142,857

142857, the six repeating digits of 1/7, 0.142857, 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.

Main articles: Repeating decimal and Cyclic number

← 142856 142857 142858 →
List of numbers — Integers ← 10^0 10^1 10^2 10^3 10^4 10^5 10^6 10^7 10^8 10^9
Cardinal	one hundred forty-two thousand eight hundred fifty-seven
Ordinal	142857th (one hundred forty-two thousand eight hundred fifty-seventh)
Factorization	3^3 × 11 × 13 × 37
Divisors	1, 3, 9, 11, 13, 27, 33, 37, 39, 99, 111, 117, 143, 297, 333, 351, 407, 429, 481, 999, 1221, 1287, 1443, 3663, 3861, 4329, 5291, 10989, 12987, 15873, 47619, 142857
Greek numeral	${\displaystyle {\stackrel {\iota \delta }{\mathrm {M} }}}$͵βωνζ´
Roman numeral	CXLMMDCCCLVII
Binary	1000101110000010012
Ternary	210202220003
Quaternary	2023200214
Quinary	140324125
Senary	30212136
Octal	4270118
Duodecimal	6A80912
Hexadecimal	22E0916
Vigesimal	HH2H20
Base 36	328936

142,857 is a Kaprekar number[5] and a Harshad number (in base 10).

Calculation

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 multiplying by an integer greater than 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 only six digits are 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

Multiplying by a multiple of 7 will result in 999999 through this process:

142857 × 7⁴ = 342999657

342 + 999657 = 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

…

1/7 as an infinite sum

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

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

Each term is double the prior term shifted two places to the right. This is can be proved by applying the identity for the sum of a geometric sequence:

${\displaystyle \sum _{k=1}^{\infty }{\frac {7}{50^{k}}}=7\cdot \sum _{k=1}^{\infty }\left({\frac {1}{50}}\right)^{k}=7\cdot {\frac {1}{50-1}}={\frac {7}{49}}={\frac {1}{7}}}$

Another infinite sum is

${\displaystyle {\begin{aligned}{\frac {1}{7}}&=0.1+0.03+0.009+0.0027+0.00081+0.000243+0.0000729+\cdots \\[6pt]&={\frac {3^{0}}{10^{1}}}+{\frac {3^{1}}{10^{2}}}+{\frac {3^{2}}{10^{3}}}+{\frac {3^{3}}{10^{4}}}+{\frac {3^{4}}{10^{5}}}+\cdots +{\frac {3^{N-1}}{10^{N}}}+\cdots \\[6pt]&=\sum _{k=1}^{\infty }{\frac {3^{k-1}}{10^{k}}}=3^{-1}\cdot \sum _{k=1}^{\infty }\left({\frac {3}{10}}\right)^{k}={\frac {1}{3}}\cdot {\frac {3}{10-3}}={\frac {1}{7}}\end{aligned}}}$

Other bases

In some other bases, six-digit numbers with similar properties exist, given by base⁶ − 1/7. For example, 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.[6]

References

1. "Cyclic number". The Internet Encyclopedia of Science. Archived from the original on 2007-09-29.
2. Ecker, Michael W. (March 1983). "The Alluring Lore of Cyclic Numbers". The Two-Year College Mathematics Journal. 14 (2): 105–109. JSTOR 3026586.
3. "Cyclic number". PlanetMath. Archived from the original on 2007-07-14.
4. Hogan, Kathryn (August 2005). "Go figure (cyclic numbers)". Australian Doctor. Archived from the original on 2007-12-24.
5. "Sloane's A006886: Kaprekar numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-03.
6. Ouspensky, P. D. (1947). "Chapter XVIII". In Search of the Miraculous: Fragments of an Unknown Teaching. London: Routledge.

• Leslie, John (1820). The Philosophy of Arithmetic: Exhibiting a Progressive View of the Theory and Practice of…. Longman, Hurst, Rees, Orme, and Brown. ISBN 1-4020-1546-1.
• Wells, D. (1997). The Penguin Dictionary of Curious and Interesting Numbers (revised ed.). London: Penguin Group. pp. 171–175. ISBN 978-0-140-26149-3.