Math nerds: Why does this happen?

[apblog]

So I came across this strange bit of math today.

I needed to know the frequency of a 26ns period.  Which turns out to be:

38,461,538.461538(repeating).   Ok, not that strange.  Now multiply it by 4:

153,846,153.846153(repeating).  Same digits repeating, same order.  Multiply by 4 again:

615,384,615.384615(repeating).  And again:

2,461,538,461.538461(repeating).  Now there is a leading 2, but the same digits repeat.

How does this work?  Why four?  Why 26? 

Just curious if there is an interesting math principle at work here.  Just dumb coincidence?

I'm not that great at math so apologies if this is something really basic.

Thanks!

[Decoman]

I don't know much about math but this seems familiar. Maybe related to 'cyclic numbers'
https://en.wikipedia.org/wiki/Cyclic_number



Maybe fractions and multiplication in a way being two sides of the same coin so to speak, in a decimal number system. *shrugs*
In the video, it is stated that some third of all the prime numbers "will produce cyclic numbers".

At the end of the video, the guy mention something called 'enneagram'.
https://en.wikipedia.org/wiki/Enneagram_(geometry)

Various related youtube channels:
numberphile
computerphile
sixtysymbols
singingbanana

[RoGeorge]

Any periodic (repeating number) multiplied with any other number will produce a similar result, because multiplication is repeated addition.

As a general rule, combining any pattern with itself will also give a pattern.

[chriswebb]

So I came across this strange bit of math today.

I needed to know the frequency of a 26ns period.  Which turns out to be:

38,461,538.461538(repeating).   Ok, not that strange.  Now multiply it by 4:

153,846,153.846153(repeating).  Same digits repeating, same order.  Multiply by 4 again:

615,384,615.384615(repeating).  And again:

2,461,538,461.538461(repeating).  Now there is a leading 2, but the same digits repeat.

How does this work?  Why four?  Why 26? 

Just curious if there is an interesting math principle at work here.  Just dumb coincidence?

I'm not that great at math so apologies if this is something really basic.

Thanks!

What you are looking at is just a quirk of dividing by 26 (really 13).

1 000 000 000 / 26 = 38,461,538.461538 | 1 000 000 000 mod 26 = 12

4 000 000 000 / 26 = 153,846,153.846153 | 4 000 000 000 mod 26 = 22

16 000 000 000 / 26 = 615,384,615.384615 | 16 000 000 000 mod 26 = 10

64 000 000 000 / 26 = 2,461,538,461.538461 | 64 000 000 000 mod 26 = 14

So none of these numbers are congruent mod 26. And definitely not congruent mod 13.

While one of these repeating terms may represent a cyclical number (haven't found one yet), 1/26 isn't cyclical nor are the others i checked.

You can look at the different fractions of 26 from 0/26 to 26/26 to see what the repeating decimals will be.

And when you get to a multiple of 13 like 63 999 999 999 you get either .5 as the decimal or nothing because its divisible.

I don't really see anything interesting here, because its not like it is ALWAYS the same. So mostly I'd just attribute this to a quirky decimal representation. But it's been a while since number theory.

[apblog]

Thanks everyone.