A small fraction. A surprisingly large story.
Where did
the pattern go?
A decimal starts counting in powers of three. Then it seems to lose the thread. Let’s slow it down and look inside.
Follow the arithmeticRead the decimal in groups of three.
729 still fits into three digits. So why did it become 731?
The pattern kept going.
Its contributions overlapped.
The clue is a small gap: 997 is three less than 1000. That gives us a way to assemble its reciprocal from powers of three:
These are amounts we add, not labels we simply join together. Each amount is smaller than the one before it, even as its numerator grows. Eventually, the addition reaches back into earlier groups.
Unfold. Then settle.
A wider written term keeps its original place value. More ink does not mean more decimal places.
729 + 2 − 1000 × 0 = 731
Two whole units arrive from the remaining tail. No earlier carry needs exchanging yet.
Long integers are shortened in this view. Expand the exact values below to inspect every digit.
Inspect this step and its exact balance
The companion proof-status atlas labels the geometric-series identity (series_q_weighted_identity) and incoming-carry formula (incoming_carry_position_formula) reproved-here. This example uses (base, N, m, B, q, k, L) = (10, 997, 3, 1000, 1, 3, 166), where B = qN + k and L counts groups in a cycle.
The incoming carry accounts for the entire remaining tail at this boundary. In later groups, we must also remove the earlier carry’s contribution.
- Raw term
- 729
- Incoming carry
- 2
- Previous carry
- 0
- Ordinary division
- 1000 × 729 = 997 × 731 + 193
After 7 terms, the raw prefix leaves the exact balance 2187 / (997 × 1000⁷). Settling it adds 2 to the prefix’s integer numerator and leaves remainder 193.
For this unit numerator, every displayed prefix is checked using the integer identity D × A + xM = BM, where A is the raw prefix read as one weighted integer and xM is its remaining lifted state.
Make whole groups.
Keep the leftover.
Ordinary long division carries a small piece of state forward: the remainder. At the first change, that remainder is 729.
Because 1000 = 997 + 3, tripling the remainder reveals the next move. Complete groups of 997 contribute to what we print; the leftover continues the calculation.
At this first boundary:
2187 = 2 × 997 + 193.
We print 729 + 2 = 731. The next remainder is 193.
3 × 729 = 2187
This split uses the bounded remainder. Its local group count is different from the full growing carry in the comparison above.
Something can grow forever
and still keep coming back.
The powers of three have unlimited room to grow. A remainder after division by 997 has only 997 possible values: 0 through 996. For this fraction, the nonzero remainders trace a repeating route.
When a remainder returns, the same next step follows. That is why the decimal repeats. The clean powers at its beginning were already part of the cycle.
See the whole power and check the period
729
The proof-status atlas records digit periodicity (digit_periodicity) as reproved-here and the order of a power (power_order_formula) as classical.
The shortest cycle measured in three-digit groups is 166 steps. That spans 498 decimal places. The shortest decimal-digit cycle is also 166 steps, so those 498 places contain three copies of it.
ord₉₉₇(1000) = ord₉₉₇(10) / gcd(ord₉₉₇(10), 3) = 166.
Keeping both quotient and remainder recovers the growing integer exactly. Keeping only the remainder forgets which whole multiple was removed, while retaining everything needed for the next division step.
The growing terms reveal the construction.
The remainders reveal the return.
What if multiplication
were something you could see?
Start with another simple list: 1/9999 = 0.0001 0001 0001 …. We have chosen four-digit groups here because 9999 is one less than 10000.
Squaring pairs every term in one copy with every term in another. A pair from positions 2 and 3 lands in position 5. The pairs form a grid; pairs with the same sum lie along a diagonal.
Counting numbers count how many products arrive at each place. Cubing adds a third choice. The resulting slices contain triangular numbers: 1, 3, 6, 10, …
Open the geometry of multiplication
For the square, the raw four-digit groups begin 0000, 0001, 0002, 0003, …. For the cube, they begin 0000, 0000, 0001, 0003, 0006, 0010, ….
The initial zeros have a reason: two positive positions cannot sum to 1, and three cannot sum to 1 or 2. These early coefficients fit unchanged; later contributions require carrying too.
From a grid to a coefficient
Each factor is the convergent geometric series Σᵢ≥₁ B⁻ⁱ, with B = 10000. Multiplication collects ordered pairs i + j = s, or ordered triples i + j + k = s. There are s − 1 pairs and (s − 1)(s − 2)/2 triples at place s, once their respective starting positions are reached.
This is the Cauchy product of absolutely convergent series. Collecting contributions produces raw coefficients; carrying normalizes those coefficients into ordinary base-B groups. They are separate operations.
One calculation.
Several exact descriptions.
This essay explores a way of explaining established arithmetic. The growing-term notation is an experimental presentation. We make no claim to a new division algorithm or to novelty of the underlying generating-series identities.
For the status of individual results, see the project’s proof-status atlas. The reciprocal-series, incoming-carry, and digit-periodicity claims used above are reproved-here; the power-order formula is classical. The interactive exposition is implemented-here. The general lifted-state formulation and finite-prefix certificates below are explanatory algebraic derivations; their separate formalization remains open.
The general carry identity
Let B ≥ 2 and D ≥ 2 be integers, and choose coherent integer states x and x′ such that Bx − x′ is divisible by D. Define the raw emission a = (Bx − x′)/D. Write the Euclidean divisions x = Dq + r and x′ = Dq′ + r′, with 0 ≤ r, r′ < D.
w = a + q′ − Bq
Br = Dw + r′
The second line is ordinary long division. The first describes exactly how any coherent integer lift yields its printed group. Taking x = r and x′ = r′ gives the canonical lift with both quotient coordinates zero. A recurrence-adapted lift instead exposes simple growing terms.
For a monic polynomial P(X) = Xd − c1Xd−1 − … − cd with integer coefficients, d ≥ 1, and D = P(B) > 1, reduce UXm modulo P and evaluate at B. These lifted states produce the seeded recurrence, while Euclidean normalization produces the same groups as division of U/D. Here U is an integer with 0 < U < D, and the initial state is x0 = U.
Starting from x0 = U in [0, D), the first changed group occurs at the first step whose next lifted state leaves [0, D), if such a step occurs. Until that exit, both quotient coordinates are zero; at the exit, q′ becomes nonzero while q is still zero. This need not be exactly one group before a raw term overflows. Signed states require floor division and may produce borrows.
Why a finite view can still be exact
After M terms, let A = a0BM−1 + ⋯ + aM−1 be the raw prefix read as one weighted integer, and xM its remaining lifted state. With initial state x0 = U:
D × A + xM = U × BM
The fraction U/D is exactly A/BM + xM/(D × BM). Writing xM = DqM + rM, with 0 ≤ rM < D, settles the prefix to (A + qM)/BM and leaves rM/(D × BM).
The display can therefore stop without pretending the unwritten balance is zero. This finite identity needs no infinite-series convergence assumption. Interpreting the raw terms as an infinite sum does require convergence; the positive examples on this page satisfy it.
Sources, scope, and further reading
Recurrence generating functions and reciprocal digit patterns have a substantial history. These primary references place the examples in that context:
- C. T. Long (1981), “The Decimal Expansion of 1/89 and Related Results” — Fibonacci expansions and finite tails.
- G. Köhler (1985), “Generating Functions of Fibonacci-Like Sequences and Decimal Expansions of Some Fractions” — recurrence-based expansions.
- Jin-Zai Lee and Jia-Sheng Lee (1987), “A Complete Characterization of B-Power Fractions That Can Be Represented as Series of General n-Bonacci Numbers” — related recurrence constructions.
- Marjorie Bicknell-Johnson (1989), “Retrograde Renegades and the Pascal Connection: Repeating Decimals Represented by Fibonacci and Other Sequences Appearing from Right to Left” — recurrence sequences read from the right of repeating decimals.
Every interactive step uses exact integer arithmetic and is checked against ordinary long division. These checks support the implementation; the identities above provide the mathematical argument. The page does not claim that the full recurrence framework has been formally verified in Lean.