142857 (number)

Main articles: Repeating decimal and Cyclic number
"1/7 (number)" redirects here. For the number 7, see 7 (number).
 ← 142856 142857 142858 →
Cardinal one hundred forty-two thousand eight hundred and fifty-seven
Ordinal 142857th
(one hundred forty-two thousand eight hundred and fifty-seventh)
Factorization 33· 11 · 13 · 37
Roman numeral CXLMMDCCCLVII
Binary 1000101110000010012
Ternary 210202220003
Quaternary 2023200214
Quinary 140324125
Senary 30212136
Octal 4270118
Duodecimal 6A80912
Vigesimal HH2H20
Base 36 328936

142857 is the six repeating digits of 1/7, $0.\overline{142857}$, and is the best-known cyclic number in base 10.[1][2][3][4] If it is multiplied by 2, 3, 4, 5, or 6, the answer will be a cyclic permutation of itself, and will correspond to the repeating digits of 2/7, 3/7, 4/7, 5/7, or 6/7 respectively.

Calculations

1 × 142,857 = 142,857
2 × 142,857 = 285,714
3 × 142,857 = 428,571
4 × 142,857 = 571,428
5 × 142,857 = 714,285
6 × 142,857 = 857,142
7 × 142,857 = 999,999

If you multiply by an integer greater than 7 that is not divisible by 7, there is a simple process to get to a cyclic permutation of 142857. By adding the rightmost six digits (ones through hundred thousands) to the remaining digits and repeating this process until you have only the six digits left, it will result in a cyclic permutation of 142857:

142857 × 8 = 1142856
1 + 142856 = 142857
142857 × 815 = 116428455
116 + 428455 = 428571
1428572 = 142857 × 142857 = 20408122449
20408 + 122449 = 142857

This will work even if the product is greater than 12 digits, in which case there are two methods of getting back to the original six digits. You can either split it up every six digits (which will result in the original six digits followed by one or more zeros):

142857 x 47185939 = 6740841687723
6 + 740841 + 687723 = 1428570

Or you can add the final six digits to the rest of the digits, and do that process recursively (which will result in the original six digits with no additional zeros):

142857 x 47185939 = 6740841687723
6740841 + 687723 = 7428564
7 + 428564 = 428571

Multiplying by a multiple of 7 will result in 999999 through the above processes:

142857 × 74 = 342999657
342 + 999657 = 999999
142857 x 7777777 = 1111109888889
1111109 + 888889 = 1999998
1 + 999998 = 999999
142857 x 7777777 = 1111109888889
1 + 111109 + 888889 = 999999

If you square the last three digits and subtract the square of the first three digits, you also get back a cyclic permutation of the number.

8572 = 734449
1422 = 20164
734449 − 20164 = 714285

It is the repeating part in the decimal expansion of the rational number 1/7 = 0.142857. Thus, multiples of 1/7 are simply repeated copies of the corresponding multiples of 142857:

1 ÷ 7 = 0.142857
2 ÷ 7 = 0.285714
3 ÷ 7 = 0.428571
4 ÷ 7 = 0.571428
5 ÷ 7 = 0.714285
6 ÷ 7 = 0.857142
7 ÷ 7 = 0.999999 = 1
8 ÷ 7 = 1.142857
9 ÷ 7 = 1.285714

In base 10, 142,857 is a Harshad number and a Kaprekar number.

1/7 as an infinite sum

There is an interesting pattern of doubling, shifting and addition that gives 1/7.

\begin{align}1/7\ & = 0.142857142857142857\ldots \\[6pt] & = 0.14 + 0.0028 + 0.000056 + 0.00000112 + 0.0000000224 + 0.000000000448 + 0.00000000000896 + \cdots \\[6pt] & = \frac{14}{100} + \frac{28}{100^2} + \frac{56}{100^3} + \frac{112}{100^4} + \frac{224}{100^5} + \cdots + \frac{7\times2^N}{100^N} + \cdots \\[6pt] & = \left( \frac{7}{50} + \frac{7}{50^2} + \frac{7}{50^3} + \frac{7}{50^4} + \frac{7}{50^5} + \cdots + \frac{7}{50^N} + \cdots \right) \\[6pt] & = \sum_{k=1}^\infty \frac{7}{50^k} \end{align}

Each term is double the prior term shifted two places to the right.

Other bases

In some other bases, six-digit numbers with similar properties exist, given by (base6 − 1)/7. E.g. in base 12 it is 186A35 and base 24 3A6LDH.

Connection to the enneagram

The 142857 number sequence is used in the enneagram figure, a symbol of the Gurdjieff Work used to explain and visualize the dynamics of the interaction between the two great laws of the Universe (according to G. I. Gurdjieff), the Law of Three and the Law of Seven. The movement of the numbers of 142857 divided by 1/7, 2/7. etc., and the subsequent movement of the enneagram, are portrayed in Gurdjieff's sacred dances known as the movements.

References

1. ^ "Cyclic number", The Internet Encyclopedia of Science
2. ^ Michael W. Ecker, "The Alluring Lore of Cyclic Numbers", The Two-Year College Mathematics Journal, Vol.14, No.2 (March 1983), pp. 105–109
3. ^ Cyclic number, PlanetMath
4. ^ Hogan, Kathryn (August 2005). "Go figure (cyclic numbers)". Australian Doctor. Archived from the original on 24 December 2007.