SEQUENCE
NUMBERS THAT COUNT:
We already saw the sequence number
998,001. It produced a sequence of
numbers that counted (without any errors) from “000” to “997”, in three digit
strings. To understand this better it
helps to know that 998,001 = 9992.
So let’s first look at its predecessors.
92 = 81. 81 is a sequence number that produces a
counting sequence.
1/81 = 0.012345679…
It counts from 0 to 7
(accurately), using one digit strings.
|
992 =
9,801. 9,801 is a sequence number that
produces a counting sequence.
1/9801 =
0.
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 99 00 01 02 ...
You will notice that it
counts (in two digit strings) from “00” to “97” (without error), then it
skips 98, does 99, and begins to repeat the whole sequence again.
|
9992 =
998,001. We have done this sequence
number previously. In summary it
counted in three digit strings, from “000” to “997” (accurately).
|
So, if I were a mathematician and
recognized this pattern, I might try 9,9992 next.
9,9992 =
99,980,001. I will check to see if it
produces a known number sequence.
1/99980001 =
0.
0000 0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024 0025 0026 0027 0028 0029 … |
Well, I am not able to show this whole
sequence using Wolfram Alpha. However,
after seeing the behavior of the previous sequence numbers I would guess that
this sequence counts from “0000” to “9997”.
And that further sequences can be produced to count to 99,997, 999,997,
9,999,997 and beyond. This could also be
the basis of a proof to show that we could find a sequence number that could
count as high as we needed it to count because there are an infinite number of
sequences that fit this pattern.
Comment: 81 is a sequence number that produces
a digital sequence that counts from 0 to 7 accurately. If I want a sequence that counts to 10, I
can use the sequence number 9801 (which produces a sequence that can count to
97 accurately). If I want to count
higher I can always find a sequence number that can count from 0 to whatever
number I specify. But we may also be
able to find a sequence number that counts from zero to any desired number.
Suppose I wanted a sequence that counts
from zero to 10, that did not skip any number, and did not keep on counting
past 10?
I can simple create the string of digits
I desire: “012345678910” and divide by a string of nines of the same digit
length – in this case 12 nines. But
then I have some work to do because 012345678910/999999999999 does not
simplify to a unit fraction. But this
fraction can be re-written as the sum of two or more unit fractions whose sum
will produce the string I want. In
this case I will need nine such fractions.
(If you are fraction-phobic please keep reading – we have calculator
and computers to do the arithmetic.)
It looks like a lot of work for such a
simple problem – but it does show that we can produce the “right” solution,
not just the “almost right” solution.
012345678910/999999999999 =
1/82 + 1/6643 + 1/44322935 + 1/3342126505735204
+
1/13777776364571245538106741530564 +
1/2837373133928592270994420052649404800004803647 26211938769978064 +
1/1211639332670151470619177485404809821093062603 140049804835240136696283172322539452816499055701 65476145628085154221859579459680 +
1/2291096678667782109419243152987347237995996443 884534813699339825439529970174908326181631920130 250818574590841574846920318160284278301627446520 767380402960694182076308529613867661634657040511 751782409282818590382632925014130579181000298831 3936928383139
+
1/5611132542106166045308830073608736935506155483 654607537208862712418700286711683287171939572047 317508097262422710923170314803350692509395180693 759783824248926790350783993785695711223184987137 890142972078853021019188233737784908200416410664 880546014671839411952610368072183227765982502400 291692137776402613562963719539071697400813720507 839577544288810854625242673675137216021748514095 434197759317668645621187460262243450067595807113 572261544833521442175314847918225229985218510678 92475647151900456592480 =
0.
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 …
This fraction is a repeating decimal with
a period of 12.
It counts from 0 to 10, then does it
again, and again – just to show you it can.
AND it never skips a number.
It gives you something to think about.
|
So we have two ways to create sequence
numbers that count as high as we want or need them to..
First we can use 81, 9801, 998001,
99980001, 9999800001, … (just add nines an zeros in equal numbers until you get
to one that will count high enough for you).
Second, we can create a custom Sequence
Number by using the method shown above. This is not easy but it does work - theoretically.
David
1/81 =
ReplyDelete0 . 0 1 2 3 4 5 6 7 9 0 1 2 3 4 5 6 7 9 ...
1/891 =
0 . 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 9 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 9 ...
1/8991 =
ReplyDelete0 . 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 9 ...
1/89991 =
ReplyDelete0 . 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6 7 7 7 7 8 8 8 9 ...
1/9801 =
ReplyDelete0 . 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 99 00 01 02 03 ...
1/989901 =
ReplyDelete0 . 00 00 01 01 02 02 03 03 04 04 05 05 06 06 07 07 08 08 09 09 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 43 43 44 44 45 45 46 46 47 47 48 48 49 49 50 50 51 51 52 52 53 53 54 54 55 55 56 56 57 57 58 58 59 59 60 60 61 61 62 62 63 63 64 64 65 65 66 66 67 67 68 68 69 69 70 70 71 71 72 72 73 73 74 74 75 75 76 76 77 77 78 78 79 79 80 80 81 81 82 82 83 83 84 84 85 85 86 86 87 87 88 88 89 89 90 90 91 91 92 92 93 93 94 94 95 95 96 96 97 97 98 99 ...
1/98999901 =
ReplyDelete0 . 00 00 00 01 01 01 02 02 02 03 03 03 04 04 04 05 05 05 06 06 06 07 07 07 08 08 08 09 09 09 10 10 10 11 11 11 12 12 12 13 13 13 14 14 14 15 15 15 16 16 16 17 17 17 18 18 18 19 19 19 20 20 20 21 21 21 22 22 22 23 23 23 24 24 24 25 25 25 26 26 26 27 27 27 28 28 28 29 29 29 30 30 30 31 31 31 32 32 32 33 33 33 34 34 34 35 35 35 36 36 36 37 37 37 38 38 38 39 39 39 40 40 40 41 41 41 42 42 42 43 43 43 44 44 44 45 45 45 46 46 46 47 47 47 48 48 48 49 49 49 50 50 50 51 51 51 52 52 52 53 53 53 54 54 54 55 55 55 56 56 56 57 57 57 58 58 58 59 59 59 60 60 60 61 61 61 62 62 62 63 63 63 64 64 64 65 65 65 66 66 66 67 67 67 68 68 68 69 69 69 70 70 70 71 71 71 72 72 72 73 73 73 74 74 74 75 75 75 76 76 76 77 77 77 78 78 78 79 79 79 80 80 80 81 81 81 82 82 82 83 83 83 84 84 84 85 85 85 86 86 86 87 87 87 88 88 88 89 89 89 90 90 90 91 91 91 92 92 92 93 93 93 94 94 94 95 95 95 96 96 96 97 97 97 98 98 99 ...