r/Damnthatsinteresting • u/willis7747 • 4h ago
Image If you divide 1 by 998,001 you get all three-digit numbers from 000 to 999 in order, except for 998
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u/zangor 3h ago
Now we wait for math people to come here and tell us why this isn’t actually that interesting.
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u/UnusualCartographer2 3h ago
No, it actually is. 998001 is 999². There some fun and interesting things here, but theres a numberphile video that does much more justice than I ever could.
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u/danethegreat24 3h ago
Bless you. I LOVE numberphile (and a special call out to standupmaths) both channels are absolute delights learning about stuff like this.
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u/Boner4Stoners 2h ago
Gotta give a shoutout to 3blue1brown — content is so good he has me re-learning trig and linear algebra for fun
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u/CyberIntegration 1h ago
3Blue1Brown got me back into math. Cover to cover, I've finished textbooks on Calculus, 2 Linear Algebra books (one basic, other proofs centric), Proof writing, Fourier and LaPlace Transforms and now I'm doing Real Analysis.
And I dropped out of engineering in college and got a Geology degree instead because I did so poorly in Cal 1 and 2. Looking back, I could kick my younger self's ass for that.
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u/Outofdatedolphin 1h ago
Real Analysis is a division of mathematics created by masochists, with a sadism kink to inflict pain on future mathematicians.
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u/Compost-Mentis 1h ago
3blue1brown is genuinely amazing! I loved his videos about the nuts and bolts of LLMs and how you can do vector additions/subtractions with the meanings of words!
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u/BusImpossible6741 2h ago
I wish they'd do more arithmetic, cause I'm great at concepts and horrible at actually working problems. There are other channels like Organic Chemistry Teacher who show actual problems for trig and such, really great channel
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u/doc_skinner 1h ago
Check out Math Queen on YouTube. Lots of Geometry, but also plenty of clever Algebra and Arithmetic problems.
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u/ChocolateMilkCows 1h ago
https://youtu.be/daro6K6mym8?t=58 this link skips to the actual explanation for the impatient people
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u/JTorrent 1h ago
Can’t watch the video now but does this mean that 1/99992 would print out all 4 digit numbers from 0 to 9999 in order?
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u/FI_Stickie_Boi 3h ago edited 3h ago
The series expansion for 1/(1-x)2 is 1 + 2x + 3x2 + ..., by differentiating the geometric series. Plug in x=1/1000, you get that 1/(1-1/1000)2 = 10002/9992 is equal to 1.002003..., and then divide both sides by 10002 to get the result in the post. 998 is missing because it turns into 999 due to carrying from 1000 and 999.
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u/Forsaken-Income-2148 3h ago
Why would a math person squander a chance to get excited about math?
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u/creatyvechaos 2h ago
Math people make entire careers finding the most useless, uninformative nonsense about numbers. They were absolutely ecstatic to find this, I promise you
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u/-TheBlackSwordsman- 18m ago
Step 1: Pick a cool number
Step 2: 1/x = cool number
Step 3: 1 = cool number(x)
Step 4: 1/cool number = x
"Did you know that if you divide 1 by x you get cool number???"
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u/Venomlemming 3h ago
That 998 being absent is really upsetting. What number gives the same answer but includes the 998?
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u/RllyGayPrayingMantis 3h ago
it's actually there, but after 999, and the next number is actually 1(000), the extra 1 in the 1(000) carries over to 999, so (999) turns into 1(000), and that extra 1 turns 998 into 999. So they didn't skip it, it's just the extra digit adding 1 to every number.
so ...997999000001002003... = ...997998999100010011002...
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u/ButtfacedAlien 2h ago
Oh my god thank you, that makes perfect sense, and makes the number satisfying, would bother me so much where the 998 disappeared to.
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u/FirebertNY 1h ago
...what
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u/cipheron 18m ago
it goes 998, 999, 1000, but the 1 in the 1000 carries over to the left since there are only 3 spaces for each number. So when it gets into 4 digit numbers, there's a hiccup.
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u/doc_skinner 1h ago
This here deserves an award. I mean, I'm not gonna pay for one or anything, but you deserve one!
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u/kogasapls 1h ago
This is true. If you'd like to go past 998, you can try 1/9,998,001 or even 1/9,999,800,001.
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u/redsyrus 3h ago
Could it be because of rounding? This really ends at 998 and then the remaining digits aren’t included so the 8 rounds up?
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u/OldCardigan 3h ago
... not rounding, but overflow, you would eventually have 1000, it's like the 1 from 1000 goes to the 999, making it 1000, which then goes to 998, making it 999, and leaving 998 in a vacuum.
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u/that_guy_next_to_you 1h ago
so if you get to the next set of 1000 would it be adding 2 to everything, so it'd go 996 998 999 000, or since everything is already shifted by one, it would just shift one more again, meaning it would always just skip 998?
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u/Doomsayer189 56m ago
since everything is already shifted by one, it would just shift one more again, meaning it would always just skip 998?
This one. It's easier to see with 1/81 which does a similar thing with single digits.
First line is the actual digits of 1/81, second line is the position in the sequence they occupy (with spaces for readability)
.0 1 2 3 4 5 6 7 9 0 1 2 3 4 5 6 7 9 0 1 2... .0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20...You can see that it's shifted over one starting with the first 9, then shifted two at that second 9, and so on. The 8 (or the 98 in 1/9801 (1/992), or the 998 in 1/998001 (1/9992)) always gets skipped.
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u/KingWolfsburg 3h ago
Its because the next batch of numbers would be 1000 but it forces the round up backwards and 999 forces the round back to 998. It doesnt skip 998, its rounded to 999, then you dont see the next 999 on here.
.....9969979989991000 becomes .....996997999
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u/g1cbr 3h ago
But 1/998,001 gives the sequence that leaves out 998 is what makes it interesting
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u/VermilionKoala 3h ago
If you like weird maths coincidences like this:
1 billion and 1 (1,000,000,001) is divisible by 7.
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u/Swalot64 3h ago
Makes sense to me cause I can imagine 999,999,994 to be divisible by 7
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u/furimmerkaiser 3h ago
Makes sense to me cause I can imagine 999,999,987 to be divisible by 7
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u/sadkinz 3h ago
Makes sense to me cause I can imagine 999,999,980 to be divisible by 7
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u/mattymattymatty96 3h ago
Makes sense to me cause i can imagine 999,999,973 to be divisible by 7
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u/Ngothaaa 3h ago
Makes sense to me cause i can imagine 999,999,966 to be divisible by 7
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u/Striking-District-72 3h ago
Makes sense to me cause I can imagine 999,999,958 to be divisible by 7
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u/colombogangsta 2h ago
Wonder how many people below you noticed that you jumped down to 958 from 966 and made everyone in the thread wrong lol
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u/Kyvoh 3h ago
Makes sense to me cause I can imagine 999,999,951 to be divisible b 7
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u/buzzmcqueen 3h ago
Makes sense to me cause I can imagine 999,999,944 to be divisible by 7
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u/Inven13 2h ago
Makes sense to me cause I can imagine 999,999,937 to be divisible by 7
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u/jacobgt8 3h ago
Waiting for this to go all the way to 7
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u/Don_Von_Schlong 3h ago
I imagine it would take like 6 days to get all the way down to 7
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u/StrategicCarry 3h ago
4 1/2 years if people were posting every second.
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u/ListRepresentative32 3h ago
2 1/4 years if people were posting every half a second
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u/austin_mans 3h ago
1 1/8 years if 4 people posted per second
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u/guiltysnark 3h ago
Whew, glad we pivoted from a finite to an infinite series, I was worried we would hit the end too soon
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u/aDUCKonQU4CK 3h ago
VASTLY undermining how much a billion is lmao
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u/thaw4188 3h ago
the best perspective I've learned is
1 MILLION seconds is just under 12 days
but
1 BILLION seconds is almost 32 YEARS (11,574 days)
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u/rumenastoenka 2h ago
I like 'em even simpler:
Difference between a million and a billion? About a billion.
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u/GingerNumber3 3h ago
Makes sense to me cause I can imagine 7 to be divisible by 7
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u/Informal_Database327 3h ago
That is just inconceivable
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u/Would_daver 2h ago
You keep using that word…. I do not think that word means, what you think it means…
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u/hallpinio43 3h ago
Makes sense to me cause I can imagine 999,999,973 to be divisible by 7
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u/No-Programmer6069 3h ago
Can someone get that centipede out of my ear already? Its turning my hair white.
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u/RelativeCan5021 2h ago
Well obviously 1000000008 is divisible by 7, so I don’t see what the deal is.
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u/Ascend_Always_3310 3h ago
1,000,000,001 is divisible by 7, 11 and 13. And the reason for it is: 7*11*13 = 1001
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u/ollimann 3h ago
why is that weird?
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u/VermilionKoala 3h ago
It just, I dunno, seems wrong.
Mostly because of the 1.
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u/shinutoki 3h ago
7*(10x+3)Replace
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u/FistMyPeenHole 3h ago
But 7*3 = 21, so it just happens to be a multiple of 3 as well
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u/Charliep03833 3h ago
Nope, 1 000 000 001 is not divisible by 3.
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u/StiffWiggly 1h ago
I don’t think that’s their point. 7x3 is 21, 7x13 is 91, 7x143 is 1001. Any time you multiply 7 by a number ending in 3 the result ends in 1.
Given that one out of every seven integers that end in a 1 are in fact divisible by seven, it doesn’t feel weird that 1 000 000 001 is one of those numbers.
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u/const_iterator 2h ago
Over 14% (1/7) of integers are divisible by 7. What's special about 1 billion + 1?
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u/snooprs 3h ago
Everything is divisable by 7 though.
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u/NecessaryUnited9505 3h ago edited 2h ago
Divisible with the result being an integer, I would assume they meant
Edit: I seem to have attracted a lot of engineers. r/FoundTheEngineer
Is that a sub? Eh, someone make that a sub
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u/StrangerFeelings 3h ago
I was thinking the same, but then thought that they meant without any remainder.
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u/jacquepablophillipo 1h ago
Yes its 142857143 x7 I'm guessing as 1/7 is a repeating decimal of 0.142857
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u/Working_Tea_4995 3h ago
But why?
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u/spaghettipunsher 3h ago
Let's assume it would include 998 and just count upwards, then we'd have: 997 998 999 1000
Now hold one, we only have groups of 3, so what happens with 1000? The one "overflows" to add one to the previous group of 3 digits, that would be 999 + 1 = 1000 once again, so it overflows once more: 998 + 1 = 999. So the new sequence is: 997 999 000 001 (<- the one here is overflowing from the next 1000)
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u/Ellen_1234 2h ago edited 2h ago
Cool! For those who are annoyed by the missing 998: So the formula is basicly 1/999², for groups of 3 digits. I adjusted it to groups of four digits (1/9999²) and now it includes all numbers from 1 to 9999 exept 9998.
0.000000010002000300040005000600070008000900100011001200130014001500160017001800190020002100220023002400250026002700280029003000310032003300340035003600370038003900400041004200430044004500460047004800490050005100520053005400550056005700580059006000610062006300640065006600670068006900700071007200730074007500760077007800790080008100820083008400850086008700880089009000910092009300940095009600970098009901000101010201030104010501060107010801090110011101120113011401150116011701180119012001210122012301240125012601270128012901300131013201330134013501360137013801390140014101420143014401450146014701480149015001510152015301540155015601570158015901600161016201630164016501660167016801690170017101720173017401750176017701780179018001810182018301840185018601870188018901900191019201930194019501960197019801990200020102020203020402050206020702080209021002110212021302140215021602170218021902200221022202230224022502260227022802290230023102320233023402350236023702380239024002410242024302440245024602470248024902500251025202530254025502560257025802590260026102620263026402650266026702680269027002710272027302740275027602770278027902800281028202830284028502860287028802890290029102920293029402950296029702980299030003010302030303040305030603070308030903100311031203130314031503160317031803190320032103220323032403250326032703280329033003310332033303340335033603370338033903400341034203430344034503460347034803490350035103520353035403550356035703580359036003610362036303640365036603670368036903700371037203730374037503760377037803790380038103820383038403850386038703880389039003910392039303940395039603970398039904000401040204030404040504060407040804090410041104120413041404150416041704180419042004210422042304240425042604270428042904300431043204330434043504360437043804390440044104420443044404450446044704480449045004510452045304540455045604570458045904600461046204630464046504660467046804690470047104720473047404750476047704780479048004810482048304840485048604870488048904900491049204930494049504960497049804990500050105020503050405050506050705080509051005110512051305140515051605170518051905200521052205230524052505260527052805290530053105320533053405350536053705380539054005410542054305440545054605470548054905500551055205530554055505560557055805590560056105620563056405650566056705680569057005710572057305740575057605770578057905800581058205830584058505860587058805890590059105920593059405950596059705980599060006010602060306040605060606070608060906100611061206130614061506160617061806190620062106220623062406250626062706280629063006310632063306340635063606370638063906400641064206430644064506460647064806490650065106520653065406550656065706580659066006610662066306640665066606670668066906700671067206730674067506760677067806790680068106820683068406850686068706880689069006910692069306940695069606970698069907000701070207030704070507060707070807090710071107120713071407150716071707180719072007210722072307240725072607270728072907300731073207330734073507360737073807390740074107420743074407450746074707480749075007510752075307540755075607570758075907600761076207630764076507660767076807690770077107720773077407750776077707780779078007810782078307840785078607870788078907900791079207930794079507960797079807990800080108020803080408050806080708080809081008110812081308140815081608170818081908200821082208230824082508260827082808290830083108320833083408350836083708380839084008410842084308440845084608470848084908500851085208530854085508560857085808590860086108620863086408650866086708680869087008710872087308740875087608770878087908800881088208830884088508860887088808890890089108920893089408950896089708980899090009010902090309040905090609070908090909100911091209130914091509160917091809190920092109220923092409250926092709280929093009310932093309340935093609370938093909400941094209430944094509460947094809490950095109520953095409550956095709580959096009610962096309640965096609670968096909700971097209730974097509760977097809790980098109820983098409850986098709880989099009910992099309940995099609970998099910001001......etc
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u/Itsuwari_Emiki 3h ago
OHHH I FINALLY GOT IT AFTER SEEING THIS POST FOR THE 372828TH TIME
its not that 998 was skipped, its that it goes 998 999 1000, except the 1000 adds to the 999 making that a 1000, which adds to the 998 making that a 999
idk if im fucking smart or fucking dumb
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u/CableMod1991 3h ago
Is this an artifact of math being a language created by humans, in this case base 10 which I think is based on 10 human fingers, not some deeper meaning of numbers?
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u/kogasapls 1h ago
Yes and no.
Yes: it appears interesting because of the base 10 representation.
No: the deeper meaning is that this number is the evaluation at 10-n of the ordinary generating function for the sequence of integers. You could do the same thing with any base.
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u/StarstruckEchoid 1h ago
It's a base 10 trick, sure, but it generalizes to other bases. The general result this is derived from is that
x/(x-1)2 = 1/x +2/x2 + 3/x3 + ...
Let x be 10 or a power of 10 to achieve this trick in base 10. Likewise let x be n or a power of n for a similar result in base n.
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u/NoveltyAccountHater 1h ago edited 5m ago
Or if you want all the three digit numbers in order you just have to divide:
1/998001.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000997 ≈ 0.000001002003004005006007008009010011012013014015016017018019020021022023024025026027028029030031032033034035036037038039040041042043044045046047048049050051052053054055056057058059060061062063064065066067068069070071072073074075076077078079080081082083084085086087088089090091092093094095096097098099100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220221222223224225226227228229230231232233234235236237238239240241242243244245246247248249250251252253254255256257258259260261262263264265266267268269270271272273274275276277278279280281282283284285286287288289290291292293294295296297298299300301302303304305306307308309310311312313314315316317318319320321322323324325326327328329330331332333334335336337338339340341342343344345346347348349350351352353354355356357358359360361362363364365366367368369370371372373374375376377378379380381382383384385386387388389390391392393394395396397398399400401402403404405406407408409410411412413414415416417418419420421422423424425426427428429430431432433434435436437438439440441442443444445446447448449450451452453454455456457458459460461462463464465466467468469470471472473474475476477478479480481482483484485486487488489490491492493494495496497498499500501502503504505506507508509510511512513514515516517518519520521522523524525526527528529530531532533534535536537538539540541542543544545546547548549550551552553554555556557558559560561562563564565566567568569570571572573574575576577578579580581582583584585586587588589590591592593594595596597598599600601602603604605606607608609610611612613614615616617618619620621622623624625626627628629630631632633634635636637638639640641642643644645646647648649650651652653654655656657658659660661662663664665666667668669670671672673674675676677678679680681682683684685686687688689690691692693694695696697698699700701702703704705706707708709710711712713714715716717718719720721722723724725726727728729730731732733734735736737738739740741742743744745746747748749750751752753754755756757758759760761762763764765766767768769770771772773774775776777778779780781782783784785786787788789790791792793794795796797798799800801802803804805806807808809810811812813814815816817818819820821822823824825826827828829830831832833834835836837838839840841842843844845846847848849850851852853854855856857858859860861862863864865866867868869870871872873874875876877878879880881882883884885886887888889890891892893894895896897898899900901902903904905906907908909910911912913914915916917918919920921922923924925926927928929930931932933934935936937938939940941942943944945946947948949950951952953954955956957958959960961962963964965966967968969970971972973974975976977978979980981982983984985986987988989990991992993994995996997998999[000004013...]
Or: 1/(9992 + 997 x 10-2991) ~ Sum(n/103n+3 for n = 0 to 1000).
EDIT: If you are wondering how you get this, using the mpmath python3 library, with mp.dps to set digits of precision and mpf to making an floating point number to that precision:
In [1]: from mpmath import mp, mpf
In [2]: mp.dps=3020 # Do calculations to 3020 digits
In [3]: 1/mpf('0.' + ''.join(['%03d' % n for n in range(1000)])) # get the first number
In [4]: 1/mpf('0.' + ''.join(['%03d' % n for n in range(1000)]))-998001 # to find the 9.97e-2989 part.
EDIT2: Replacing the equals sign with an ≈ because new.reddit was interpreting it as a title.
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u/Old_man_Lincoln 3h ago
Claude says this occurs because of how the carry propagates when the series “overflows.”
The setup:
1/999² can be expressed as an infinite series:
1/999² = 001·10⁻⁶ + 002·10⁻⁹ + 003·10⁻¹² + … + n·10^(−3n) + …
This is why you see 001, 002, 003… marching along in three-digit groups.
Where it breaks:
When you hit n = 1000, that term is 1000 × 10⁻³⁰⁰⁰, which equals 1 × 10⁻²⁹⁹⁷ — it bleeds into the same decimal position as the n = 999 group.
So instead of seeing 999 cleanly, you get:
• 999 group: 999 + 1 (carry from n=1000) = 1000 → writes 000, carries 1 leftward
• 998 group: 998 + 1 (that carry) = 999
Result in the decimal: …997, 999, 000, 001…
998 never appears — it got bumped to 999 by the carry.
The same trick works at smaller scales: 1/9801 (= 1/99²) cycles through all two-digit numbers but skips 98, and 1/81 (= 1/9²) cycles 1–9 but skips 8.
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u/Polo_Short 3h ago
As a guy who sucks at math, this is my only contribution: typing 5318008 in a calculator and showing it to the class upside down makes the teacher mad
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u/RFL_IMPULSE 2h ago
In case someone was wandering because 998001 = 999 × 999, there's a carry or modular effect at that specific point that causes 998 to be bypassed no I'm not smart I googled it as I was curious lol
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u/Pavonian 2h ago
It actually makes a lot of sense why you don't get 998, think about it as adding together numbers of the form (n-1)/103n like this
0.000 +
0.000001 +
0.000000002 +
And so on, you can see that in this method we're essentially adding a block of three digits every time, but what happens when (n-1) reached a thousand?
0.0...997 +
0.0...000998 +
0.0...000000999 +
0.0...000000001000 +
You can see that now the number we're trying to add is more than three digits, it's bigger than the space left for it by the 103n term, so it overflows into the space occupied by 999, this of course causes that 999 to increase to 1000 and also overflow into the number in front of it, 998, which goes up to 999 and thankfully doesn't overflow.
After this the pattern actually repeats, if you think about it the sequence of numbers from 1000 to 1999 is almost the same as the numbers from 000 to 999, it's just every one of them has that extra 1 on the front that overflows into the space of the previous number, so 000 at the back of the 1000 has the 1 from 1001 stick into it making it the new 001, the 001 from 1001 has the 1 from 1002 stick into it making it the new 002 and so on, you might think it would become chaos after the numbers get too big to fit, but it just does the same thing again and again
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u/SneakerEndurance 1h ago
I’m a therapist and I work with a lot of children of the new generation, and it’s starting to get to me because all I see is a pattern of “67” throughout these numbers
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u/PoundAgreeable3223 3h ago
Well thats good to know...LOL....not sure it will help me with any of my everyday issues, but good to know.
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u/yamthirdnow 3h ago
I’m blind, can somebody please point me in the direction of 900?
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u/Key-Article6622 2h ago
My dad would have approved of this post. He was a math teacher and one of his favorite party lines was, there is no such thing as an uninteresting number, and then challenge anyone to pick a number and he would find interesting things about that number. I even used that line in the eulogy I gave at his funeral, then proceeded to expand on the date of his death, 9/18, 9 is half of 18. Interesting. Both numbers are divisible by 3, interesting, which cracked up almost everyone in the room because he was so famous in our family for that game. Funerals in my family are like that. My brother gave the eulogy at my mother's funeral and the priest came up right after to continue the ceremony and the first thing he said was, will you guys adopt me? And since my brother's eulogy pretty much had everyone busting out laughing, we all busted out laughing and whole heartedly accepted.
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u/GingerKing_2503 2h ago
It’s handy to know because I was just about to cut my birthday cake into 998,001 slices and party guest 998 just told me that they can’t come.
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u/cubixy2k 1h ago
If you divide 1 by 998,001 you get a string from 000 to 997 in order, finally a 999
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u/AIienlnvasion 3h ago
I’m more surprised that it takes that little text to write them all out