceeds d, the number of digits apart you want the elements of the sequence to be, the result will break down,” he says. As an example, here’s the d=2 case:

Q= 1/9899

= 0.00 01 01 02 03 05 08 13 21 34 55 90 46 36 83 20 03 23 26 49 76 26 02 28 30 58 89 48 37 86 24 10 34 44 79 24 03 27 30 57 88 46 34 81 15 97 13 10 23 33 56

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Which, as Greenberg points out, rapidly goes to pieces.

“You’ll note that the ‘90’ value in bold is wrong (34+55=89, not 90). It breaks down because the next number needs to be 3 digits but there are only 2 digit gaps between the values,” he says. “So even in the original case where they use d=24, the pattern would break down eventually (it would just take a long time for that to happen, since you’d need the next Fibonacci number to exceed 24 digits).”

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Greenberg says the Fibonacci numbers are not unique in having this property that they can be embedded inside the base 10 digits of rational numbers (i.e. written as an integer divided by another integer), though for many infinite sequences this would be impossible so there is something somewhat special about the Fibonacci numbers here.

“If you want to embed any finite number of digits within a rational number’s digits (as opposed to an infinite sequence like the Fibonacci sequence), this is pretty easy,” he says. “But it’s generally not going to lead to a nice, simple rational number, so usually it won’t look cool.”

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For instance,if you want to embed the sequence:

{0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

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you’ll get the very boring huge ratio

Q = 10120123012340123450123456012345670123456780123456789 / 10000000000000000000000000000000000000000000000000000000

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= 0.0010120123012340123450123456012345670123456780123456789

You can find another cool explanation here and a good threaded conversation about the problem here.

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H/t Futility Closet via Kottke!

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