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142,857 is the best-known cyclic number in base 10.[1][2][3][4] If you multiply the number by 2, 3, 4, 5, or 6, the answer will be a cyclic permutation of itself. A cyclic number is an integer in which cyclic permutations of the digits are successive multiples of the number. ...
In mathematics, multiplication is an elementary arithmetic operation. ...
A cyclic permutation is a permutation that shifts all elements of given ordered set by a fixed offset, with the elements shifted off the end inserted back at the beginning in the same order, i. ...
- 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 bigger than 7, there is a simple process to get to a cyclic permutation of 142857. By adding the first 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 This is a list of articles about numbers (not about numerals). ...
One hundred thousand (100000) is the natural number following 99999 and preceding 100001. ...
One million (1,000,000), or one thousand thousand, is the natural number following 999,999 and preceding 1,000,001. ...
Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. ...
In set theory, ordinal, ordinal number, and transfinite ordinal number refer to a type of number introduced by Georg Cantor in 1897, to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. ...
...
The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. ...
In mathematics and computer science, hexadecimal, base-16, or simply hex, is a numeral system with a radix, or base, of 16, usually written using the symbols 0–9 and A–F, or a–f. ...
The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. ...
- 142857 × 8 = 1142856
- 1 + 142856 = 142857
- 142857 × 815 = 116428455
- 116 + 428455 = 428571
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 142857 14... Thus, multiples of 1/7 are simply repeated copies of the corresponding multiples of 142857: A numeral is a symbol or group of symbols that represents a number. ...
In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ...
- 1 ÷ 7 = 0 . 142857 142857 142857 14...
- 2 ÷ 7 = 0 . 285714 285714 285714 28...
- 3 ÷ 7 = 0 . 428571 428571 428571 42...
- 4 ÷ 7 = 0 . 571428 571428 571428 57...
- 5 ÷ 7 = 0 . 714285 714285 714285 71...
- 6 ÷ 7 = 0 . 857142 857142 857142 85...
- 9 ÷ 7 = 1.2857 142857 142857...
- 22 ÷ 7 = 3 . 142857 142857..., an ancient approximation to Pi.
Simply combining the digits in different ways and dividing by 7 or a multiple of 7 When a circles diameter is 1, its circumference is π. Pi or π is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ...
- 142 + 857 = 999
- 999 ÷ 7 = 142.7 142857 142857 142857 14...
- 14 + 28 + 57 = 99
- 99 ÷ 7 = 14. 142857 142857 142857...
- 1 + 4 + 2 + 8 + 5 + 7 = 27
- 27 ÷ 7 = 3.857 142857 142857 142857...
- 857 - 142 = 715
- 715 ÷ 7 = 102. 142857 142857 142857...
- 1 × 4 × 2 × 8 × 5 × 7 = 2240
- 2240 ÷ 49 = 45.7 142857 142857 142857...
- 1 + 42 + 857 = 900
- 900 ÷ 7 = 128.57 142857 142857 142857...
- 1 × 4 ÷ 2 × 8 ÷ 5 × 7 = 22.4
- 22.4 ÷ 49 = 0.457 142857 142857 142857...
In base 10, 142,857 is a Harshad number and a Kaprekar number. A Harshad number, or Niven number, is an integer that is divisible by the sum of its digits in a given number base. ...
In mathematics, a Kaprekar number is a number that, in a given base, when squared, can be split into two numbers with the same number of digits as the original number which add up to the original number again. ...
 as π
This ancient approximation to π has also been associated with Jewish mysticism. In particular, it is known in certain circles of Kabbalists such as the "Mediogegnians", that the twenty two characters of the Hebrew language represent a complete circumference that when divided by seven (the sacred number of cycles) produces the Kabbalistic π, also known within this circle of practitioners as the "perfect π".
22 / 7 = 3.142857142857142857... π = 3.141592653589793238...
It is interesting to know that the difference between these two numbers is represented by,
References - ^ "Cyclic number", The Internet Encyclopedia of Science
- ^ Michael W. Ecker, "The Alluring Lore of Cyclic Numbers", The Two-Year College Mathematics Journal, Vol.14, No.2 (March 1983), pp.105-109
- ^ Cyclic number, PlanetMath
- ^ Kathryn Hogan, "Go figure (cyclic numbers)", Australian Doctor, August 2005
- Leslie, John. "The Philosophy of Arithmetic: Exhibiting a Progressive View of the Theory and Practice of ....", Longman, Hurst, Rees, Orme, and Brown, 1820, ISBN 1-4020-1546-1
- Wells, D. The Penguin Dictionary of Curious and Interesting Numbers Revised Edition. London: Penguin Group. (1997): 171 - 175
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