142,857

← 142856 142857 142858 →
List of numbers — Integers ← 100 101 102 103 104 105 106 107 108 109
Cardinal	one hundred forty-two thousand eight hundred fifty-seven
Ordinal	142857th (one hundred forty-two thousand eight hundred fifty-seventh)
Factorization	33 × 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
Octal	4270118
Duodecimal	6A80912
Hexadecimal	22E0916

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.

142,857 is a Kaprekar number^[5] and a Harshad number (in base 10).^[6]

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:^{[citation needed]}

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.^{[citation needed]}

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[]

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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.^{[citation needed]} For example, in base 12 it is 186A35 and base 24 3A6KDH.

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.^[7]

Other properties[]

The 142857 number sequence is also found in several decimals in which the denominator has a factor of 7. In the examples below, the numerators are all 1, however there are instances where it does not have to be, such as 2/7 (0.285714).

For example, consider the fractions and equivalent decimal values listed below:

1/7 = 0.142857...

1/14 = 0.0714285...

1/28 = 0.03571428...

1/35 = 0.0285714...

1/56 = 0.0157857142...

1/70 = 0.0142857...

The above decimals follow the 142857 rotational sequence. There are fractions in which the denominator has a factor of 7, such as 1/21 and 1/42, that do not follow this sequence and have other values in their decimal digits.

References[]