decimal to fraction

John E. Franklin · 2014-02-17 08:02:03

Given the following number: 11.12346879264332643326433264332643326433(26433...)

Can someone help me convert this to a fraction with the numerator and denominator both counting numbers?

Any hints welcome?

Or a complete exercise would be awesome too.

I haven't looked into it yet, so maybe you can beat me to it, otherwise I'll start researching soon...

anonimnystefy · 2014-02-17 08:22:49

Hi JEF

John E. Franklin · 2014-02-17 08:33:41

thank you very much! You beat me to it! I just got to 26/99 = 0.262626262626(26...), so thanks a bunch.
I don't have a use for it today, but I was watching Norman Wildberger doing math on youtube and he mentioned it could be done and in college I had seen the 9999 trick used once in a lecture, but had forgotten the details.

anonimnystefy · 2014-02-17 08:47:49

You're welcome!

John E. Franklin · 2014-02-17 09:15:22

Along the same lines as repeating decimals. Do you think the 9999 or 999999 or 9999999999 numbers have prime factors that might be somehow interesting because we know that certain fractions like 3/7 repeats in fewer than 7 digits?

Info I found so far. 999 x 1001 = 999999

9 = 3 x 3
99 = 3 x 3 x 11
999 = 37 x 3 x 3 x 3
9999 = 99 x 101 = 3 x 3 x 11 x 101
99999 = 271 x 41 x 3 x 3
999999 = 999 x 1001 = (37 x 3 x 3 x 3) x (7 x 11 x 13)
9999999 = Help me??
Can anyone help do more longer ones of these??? (any source welcome!)

bobbym · 2014-02-17 09:31:52

Hi John;

You are working with 9 times a repunit (1111...). To factor them you will need to use a program. You can find some on the internet or wolfram alpha.

John E. Franklin · 2014-02-17 09:54:20

Thanks bbbm! I forgot about wolfram alpha, haven't used it in a year.
seven 9's = 3 x 3 x 239 x 4649
eight 9's = 3 x 3 x 11 x 73 x 101 x 137
nine 9's = 3 x 3 x 3 x 3 x 37 x 333667
ten 9's = 3 x 3 x 11 x 41 x 271 x 9091
eleven 9's = 3 x 3 x 21649 x 513239
twelve 9's = 3 x 3 x 3 x 7 x 11 x 13 x 37 x 101 x 9901
thirteen 9's = 3 x 3 x 53 x 79 x 265371653
Well, I'm a little disappointed at those huge numbers like 21649 and 265371653, but whatever!?

fifty-nine 9's = 3 x 3 x 2559647034361 x 4340876285657460212144534289928559826755746751
The odd numbers of nines seem to have longer answers, wow I can't believe this 45 digit number
they know is prime: 4340876285657460212144534289928559826755746751

The thing I don't get is like if you take a fraction like say numerator and denominator are between
eighty and two hundred each, then you get these repeating decimals I've been told that are quite long
at for some examples. How is one with this 99999999 method of conversion going to then factor the
fraction into simplest terms. I suppose it must work, but I can't imagine it working with the data above.
Is my brain jumping to conclusions here?? What is your take on this??
Maybe I should try an example with Wolfram like 89/151 and see what I learn...
Well it repeats on a 75 digit period and Wolfram Alpha can go backwards with the following!!! and reduce it!!! wow!!! nice software!!
589403973509933774834437086092715231788079470198675496688741721854304635761/999999999999999999999999999999999999999999999999999999999999999999999999999
and so I tried this out:
999999999999999999999999999999999999999999999999999999999999999999999999999/6622516556291390728476821192052980132450331125827814569536423841059602649 and I got 151!!!!!!
89 times this 6622516556291390728476821192052980132450331125827814569536423841059602649 is the 589403....numerator or repeating part.
So let's see if I learned anything.... I divided 89 by 151 and got a 75 repeating period decimal starting with 58940397....35761 as shown above. Then I divided that 75 digit number with Wolfram by 75 nine's and I got my fraction back, that was startling that wolfram knew that, but when I entered 75 nines into wolfram alpha, it was too long to give me all the prime factors for whatever reason. So next I divided the 75 period number 5894....761 by 89 just to see if Wolfram could do anything with it, and low and behold it gave me a 73 digit number shown above starting 66225165...2649 and so I took that and divided 75 nines by that number and I got my 151 exactly back!!!!!
Welp. That sure was a confusing little experiment. I don't really know what to say.... Anyone have any comments on this???

Last edited by John E. Franklin (2014-02-17 10:32:14)

John E. Franklin · 2014-02-17 10:50:31

Hey, get this!!!! I divided seventy-five nine's by 151 and it came out no remainder with wolfram alpha,
so I take this to mean 151 is a factor. Now isn't this something important here we are stumbling upon?
For example if I divide other fractions similar to 89/151 with wolfram, then might I find more factors
that are my denominators for large repeating 99999999-numbers?? Because in this case, wolfram
wasn't displaying prime factors anymore somewhere between 50 and 75 digit numbers, but I
managed to squeeze out the 151 number. Oh well I knew there was something here to learn, and
I think that was it right there. Take for example six nines (999999), I predict you can divide that by 7
because 1/7 is repeating on a period of six digits... Wow, and since this is a good tiny find, please
don't move this post up into the other one, it'll just make it more math.

I'm not done baffling you with the math math, because just imagine, what could be done if we
change to hundreds of other number bases!! Now the number that is one less than 100000 or 1000000,
for example, will be not what we now call 999, but in base-7, it would be 343 - 1 or 342. See
what I'm up to??? Fractions in many bases could help aid in finding prime factors of 999 or

Comments always welcome!!

Okay... I filled up a page on my pad of paper doing division in base-4. I divided 156 by 74 or 2130/1022 in base-4.
I got

So it repeats every 18 digitis. So I'm totally guessing
as usual for fun (math is fun

) (why not guess!? an fix it later) that 4**18 - 1 is divisible by 74, but actually it
turns out wolfram alpha says 74/2 is a prime factor and 2 is not since it was odd. Oh well, good first try anyway.
The 3333333333333-method in base-4 might work like the 999999-method in base-10. Any ideas how to
do that one? I'm not up for that division tonight.

Last edited by John E. Franklin (2014-02-17 12:27:42)

John E. Franklin · 2014-02-18 10:56:51

Today I did another one in base-4 by hand.
Here it is:

The long hand division revealed this quotient with a repeating decimal of length 9:

From this I am guessing that 4^9 - 1 is divisible by 19, and wolfram alpha says "yes".

The same fraction is base-10 can be done with wolfram: 13/19 and
then you get a repeating part 18 long instead of 9 in base-4.
And 10^18 - 1 is also divisible by 19, the denominator, says wolfram alpha.

So this is pretty neat, if it always works, I'm not sure.
Anyone know if this always works in all bases or always in base-10??

Last edited by John E. Franklin (2014-02-18 11:04:25)

Math Is Fun Forum

#1 2014-02-17 08:02:03

decimal to fraction

#2 2014-02-17 08:22:49

Re: decimal to fraction

#3 2014-02-17 08:33:41

Re: decimal to fraction

#4 2014-02-17 08:47:49

Re: decimal to fraction

#5 2014-02-17 09:15:22

Re: decimal to fraction

#6 2014-02-17 09:31:52

Re: decimal to fraction

#7 2014-02-17 09:54:20

Re: decimal to fraction

#8 2014-02-17 10:50:31

Re: decimal to fraction

#9 2014-02-18 10:56:51

Re: decimal to fraction

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