M :)Dividing by 7: A Pattern Revealed

[JPC]

I realized a pattern in the division of an integer n by 7.

the repeating patterns :
1/7 : 142857
2/7 : 285714
3/7 : 428571
4/7 : 571428
5/7 : 714285
6/7 : 857142

in each one, the are just permutations of the numbers : 1, 2, 4, 5, 7, 8

from left to right :
: +1, +2, +1, +2, +1

Is there a significance to this ?

 

[praharmitra]

yes, there is! you have discovered what are called CAROUSEL NUMBERS...

these are numbers such as 142857...and many more, with the amazing property that each multiple is cyclic permutation of the nuumber itself..

these numbers are generated in the following way..

take a prime p...and find the repeating decimal of 1/p...if that contains (p-1) digits, then the repeating decimal is a carousel number.

for example 1/7 = 0.142857... contains 6 decimals, meaning 142857 is a carousel number

similarly 1/13 = 0.0588235294117467, and so on contains 16 digits...this means that it too is a carousel number...try multiplying it with numbers from 1 to 16 and see what u get...note the importance of the zero in the beginning!

 

[JPC]

similarly 1/13 = 0.0588235294117467, and so on contains 16 digits...this means that it too is a carousel number...try multiplying it with numbers from 1 to 16 and see what u get...note the importance of the zero in the beginning!

you mean 1/17 and not 1/13 ?

i never realized there was a pattern for all numbers of the type ' 1 / prime'
I found out because of a program that calculates as many decimals i want that i made

but, is there a way to use this property to calculate a number of the the type ' 1 / prime'
for example : ''1/23' without a calculator, and without doing a a division

 

[praharmitra]

you mean 1/17 and not 1/13 ?

i never realized there was a pattern for all numbers of the type ' 1 / prime'
I found out because of a program that calculates as many decimals i want that i made

but, is there a way to use this property to calculate a number of the the type ' 1 / prime'
for example : ''1/23' without a calculator, and without doing a a division

ya well, sorry...i mean 1/17...

however, be sure that not all primes have that property...only some...

see, 1/11 = 0.09090909...so it doesn't have 10 repeating decimals...

without a calc?? no clue..but will think about it

 

[JPC]

like for example

1/7 : 142857
2/7 : 285714
3/7 : 428571
4/7 : 571428
5/7 : 714285
6/7 : 857142

i see that all the numbers are following an order :
1 - > 4 -> 2 - > 8 - > 5 - > 7 -> come back to 1 -> ect
its just the starting number that changes

and the starting number is in order from lowest to bigest : 1, 2, 4, 5, 7, 8
1/7 : 142857
2/7 : 285714
3/7 : 428571
4/7 : 571428
5/7 : 714285
6/7 : 857142

So, for example, if you know 1/7, you can determine any n/7 very fast

 

[Kaimyn]

Heh, 5 is a carousel number... In theroy:

$$1/2$$ = 0.5
0.5*10^(2-1) = 5
5*n=5

n=an integer $$\geq$$1, but <2.
As the only integer can be 1, then 5 is a carousel number because it "rearranges" to make a number that uses the same numbers:

5*1=5

But then again, it doesn't recurr, so is it a carousel number?

 

[DeaconJohn]

yes, there is! you have discovered what are called CAROUSEL NUMBERS...

these are numbers such as 142857...and many more, with the amazing property that each multiple is cyclic permutation of the nuumber itself..

Praharmitra,

I didn't know that!

(Or, if I did, I'd completely forgotten aboput them, and that amounts to the same thing.)

Thanks for telling us about them.

If, perchance, I did know about them, I sure didn't know as much as is reported in the following link.

http://mathforum.org/orlando/klatt.orlando.html

According to this link, there are all kinds of inteeresting open questions about them. (The link claims they are open; I think they are interesting.)

DJ

 

[DeaconJohn]

Heh, 5 is a carousel number... In theroy:

$$1/2$$ = 0.5
0.5*10^(2-1) = 5
5*n=5

n=an integer $$\geq$$1, but <2.
As the only integer can be 1, then 5 is a carousel number because it "rearranges" to make a number that uses the same numbers:

5*1=5

But then again, it doesn't recurr, so is it a carousel number?

Not according to the following link,

http://mathforum.org/orlando/klatt.orlando.html

 

[tiny-tim]

vedic mathematics

but, is there a way to use this property to calculate a number of the the type ' 1 / prime'
for example : ''1/23' without a calculator, and without doing a a division

Hi JPC!

There are several fun books on vedic mathematics which will give you a simple mental arithmetic way of doing it.

 

[JPC]

do you have the names ?

 

[DeaconJohn]

do you have the names ?

google on vedic mathematics. you'll find a westerner who did a similar thing. it appears to me that the westerner's exposition is easier to understand. unless you want to learn a lot of quotes from ancient vedic literature.

 

[praharmitra]

hey deaconjohn, that was a very informative link...thnx...

i myself did some research on carousel numbers, trying to prove, atleast by example that every number can be written as "partial carousel number"(a word made up by me) by adding a few zeros before it...and multiplying by appropriate numbers.

 

[DeaconJohn]

hey deaconjohn, that was a very informative link...thnx...

You're welcome. It's amazing how minor variations in a google search - even including the specific IP address from which you search - can vastly influence the results.

i myself did some research on carousel numbers, trying to prove, atleast by example that every number can be written as "partial carousel number"(a word made up by me) by adding a few zeros before it...and multiplying by appropriate numbers.

Hey. write it up so we PhysicForum-ites can see it!

 

[Gokul43201]

google on vedic mathematics. you'll find a westerner who did a similar thing. it appears to me that the westerner's exposition is easier to understand. unless you want to learn a lot of quotes from ancient vedic literature.

Trachtenberg?

 

[DeaconJohn]

Trachtenberg?

Yeah, That's it. DJ