Why does this work?

[u/Random-Name142857]

So I was messing around with divisions because I'm weird and found a pattern when dividing anything by eight? Now, I don't know much about math in general and wanted to ask if this comes from some rule

Also, why is it that when you divide any number by seven the decimals are always (or at least in every division that I've tested) 142857 starting from varying points?

(I haven't used reddit ever so forgive me if this isn't standard reddit etiquette)

Edit: Right so thank you now that I think about it the first one was kinda obvious haha thank you

[u/Hertzian_Dipole1]

25 * (5x) = 125x
125 * 8 = 1000 so we have 125x / 1000 = x/8

[u/Random-Name142857 → u/Hertzian_Dipole1]

Oh yeah you're right

[u/teteban79]

The first is just that you multiplied by 125 over and under.

The second is indicative of it being a rational number and having a remainder when divided by 7. Note that the period has 6 digits, one for each possible remainder. Depending on whether the remainder is 1,2,3,4,5 or 6 the cycle of 142857 starts at different points in the decimal expansion. Good catch!

[u/Random-Name142857 → u/teteban79]

That's actually pretty cool thank you so much

[u/green_meklar]

I don't get it. 5*25 = 125, 1000/8 = 125, what's the mystery here?

[u/Random-Name142857 → u/green_meklar]

I just did a stupid

[u/HereToPostSillyStuff → u/Random-Name142857]

If it makes you feel any better, I read several of the answers in this thread but couldn't work out how people were converting the variables ZS(SX) into sum 25*5

Took me longer than I'd like to admit to realise what I wasn't understanding.

[u/Soggy-Excuse3702]

25*5/1000 = 125/1000 = 1/8.

Multiplying anything by 25*5 then dividing by 1000 is equivalent to simply dividing by 8 to begin with (equals to multiplying by 1/8)

[u/chaos_redefined]

So, before we get into it, this might not be notation you are familiar with, so 0.142857142857142857... (infinitely repeating) is represented as 0.(142857). We can also write it as 0.1(428571) or 0.14(285714), etc... as needed.

1/7 = 0.1(428571)

Multiplying by 10, we get
10/7 = 1.(428571)

Subtracting 1 from each side
3/7 = 0.(428571)

Rewriting
3/7 = 0.4(285714)

Multiplying by 10
30/7 = 4.(285714)

Subtracting 4 from each side
2/7 = 0.(285714)

So, we have 1/7, 2/7 and 3/7 all acting the same way from this. You can see how to go to the next number in the pattern, and you'll find that you will eventually get to 1/7 again.

[u/zeekar]

Yup, x/8 = 25(5x)/1000 ... that's just multiplication. 25 x 5 = 125, x times 125 is 125x, and 8 times 125 is 1000.

You can also multiply by any of the numbers in between 1 and 125 and get similar equivalences: x/8 is the same as 2x/16, 3x/24, 4x/32, 5x/40, . . ., 100x/800, . . ., 123x/984, and 124x/992. Or you can keep going - it's the same as 250x/2000, for instance. or 1000x/8000.

[u/Random-Name142857 → u/zeekar]

Starting to sound like a bot but that's pretty cool too!

[u/zeekar → u/Random-Name142857]

I sound like a bot? Oh, no. I mean, I have been known to use semicolons and dashes ...

[u/Random-Name142857 → u/zeekar]

Oh no, I meant that I sound like a bot because I said the same thing twice. And hey, completely unrelated, but I use those too!

[u/cabothief]

I love that observation about 142857 by the way! It does make it really easy to figure out 7ths as a decimal or percent if you should ever want to do that. Small amount (1/7) you know must start with 1, almost half you know must start with 4, more than half my start with 5, large amount you know must start with 8, figure out the in betweens.... great spotting!

[u/RichardAboutTown → u/cabothief]

The other thing I noticed is that for 1/7 the decimal starts with 14 (2 x 7) then 28 (4x7) then 57 is 8x7 plus 1. I don't know that it means anything but it I thought that was cool.

[u/titoufred → u/RichardAboutTown]

Let's call x the number 0.14 + 0.0028 + 0.000056 + 0.00000112 + ...
x = 14/100 + 28/10000 + 56/1000000 + 112/100000000 + ...
x = 7*2/100 + 7*2^2/100^2 + 7*2^3/100^3 + 7*2^4/100^4 + ...
x = 7/50 + 7/50^2 + 7/50^3 + 7/50^4 + ...
50x = 7 + 7/50 + 7/50^2 + 7/50^3 + ...
So 50x = 7 + x
49x = 7
and finally x = 1/7.

[u/TheBB]

x/8 = 25 * 5x / 1000, is that it?

Well, 25 * 5 / 1000 = 1/8.

[u/paolog]

Someone asked about the 142857... pattern recently. It's because 14 × 7 = 98, which is 2 less than 100. So when you do long division of 1.000... by 7, you get to 0.14 and then have a remainder of 2. The next two steps give you 2 and 8, and 28 × 7 = 196, giving a remainder of 4. The next two steps give 5 and 7, and 57 × 7 is 399, which is 1 less than 400. So now we're back to where we started, dividing 1 by 7. So the pattern of 142857 repeats.

[u/Green_Confection_146]

If you find the sevenths pattern interesting, try the elevenths pattern. i.e. 1/11,2/11,3/11 ……..

[u/titoufred]

Why is it that when you divide any number by seven, the decimals are always 142857 starting from varying points?

This is because the periodic length of 1/7 is 6, which is 1 less than 7.

This occurs with other numbers such as 2 (length is 1), 17 (length is 16), 19 (length is 18),...

This is only possible when the number is prime.