NERDY Numbers

An Introduction to
Numerically Enriching and
Recipricalico-Denominatorishly
Y-Computational Numbers

(or simply NERDY Numbers)

First: Some History:

I have been very careful in the past to remind my students that you can’t believe everything you see on TV, or in a movie, or even on the Internet.  TV programs and movies are changed every day to attract a larger viewing audience.  The internet is so easy and inexpensive to access that anybody can post anything (made up or true).

I ran across the following note on the internet the other day.  I found in on a blog that I check regularly, and one that I generally trust to be correct and accurate.  One of the things I like about this blog is that it also includes interesting little tidbits of interesting math trivia.  (I like the other stuff too!)

Math Notes Posted in Science & Math by Greg Ross on January 8th, 2012 At www.futilitycloset.com 1/998001 = 0.000001002003004005006007008009010011012013014015016 017018019020021022023024025026027028029030031032033034 035036037038039040041042043044045046047048049050051052 053054055056057058059060061062063064065066067068069070 071072073074075076077078079080081082083084085086087088 089090091092093094095096097098099100101102103104105106 107108109110111112113114115116117118119120121122123124 125126127128129130131132133134135136137138139140141142 143144145146147148149150151152153154155156157158159160 161162163164165166167168169170171172173174175176177178 179180181182183184185186187188189190191192193194195196 197198199200201202203204205206207208209210211212213214 215216217218219220221222223224225226227228229230231232 233234235236237238239240241242243244245246247248249250 251252253254255256257258259260261262263264265266267268 269270271272273274275276277278279280281282283284285286 287288289290291292293294295296297298299300301302303304 305306307308309310311312313314315316317318319320321322 323324325326327328329330331332333334335336337338339340 341342343344345346347348349350351352353354355356357358 359360361362363364365366367368369370371372373374375376 377378379380381382383384385386387388389390391392393394 395396397398399400401402403404405406407408409410411412 413414415416417418419420421422423424425426427428429430 431432433434435436437438439440441442443444445446447448 449450451452453454455456457458459460461462463464465466 467468469470471472473474475476477478479480481482483484 485486487488489490491492493494495496497498499500501502 503504505506507508509510511512513514515516517518519520 521522523524525526527528529530531532533534535536537538 539540541542543544545546547548549550551552553554555556 557558559560561562563564565566567568569570571572573574 575576577578579580581582583584585586587588589590591592 593594595596597598599600601602603604605606607608609610 611612613614615616617618619620621622623624625626627628 629630631632633634635636637638639640641642643644645646 647648649650651652653654655656657658659660661662663664 665666667668669670671672673674675676677678679680681682 683684685686687688689690691692693694695696697698699700 701702703704705706707708709710711712713714715716717718 719720721722723724725726727728729730731732733734735736 737738739740741742743744745746747748749750751752753754 755756757758759760761762763764765766767768769770771772 773774775776777778779780781782783784785786787788789790 791792793794795796797798799800801802803804805806807808 809810811812813814815816817818819820821822823824825826 827828829830831832833834835836837838839840841842843844 845846847848849850851852853854855856857858859860861862 863864865866867868869870871872873874875876877878879880 881882883884885886887888889890891892893894895896897898 899900901902903904905906907908909910911912913914915916 917918919920921922923924925926927928929930931932933934 935936937938939940941942943944945946947948949950951952 953954955956957958959960961962963964965966967968969970 971972973974975976977978979980981982983984985986987988 989990991992993994995996997999… Thanks William

No additional explanation was given.

I saw this and noticed that it did not have an explanation about why this note was supposed to be interesting.  At first I did not see why this fraction, with it’s decimal equivalent, was special.  Then I noticed that the decimal equivalent of this fraction was counting.  After the decimal point the numbers have a pattern.  It begins with 000, then 001, 002, 003, 004, etc.  It continues up to 997, but then it skips 998, and finishes with 999.

At first I thought this was interesting.  I thought about how much work would be required to find a fraction that showed this interesting result.  Then it hit me: “I bet this is a hoax!”.

It’s easy enough to do.  Just find a fraction that starts off with 0.000001002003 – who would check the rest?  Most calculators will only show 6 or 8 significant digits.  Few people have access (or know that they have access) to programs that would actually calculate a fraction like this to thousands of significant digits.

I’ve used this trick in teaching my own math classes.  When teaching about the number pi.  Pi is a number that (when written in decimal form) continues on forever, but it does not repeat.  Some mathematicians actually complete to see who can calculate the most digits of pi (… current record …).  There are even people who complete to see who can memorize the most digits of pi (the current record is …).  That would usually prompt one of my students to ask me how many digit of pi I have memorized.  I would usually reply with a large number (like maybe 5000).  Then they would ask be to recite them.  Either they are challenging me or they are thinking “This should last until the bell rings and he will forget to give us homework”.  I would start off with 3.14159 … (most of my students only remember the 3.14 part) … and I would then start making up the rest.  The longer I would go the more amazed they would get.  But, after 30 or 40 digits I would let them in know that nobody was checking me, and I was making up the numbers.  (You just can’t trust anybody these days – not even your math teacher.)

This could be a similar case.  It looks like it is an amazing piece of trivia.  Imagine that – a fraction that can count from 000 to 997.  And it is on a blog that I usually trust to be true.  It’s hard to fact check everything that you put on a blog, and I bet this one just did not get checked.

So I decided to check it.

I knew my calculator would only give a handful of digits so I decided to give Wolfram Alpha (www.wolframalpha.com) a go.  I had heard a lot of good stuff about it.  This would be a good test for it.

1/998001 = 1.0020030040050060070080090100110120130140150160170180... × 10^-6

This was not even close to what I was hoping for.  But thin I noticed that there as a button to click that would give me “more digits”.  A few clicks later and I had:

0.000001002003004005006007008009010011012013014015016017
018019020021022023024025026027028029030031032033034035
036037038039040041042043044045046047048049050051052053
054055056057058059060061062063064065066067068069070071
072073074075076077078079080081082083084085086087088089
090091092093094095096097098099100101102103104105106107
108109110111112113114115116117118119120121122123124125
126127128129130131132133134135136137138139140141142143
144145146147148149150151152153154155156157158159160161
162163164165166167168169170171172173174175176177178179
180181182183184185186187188189190191192193194195196197
198199200201202203204205206207208209210211212213214215
216217218219220221222223224225226227228229230231232233
234235236237238239240241242243244245246247248249250251
252253254255256257258259260261262263264265266267268269
270271272273274275276277278279280281282283284285286287
288289290291292293294295296297298299300301302303304305
306307308309310311312313314315316317318319320321322323
324325326327328329330331332333334335336337338339340341
342343344345346347348349350351352353354355356357358359
360361362363364365366367368369370371372373374375376377
378379380381382383384385386387388389390391392393394395
396397398399400401402403404405406407408409410411412413
414415416417418419420421422423424425426427428429430431
432433434435436437438439440441442443444445446447448449
450451452453454455456457458459460461462463464465466467
468469470471472473474475476477478479480481482483484485
486487488489490491492493494495496497498499500501502503
504505506507508509510511512513514515516517518519520521
522523524525526527528529530531532533534535536537538539
540541542543544545546547548549550551552553554555556557
558559560561562563564565566567568569570571572573574575
576577578579580581582583584585586587588589590591592593
594595596597598599600601602603604605606607608609610611
612613614615616617618619620621622623624625626627628629
630631632633634635636637638639640641642643644645646647
648649650651652653654655656657658659660661662663664665
666667668669670671672673674675676677678679680681682683
684685686687688689690691692693694695696697698699700701
702703704705706707708709710711712713714715716717718719
720721722723724725726727728729730731732733734735736737
738739740741742743744745746747748749750751752753754755
756757758759760761762763764765766767768769770771772773
774775776777778779780781782783784785786787788789790791
792793794795796797798799800801802803804805806807808809
810811812813814815816817818819820821822823824825826827
828829830831832833834835836837838839840841842843844845
846847848849850851852853854855856857858859860861862863
864865866867868869870871872873874875876877878879880881
882883884885886887888889890891892893894895896897898899
900901902903904905906907908909910911912913914915916917
918919920921922923924925926927928929930931932933934935
936937938939940941942943944945946947948949950951952953
954955956957958959960961962963964965966967968969970971
972973974975976977978979980981982983984985986987988989
990991992993994995996997999000001002003004005006007008
009010011012013014015016017018019020021022023024025026
027028029030031032033034035036037038039040041042043044
045046047048049050051052053054055056057058059060061062
063064065066067068069070071072073074075076077078079080
081082083084085086087088089090091092093094095096097098
099100101102103104105106107108109110111112113114115116
117118119120121122123124125126127128129130131132133134
135136137138139140141142143144145146147148149150151152
153154155156157158159160161162163164165166167168169170
171172173174175176177178179180181182183184185186187188
189190191192193194195196197198199200201202203204205206
207208209210211212213214215216217218219220221222223224
225226227228229230231232233234235236 ...

Wolfram Alpha also told me that this is a repeating decimal.  It repeats after 2997 digits.  So right after the 999 (marked in bold, underlined italics above) in the count it starts over.

WOW – it is true. 

But then I started wondering why did the number 998001 have this interesting property.  I thought if I actually did the long division I would see a pattern that would explain it.  I decided that doing it by hand (all 2997 steps) would take to long, and I was not even sure I had enough paper in the house.

Aha!  www.reallylongdivision.com!  Well, apparently, there is no such web site.  As a mathematician I could not believe it there is no www.reallylongdivision.com – but on with life. 

So I went looking to see what other interesting properties this number had.  A Google search showed me that 998001 is a perfect square (it’s 999 squared).  This seemed to be more than a coincidence.

George Polya used to say if a problem was too hard for you to solve then there was a simpler problem that I could solve.  So instead of 1/998001 I started to attack 1/9801.  It was a repeating decimal with 198 digits – still to long.  How about 1/81?  Yes, that did it.

Wolfram Alpha showed me this was a repeating decimal that has a 9 digit repeat: 1/9² = 1/81 = 0.012345679…  I noticed it skipped the digit 8.  It skipped the one that comes right before the last digit 9.  After the 9 it starts to repeat itself.  So this was a fraction that could count from 0 to 7 before it made a mistake.

Back to 1/99², or 1/9801 is a repeating decimal with a period of 198 (after 198 digits it starts to repeat).  It count two digit numbers starting from 00 and going up to 97, then skips 98, and does 99.  This fraction could count from 00 to 97 before it made a mistake, and it skipped the last number before 99.  Then it starts to repeat.

Well, that’s what 1/999² or 1/998001 does … almost.  It counts three digit numbers starting at 000 and going up to 997 before it makes a mistake.  It skips 998, and does 999 – and then it starts to repeat.

Then I started to wonder if this property continued.  It worked with 1/9² with single digit numbers (0 through 7, skip 8, and do 9 and repeat).  It worked for 1/99² with two digit numbers (00 through 97, skip 98, then do 99 and repeat).  And, it worked with 1/999² with three digit numbers (000 through 997, then skip 998, do 999 and repeat).  Would it work for 1/9999²?

Well, Wolfram Alpha did well, but it would not show the whole sequence.  It did tell me that the decimal does repeat with a period of 39996.  And it showed that the pattern does continue, at least from 0000 up to 0927.  I may have to find someone down at the university to help me out with the rest of it.

Still, it is interesting.  I suspect that the property does continue.  If it does it means that I can write a fraction that uses only integers that can (when it is written as a decimal) count to … well any number I choose, no matter how larger.  My only constraint is the ability to actually do the calculation, or rather finding a computer that can do the calculation.

If I want to count up to any number less than 8, the denominator needs to be 81 (9²).  If I want to count up to any number less than 98, the denominator needs to be 9801 (99²).  If I want to count up to any number less than 998, the denominator needs to be 998001 (999²).  Do you see the pattern?

9999² = 99980001.  What would happen if I subtracted 1.  1/(9999²-1) or 1/99980000 is a repeating decimal with a period of 357.  And it does have a surprising property:

0.00000001.000200040008001600320064012802560512102420484096819363872774554910982196439287857571514302860572114422884576...

Wow!  It is counting by powers of 2.

2⁰ = 0001, 2¹ = 0002, 2² = 0004, 2³ = 0008, …  Well, at least it did up to 4096 (212) – after that the numbers start overlapping because they get larger than 4 digits.  After 4096 you should see 8192, but what is there is 8193.  That is because the next number is 16384, a 5 digit number so the first digit (1) overlaps and is added to the 8192 to get 8193.  Well, it was nice while it lasted.

But what if I did 1/(99999² – 1):

0.000000000100002000040000800016000320006400128002560051201024020480409608192163843276865537310746214924298485969719394 …

Now you can see the 08192 (2¹³) and the 16384 (2¹⁴).  And 32768 (2¹⁵) is correct also.  But the next number should be 65536, so we are facing that overlapping problem again.  Still it is interesting.  I know what you are thinking.  You are thinking that we now know how to make a fraction, using only integers, that produces a decimal that counts by powers of two up to any number that you or I choose.

Could there be more?  What if I subtracted one again?  When I calculate 1/(9999² – 2) I get:

0.00000001000200050012002900700169040809852378574238633469080150720945696348726708829032894869302809254879068362463176...

That’s 0000, then 0001, then 0002, then 0005, then 0012, and 0029.  At first I did not recognize this number sequence.  So I called upon the resources at OEIS (The Online Encyclopedia of Integer Sequences – www.oeis.org).  I did a search for “0, 1, 2, 5, 12, 29, and I found that these were the first number in the Pell sequence (similar to the Fibonacci sequence but with slightly different rules.  You can check it out at www.oeis.org/A000129.  You will notice the next two numbers in the sequence are 70 and 169 – which matches what we see in the box above.

WOW!  I feel my mathematological muscles growing.  What else is out there?

Well, 2/(99²) starts out counting the even numbers from 00 to 96, then does 99.  After that it counts the odd numbers from 01 t0 95, does 98 and then begins repeating.  It has a period of 198 and the number “97” does not appear.

And 3/(99²) starts out counting the multiples of three from 00 to 93, then does 97 and begins to repeat.  It has a period of 66.

5/9801 (5/99²) counts by fives from 00 to 90, before losing track.

1/99998 counts by powers of two.  Now I was starting to just make up numbers and wee what might show another interesting pattern.

1/99997 counts by powers of three.

1/99996 counts by powers of four.

1/99995 counts by powers of five.

1/99994 counts by powers of six.

1/99993 counts by powers of seven.

1/997002999 (1/(999³) counts by triangular numbers (see OEIS sequence A000217).

5/997002999 counts by adding multiples of 5.  It starts at 000, then adds 000 (0 x 5) to get 000, then adds 5 (1 x 5) to get 005, then adds 10 (2 x 5) to get 015, then adds 15 to get 30, and so forth ...  This continues up to 855.  See OEIS sequence number A028895.

1/996005996001 (or 1/(999⁴)) counts by tetrahedral numbers.  See OEIS sequence number A000292.

1/9995000999900004999 (or 1/(9999⁵)) counts by pentatope numbers (pentatope numbers are the 5th number on any row of Pascal’s triangle).  These numbers are also the binomial coefficients C(n, 4) (in combinatorics these are the number of ways to choose 4 items from a group of n items).  See OEIS sequence number A000332.

Not bad for an afternoon piddling around with math.  I think my curiosity will urge me on to see what else I can find.

I am attaching an appendix of this numbers which will have to be an ongoing work in progress.  I will add to it as I find more of these numbers.

Another problem – what do I call these numbers?  I have not found anything that gives a name to these kind of numbers.  I guess, until I find out otherwise, I will have to call them “Mr. B’s Super-Duper-Fantastic, Mysteriously-Mystical and Wonderfully-Mathematical Sequence Numbers.”  It sounds good to me!  But it did not catch on.  So I changed the name to NERDY numbers.

I want to thank the good people at www.futilitycloset.com for planting the mathematical seed.  And I apologize for not trusting their post at first sight.  But it did lead to some interesting mathematics.

I also want to thank the good people at www.wolframalpha.com and www.oeis.org for providing me with some technical tools and mathematical information I needed to get the seed to grow.

A mathematical friend tipped me off to this web site: http://www.asahi-net.or.jp/~kc2h0msm/mathland/math05/ repeat05.html  – which was able to point me in other directions to search that were very fruitful.  I want to say thanks to the good people that run that web site too – it really helped to get me “jump started”. 

On my next post I will talk about some of the other interesting patterns I have found.

In the mean time – remember our vets … especially the ones that did not get to come home.  They willingly paid the price to protect us. I am reminded of the bumper sticker “If you can read this – thank a teacher”.  I will add “If you can read this in English – and not in German, Russian, Japanese, Chinese, Korean or Arabic – thank a Vet.” Be sure to let the next Vet you meet just how you feel.  Give’em a big hug, a kiss on the lips, and slap their butt – then tell them that was for a Vet that did not make it back home.

David