Cyclic numbers · 1/7 = 0.142857
Cyclic Number Machine
1/7 = 0.142857 142857…, and 142857 is a cyclic number: times 2, 3, 4, 5 or 6 it gives the same six digits turned around a circle, and times 7 it gives 999999. Primes that do this, like 7, 17, 19 and 23, are called full reptend primes, the magic primes on this page. Type any whole number up to 100000 to see its exact repeating decimal, spin its ring, and check whether it is magic.
- Decimal
- Repeats
- Prime
- Magic
- Midy
- Digit sum
- n × block
- Rings
Every prime below 1000 whose 1/p repeats after p − 1 digits, worked out on this page. Tap one to load it.
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How it works
Long division that goes in circles
1. Remainders come back
Divide 1 by n the way you learned at school. Each step leaves a remainder smaller than n, so sooner or later a remainder comes back, and from then on the digits repeat. For 1/7 the remainders run 1, 3, 2, 6, 4, 5 and then 1 again, so the block is 6 digits long. The length of the block is the number of times you multiply by 10 on an n hour clock before you land on 1 again.
2. Why 2 and 5 are different
10 is 2 × 5. A number made only of 2s and 5s, like 8, divides a power of 10 exactly, so 1/8 = 0.125 ends. If n has some 2s or 5s and other factors too, like 12, you first get a few digits that never come back, 1/12 = 0.08 and then 3 forever, and the loop starts after them.
3. Magic primes have one ring
When the remainders of 1/n pass through every number from 1 to n − 1 before repeating, multiplying the block by k is the same as starting the division at remainder k. That start sits on the same loop, so you get the same digits read from a new place. Times n gives n/n = 0.999…, so the block turns into all 9s. When the loop is shorter, the remainders split into several rings of the same length, and some multiples jump to a different ring.
The picture
The inner circle has one point for each remainder, and each remainder is joined to 10 times itself, counted around the circle. It is the 10 times table circle. The outer ring holds the digits of the block, starting at the top and reading clockwise. Spin turns the ring so that the digits of k × the block start at the top.
Real vs legend
What is math and what is story
| Claim | Status |
|---|---|
| 1/7 = 0.142857 repeating. 142857 × 2, 3, 4, 5 and 6 gives the same digits turned, and 142857 × 7 = 999999. | Real |
| The full reptend primes in base 10 start 7, 17, 19, 23, 29, 47, 59, 61, 97, and each of their repeating blocks is a cyclic number. | Real |
| Nobody knows whether there are infinitely many full reptend primes. Artin's conjecture says yes, and it is still unproven. | Real |
| If Artin's conjecture holds, about 37.4% of all primes are full reptend primes. | Real |
| 1/81 = 0.012345679 repeating, the digits 0 to 9 with no 8. | Real |
| Midy's theorem: when a prime's block has an even length, its two halves add up to all 9s, like 142 + 857 = 999. | Real |
| The cyclic trick depends on base 10. In base 2, 1/7 repeats every 3 digits and 7 is not a full reptend prime. | Real |
| The enneagram symbol draws its inner lines in the order 1 4 2 8 5 7. | Real, as a symbol |
| 142857 is a sacred enneagram code with hidden power. | Legend |
| 142857 is a secret key of the universe. | No source |
Questions
Cyclic numbers FAQ
Why is 142857 a cyclic number?
142857 is the repeating block of 1/7. Long division of 1 by 7 passes through every remainder from 1 to 6 before it repeats: 1, 3, 2, 6, 4, 5. Multiplying by 2 starts the same division at remainder 2, which is 2/7 = 0.285714…, the same six digits read from a new place. Times 7 gives 7/7 = 0.999…, so 142857 × 7 = 999999.
What is a full reptend prime?
A prime p whose 1/p repeats after exactly p − 1 digits, the longest block possible. In base 10 the first ones are 7, 17, 19, 23, 29, 47, 59, 61 and 97, and each of their repeating blocks is a cyclic number. Nobody has proved there are infinitely many. Artin's conjecture says there are, and that about 37.4% of all primes are full reptend.
Why is 13 not a magic prime?
1/13 = 0.076923 repeating, so the block has 6 digits, not 12. The 12 remainders split into two rings of 6. Times 3 gives 230769, a turn of the same ring, but times 2 gives 153846, the digits of 2/13 from the other ring. A magic prime has only one ring, so every multiple is a turn.
Why does 1/81 give 0.012345679?
1/81 is 1/9 × 1/9. Squaring 0.111… adds shifted copies of it, which gives 0.0123456789 followed by 10, 11, 12 and so on. The carries from 10 onward turn the 8 9 into 9 0, so the block is 012345679, 9 digits long with no 8.
Does the trick work in other bases?
It depends on the base. 7 is magic in base 10 because 10 needs 6 steps to come back to 1 on a 7 hour clock: 1, 3, 2, 6, 4, 5. In base 2 the steps are 1, 2, 4, so 1/7 = 0.001001… in binary repeats every 3 digits and 7 is not magic there. In base 2 the magic primes start 3, 5, 11, 13.
What is Midy's theorem?
If p is a prime other than 2 and 5, and the repeating block of 1/p has an even number of digits, the two halves add up to all 9s. 142 + 857 = 999 for 7, 076 + 923 = 999 for 13, and 05882352 + 94117647 = 99999999 for 17.
Is 142857 a sacred number?
The enneagram symbol draws the path 1 4 2 8 5 7 inside a circle, and that use of the digits is real. Claims that 142857 carries hidden power or is a secret key of the universe have no source. Its behavior comes from 7 being a full reptend prime in base 10, and 17, 19, 23 and many more do the same thing with longer blocks.
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