Decimal Representation of 1/7

Factual Questions
1

I’m not exactly sure that I have a specific question, but I’m curious if anyone has anything interesting or informative about the strange decimal representation of 1/7.

1/7 = 0.142857142857…

The repeating portion consists of the digits 142857. If we break those up into three 2-digit numbers, we get 14, 28, and 57.

28 is twice 14, and 57 is almost twice 28. (Also, 14 is twice 7, the original denominator.)

If we consider doubling, say, 14.2 and rounding, we get 28. If we double 28.5 and round, we get 57 (well, we get that without rounding).

Those are definitely some curious properties, and I find it hard to believe that the relationships in the decimal expansion are just a coincidence, but I couldn’t fathom why these relationships show up in the expansion.

There’s also the fact that there are 6 repeating digits and by moving the decimal appropriately, this series of digits comprises the representation for 2/7, 3/7, 4/7, 5/7, and 6/7.

(For example, 2/7 = 0.285714…)

I don’t see this particular behavior in any other decimal expansions.

Is this truly just a coincidence? If not, is there some sort of mathematical explanation or reasoning?

Any info or thoughts would be awesome.

2

The second fact you observed explains the first one. 1/7 is 0.142857…, or approximately 0.14, so double that, 2/7, is 0.285714…, or approximately 0.28. There’s your first doubling. And double that is 4/7, 0.571428…, or approximately 0.57, which is close to 0.56, twice the approximation for 2/7.

3

“Nothing” in arithmetic is really a coincidence since it all follows from first principles. The properties of 1/n have been studied. Here’s a pdf with more than you care to know probably.

www.fq.math.ca/Scanned/11-1/brousseau.pdf‎

4

Years ago, Martin Gardner did a column on these so-called cyclic numbers. The same thing happens with 1/17, 1/19, and some other reciprocals of prime numbers. Here is a link to the Wikipedia article on cyclic numbers.

Cyclic Numbers

5

Are cyclical numbers always prime?

6

Even more fun than 1/7 is 1/49, for similar reasons:



 1/49=   
  0.     
    02      
      04         
        08
          16
            32
              64
               128
                 256
                   512
                     ...
------------------------
  0.0204081632653061...
7

Do you remember how to do long division? Assuming the dividend is less than 7, you’re going to have to add a 0 to the dividend, so ignoring the decimal point, it’ll be 10/7, 20/7, etc. Thus there are only 6 possible values for the first digit, 1, 2, 4, 5, 7, and 8. The remainder from this division will also always be 1 to 6 and you’ll add another 0, so you’ll always end up with one of these digits. It’s not hard to see why it follows a pattern. 10/7 is 1 with a remainder of 3, 30/7 is 4 with a remainder of 2, 20/7 is 2 with a remainder of 6, and so on until it makes a circle back to 10/7.

From this, it appears that division that results in non-terminating decimal representations always result in repeating patterns with a number of digits less than the divisor.

8

By cyclical, I’m guessing you mean non-terminating with a repeating pattern. In that case, no. Any divisor that doesn’t have solely 2 and/or 5 as prime factors and doesn’t divide evenly into the dividend will have a repeating pattern.

9

For a modern audience, a video may be better. Here’s a Numberphile video on the subject.

10

They are devil numbers…

11

Fun “Mathemagic” trick

Get someone to write down a 9 digit number on a piece of paper. Tell them to choose “at random” (whatever that means in this context.) Underneath you write your “random” nine digit number – 142857143

Multiply the two numbers. You can magically do it in one line writing from left to right with no working shown.
Here is how it works:
Suppose the number chosen is 987654321. In your head you duplicate this to 987654321987654321 which you then divide by 7 in one process.

It works because 143854143×7=1000000001.

Actually, 7 divides a number of similar looking numbers which directly contributes to some of the properties observed in the decimal system. (Change the base and some other number would have this honour.)

13 can also be made to work although it is a bit more tricky to divide by 13 without showing working. In the case of 13 you would write the eight digit number 76923077 since 76923077×13=1000000001

(Typing fast and have to go so excuse if mistake made…)

12

When I was a kid, I used to do math doodles during boring classes. I remember coming up with this sequence . . . but I don’t remember equating it to 1/49. That’s amazing.

13

That sounds like a rational answer.

14

Hi folks,

I just happened on this forum and have been enjoying myself so far, reading through the threads. This one in particular decided me on joining because it relates to something I’ve been thinking about a little bit lately. In short, my feeling is that any enthusiasm for the coincidental oddities that occur in our standard number system must be tempered by the acknowledgement that our base ten system is arbitrary and numbers come out dramatically different when transposed to other bases. Is it safe for me to say that 1/7 expressed in another base would probably not yield any notable repetition of digits?

15

It’ll have a finite decimal expansion in base b if and only if b is a multiple of 7.

16

In the special case of base 7, 1/7 = 0.1

17

It will yield a repetition of digits in any base that isn’t a multiple of 7.

18

In base 7, 7 will be written as 10.:stuck_out_tongue:

19

Well, quite (and those two facts are related), but we were talking about decimal fractions. :slight_smile:

20

Squares of integers

2 times 2 is 4
3 times 3 is 9
4 times 4 is 16
5 times 5 is 25
6 times 6 is 36
7 times 7 is 49
8 times 8 is 64
9 times 9 is 81
10 times 10 is 100

Differences between sequential squares of integers

4 (+5) = 9
9 (+7) = 16
16 (+9) = 25
25 (+11) = 36
36 (+13) = 49
49 (+15) = 64
64 (+17) = 81
81 (+19) = 100
100

Reduce the difference between between sequential squares of integers by 1

5 - 1 = 4
7 - 1 = **6 **
9 - 1 = 8
11 - 1 = **10
**13 - 1 = 12
15 - 1 = 14
17 - 1 = 16
19 - 1 = 18

Divide that number in half

4 / 2 = 2
6 / 2 = 3
8 / 2 = 4
10 / 2 = 5
12 / 2 = 6
14 / 2 = 7
16 / 2 = 8
18 / 2 = **9

The integers you just squared**

2 times 2 is 4
3 times 3 is 9
4 times 4 is 16
5 times 5 is 25
6 times 6 is 36
7 times 7 is 49
8 times 8 is 64
9 times 9 is 81
10 times 10 is **100

…
**