The cyclic number
Why is 142857 so special?
Real math, proven
Because 142857 is 1/7 in disguise. 1/7 = 0.142857142857..., and multiplying 142857 by 2, 3, 4, 5 or 6 gives the same six digits turned round, while times 7 gives 999999. That happens because long division by 7 passes through every possible remainder before it repeats. 142857 is the smallest cyclic number, and the only one in base 10 that does not start with a zero.
Multiply by
285,714
The long division
The reason
Long division that goes in a circle
Divide 1 by 7 by hand. At every step you carry a remainder, and dividing by 7 the remainder can only be 1, 2, 3, 4, 5 or 6. Starting from 1, the remainders run 1, 3, 2, 6, 4, 5 and then come back to 1, so the digits 1, 4, 2, 8, 5, 7 repeat forever.
Now divide 2 by 7. You start with remainder 2, which is a stop the division of 1/7 already makes, two digits in. From there the steps are identical, so 2/7 = 0.285714..., the same ring read from a different place. Every fraction k/7 up to 6/7 starts at one of those six stops, which is why 142857 × k is always a rotation.
7/7 is the exception. It is exactly 1, and 0.999999... repeating is also exactly 1, so 142857 × 7 = 999999.
The trick needs a prime whose long division visits every remainder. Mathematicians call these full reptend primes: 7, 17, 19, 23, 29, 47, 59, 61 and 97 below 100. 1/17 gives the next cyclic number, 0588235294117647, sixteen digits long, but it starts with a zero. 1/13 falls short: it repeats after only six digits, not twelve.
More tricks
Halves that add to nines
142 + 857 = 999. Split the block in half and the halves add to nines. So do its thirds: 14 + 28 + 57 = 99. This is Midy’s theorem, named after the French mathematician E. Midy: when a fraction with a prime on the bottom repeats with an even number of digits, the second half is the nines complement of the first. It holds for 1/13 too, whose block 076923 gives 076 + 923 = 999.
142857 × 142857 = 20408122449, and 20408 + 122449 = 142857. That makes it a Kaprekar number, named after the same D. R. Kaprekar who found 6174.
It is a base 10 fact. The theorem behind it works in any base, but the number changes. In base 12, 1/5 produces a four digit cyclic number that plays the same part. Count in base 12 on the base switch and 1/7 becomes a different six digit ring.
Where the legend comes in
142857 in the enneagram
The enneagram is a nine pointed figure: a circle with a triangle joining 3, 6 and 9, and a six sided line that runs 1, 4, 2, 8, 5, 7. It comes from G. I. Gurdjieff, who died in 1949 and is the only known source for the figure. That inner line is just the digits of 1/7, and the triangle is the multiples of 3. The personality types sold under the same name came later, from Oscar Ichazo in the 1950s and Claudio Naranjo in the 1970s.
A close cousin turns up in vortex math: the same six digits in a different order, 1 2 4 8 7 5, produced by doubling, next to the same 3 6 9 triangle. There the loop is the nine rule; here it is division by 7. Both are real arithmetic of base 10. Neither is a law of the universe.
Real vs legend
142857, sorted
| Claim | Status |
|---|---|
| 142857 is the repeating block of 1/7. | Real |
| × 1 to × 6 rotate the same six digits. | Real · Proven |
| 142 + 857 = 999 is a coincidence. | No · Midy’s theorem |
| 142857 is unique in the universe. | Unique in base 10 · others in other bases |
| The enneagram’s 1 4 2 8 5 7 line encodes a cosmic law. | Legend · it is 1/7 |
Questions
142857
What is the meaning of 142857?
It is the repeating part of 1/7 = 0.142857142857... Its tricks, the rotations when you multiply it and the 999999 when you multiply by 7, all follow from that. It has no other built-in meaning.
What is the smallest cyclic number?
142857. It is also the only cyclic number in base 10 that does not start with a zero. The next one, from 1/17, is 0588235294117647.
Is 142857 related to 1/7?
Yes. 1/7 = 0.142857 repeating, and 2/7, 3/7, 4/7, 5/7 and 6/7 are the same six digits starting from a different place. 2/7 = 0.285714..., for example.
Why does 142857 × 7 equal 999999?
Because 7 × 1/7 = 1, and 0.999999 repeating is exactly 1. So seven times the repeating block is a block of nines.
Are there other numbers like 142857?
Yes, one for each full reptend prime, a prime whose long division visits every remainder: 17, 19, 23, 29, 47, 59, 61 and 97 below 100. Their cyclic numbers all start with a zero.
Why is 142857 in the enneagram?
G. I. Gurdjieff drew the figure with a line through 1, 4, 2, 8, 5 and 7, the digits of 1/7, alongside a triangle through 3, 6 and 9. He is the only known source for the figure.
Sources
Where this comes from
Share
An image of the 142857 rotations. Copy link reopens your multiplier.
Poster
The remainder ring of 1/7, printed and framed.