142857

142857
142857
presentation
Roman	CXLII  DCCCLVII
dual	10 0010 1110 0000 1001
Octal	42 7011
Duodecimal	6 A809
Hexadecimal	2 2E09
Morse code	- - - - · · · - · · - - - - - - · · · · · - - · · ·
Mathematical properties
sign	positive
parity	odd
Factorization	3 3 × 11 × 13 × 37
Divider	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

142857 six repeating digits are one-seventh: . ${\displaystyle 1/7=0{,}{\overline {142\,857}}}$

Analytics

The number one hundred forty-two thousand eight hundred and fifty-seven (decimal 142,857 ) is the best-known cyclic number in the decimal system . If it is multiplied by 2, 3, 4, 5 or 6 , the result is a cyclic permutation of itself and becomes the repeating digits of 2/7 , 3/7 , 4/7 , 5/7 and 6/7 , respectively correspond.

Six-digit numbers with the same properties are also available in other bases, given by (base ⁶  - 1) / 7. Examples are 186A35 in the duodecimal system and 3A6LDH in the quadrivigesimal system (base 24).

142,857 is also the 25th Kaprekar number and a Harshad number (divisible by their checksum, both in the decimal system).

Sample 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 seven, there is a simple method of arriving at a cyclic permutation of 142857: by adding the six right-hand digits, i.e. one to hundreds of thousands, to the remaining digits and repeating this process until fewer than six digits remain, one will arrive at a cyclic permutation of 142857.

142857 × 8 = 1142856

1 + 142856 = 142857

142857 × 815 = 116428455

116 + 428455 = 428571

142857 ² = 142857 × 142857 = 20408122449

20408 + 122449 = 142857

Multiplying by a multiple of 7 through this process gives 999999.

142857 × 7 ⁴ = 342999657

342 + 999657 = 999999

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

857 ² = 734449

142 ² = 20164

734449 - 20164 = 714285

This is the repeating part in the decimal expansion (representation in the decimal system) of the rational number . Therefore, multiples of a seventh are just repeated copies of the corresponding multiples of 142857. ${\displaystyle 1/7=0{,}{\overline {142\,857}}}$

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

etc ...

1/7 as an infinite sum

There are structures that represent 1/7 as an infinite sum with the help of doubling, shifting and addition.

${\displaystyle {\begin{aligned}1/7\ &=0{,}142857142857142857\ldots \\[6pt]&=0{,}14+0{,}0028+0{,}000056+0{,}00000112+0{,}0000000224+0{,}000000000448+0{,}00000000000896+\cdots \\[6pt]&={\frac {14}{100}}+{\frac {28}{100^{2}}}+{\frac {56}{100^{3}}}+{\frac {112}{100^{4}}}+{\frac {224}{100^{5}}}+\cdots +{\frac {7\times 2^{N}}{100^{N}}}+\cdots \\[6pt]&=\left({\frac {7}{50}}+{\frac {7}{50^{2}}}+{\frac {7}{50^{3}}}+{\frac {7}{50^{4}}}+{\frac {7}{50^{5}}}+\cdots +{\frac {7}{50^{N}}}+\cdots \right)\\[6pt]&=\sum _{k=1}^{\infty }{\frac {7}{50^{k}}}\end{aligned}}}$

Each term is equal to the previous term, doubled and shifted two places to the right.

A combined shift by one place and tripling can also represent 1/7:

${\displaystyle {\begin{aligned}1/7\ &=0{,}1+0{,}03+0{,}009+0{,}0027+0{,}00081+0{,}000243+0{,}0000729+\cdots \\[6pt]&={\frac {3^{0}}{10^{1}}}+{\frac {3^{1}}{10^{2}}}+{\frac {3^{2}}{10^{3}}}+{\frac {3^{3}}{10^{4}}}+{\frac {3^{4}}{10^{5}}}+\cdots +{\frac {3^{N-1}}{10^{N}}}+\cdots \\\end{aligned}}}$

Meaning in the enneagram

→ Main article : "Geometry and numerics" in the article Enneagram

In the Enneagram (for example that of the Fourth Way by Georges I. Gurdjieff ) the nine points of the circle are connected in the sequence 142857. This shows that if you divide a natural number that is not a multiple of 7 by 7, the decimal places always contain the periodic sequence of digits 142857 or in general for every natural number that is not divisible by 7 . ${\displaystyle {\tfrac {n}{7}}}$ ${\displaystyle n}$

literature

• Leslie, John. "The Philosophy of Arithmetic: Exhibiting a Progressive View of the Theory and Practice of...." , Longman, Hurst, Rees, Orme and Brown, 1820.
• Wells, D. The Penguin Dictionary of Curious and Interesting Numbers , revised edition. London: Penguin Group. (1997): pp. 171-175

Individual evidence

1. ^ Cyclic number. In: The Internet Encyclopedia of Science. September 29, 2007, archived from the original on September 29, 2007 ; accessed on September 29, 2007 (English). 
2. ^ Michael W. Ecker: The Alluring Lore of Cyclic Numbers . In: The Two-Year College Mathematics Journal . Vol. 14, No. 2 , March 1983, p. 105-109 , JSTOR : 3026586 .
3. ^ Cyclic number. In: PlanetMath. July 14, 2007, archived from the original on July 14, 2007 ; Retrieved July 14, 2007 . 
4. ↑ Kathryn Hogan: Go figure (cyclic numbers). Australian Doctor, August 2005, archived from the original on December 24, 2007 ; accessed in August 2005 . 
5. ↑ Sloane's A006886: Kaprekar numbers. In: The On-Line Encyclopedia of Integer Sequences. OEIS Foundation, accessed June 3, 2016 .