142,857 is the
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
following 142,856 and preceding 142,858. It is both a
Kaprekar number
In mathematics, a natural number in a given number base is a p-Kaprekar number if the representation of its square in that base can be split into two parts, where the second part has p digits, that add up to the original number. For example, in ...
and a
Cyclic number
A cyclic number is an integer for which cyclic permutations of the digits are successive integer multiples of the number. The most widely known is the six-digit number 142857, whose first six integer multiples are
:142857 × 1 = 142857
: ...
.
Cyclic number
142857 is the best-known
cyclic number
A cyclic number is an integer for which cyclic permutations of the digits are successive integer multiples of the number. The most widely known is the six-digit number 142857, whose first six integer multiples are
:142857 × 1 = 142857
: ...
in base 10, being the six
repeating digits of (0.).
If 142857 is
multiplied by 2, 3, 4, 5 or 6, the answer will be a
cyclic permutation
In mathematics, and in particular in group theory, a cyclic permutation is a permutation consisting of a single cycle. In some cases, cyclic permutations are referred to as cycles; if a cyclic permutation has ''k'' elements, it may be called a ''k ...
of itself, and will correspond to the repeating digits of , , , or respectively:
: 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
: 142857
2 = 142857 × 142857 = 20408122449
: 20408 + 122449 = 142857
Multiplying by a multiple of 7 will result in 999999 through this process:
: 142857 × 7
4 = 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.
: 857
2 = 734449
: 142
2 = 20164
: 734449 − 20164 = 714285
It is the repeating part in the
decimal expansion
A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator:
r = b_k b_\cdots b_0.a_1a_2\cdots
Here is the decimal separator ...
of the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example,
The set of all ...
= 0.. Thus, multiples of are simply repeated copies of the corresponding multiples of 142857:
:
Connection to the enneagram
The 142857 number sequence is used in the
enneagram
Enneagram may refer to:
* Enneagram (geometry), a nine-sided star polygon with various configurations
* Enneagram of Personality, a model of human personality illustrated by an enneagram figure
See also
* Enneagon
In geometry, a nonagon () or ...
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
George Ivanovich Gurdjieff ( – 29 October 1949) was a philosopher, mystic, spiritual teacher, composer, and movements teacher. Born in the Russian Empire, he briefly became a citizen of the First Republic of Armenia after its formation in 19 ...
), the Law of Three and the Law of Seven. The movement of the numbers of 142857 divided by , . etc., and the subsequent movement of the enneagram, are portrayed in Gurdjieff's
sacred dances known as the movements.
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 (0.).
For example, consider the fractions and equivalent decimal values listed below:
: = 0....
: = 0.0...
: = 0.03...
: = 0.0...
: = 0.017...
: = 0.0...
The above decimals follow the 142857 rotational sequence. There are fractions in which the denominator has a factor of 7, such as and , that do not follow this sequence and have other values in their decimal digits.
References
*
*
*
{{DEFAULTSORT:142857
Integers