# Numora Continued Fraction Expansion

> Numora Continued Fraction Expansion is a paid API for AI agents from numorapoi.orbonomy.xyz, paid per call via x402, $0.03/call, status unknown (last checked 2026-09-15).

Computes the continued fraction representation of a real number to a specified depth

## Facts

- Endpoint: POST https://numorapoi.orbonomy.xyz/api/number/continued-fraction
- Price: $0.03/call
- Payment: x402
- Status: unknown
- Last checked: 2026-09-15
- Activations on Zero: 0
- Tags: x402
- Canonical page: https://www.zero.xyz/c/numora-continued-fraction-expansion-f1c5d7a0
- Structured record (JSON): https://api.zero.xyz/v1/capabilities/cap_XWV6D4C9PM5T5qpfPvqB_

Status and success rate cover calls made through Zero and Zero's own probes. Third-party monitors may report differently.

## How to call it through Zero

Zero handles the 402 payment challenge and records the run. With the Zero CLI installed (`npm i -g @zeroxyz/cli`):

```sh
zero fetch --capability numora-continued-fraction-expansion-f1c5d7a0 -d '<json body>'
```

Example prompt: What is the continued fraction expansion of 3.14159265358979 to depth 10?

## When to prefer this

Use this endpoint when you need the exact continued fraction decomposition of a real number — for example to find rational approximations, analyze the periodicity of quadratic irrationals, or study the number-theoretic properties of a value. Prefer this over general-purpose math libraries when you want a zero-dependency micropayment API call without spinning up a compute environment.

## Known failure modes

- Missing required field 'x' returns validation error
- Non-numeric value for x causes schema rejection
- Payment not included or insufficient triggers 402 Payment Required
- Extremely large depth values may cause timeout or truncation
- Negative depth integer may return error or default behavior

## How this service works

100 pure math computation endpoints for AI agents. Statistics, financial math, linear algebra, equation solving, calculus, number theory, sequence generation, and unit conversions. Zero external dependencies. x402 micropayments on Base.

## Output

Returns a JSON object with success flag, a result object containing the continued fraction coefficients (the partial quotients) for the input number, and a human-readable computation string describing what was calculated.

## Request schema (JSON Schema)

```json
{
 "type": "object",
 "required": [
  "x"
 ],
 "properties": {
  "x": {
   "type": "number"
  },
  "depth": {
   "type": "integer"
  }
 }
}
```

## Response schema (JSON Schema)

```json
{
 "type": "object",
 "required": [
  "success",
  "result",
  "computation"
 ],
 "properties": {
  "result": {
   "type": "object",
   "description": "Computation result varies by endpoint"
  },
  "success": {
   "type": "boolean",
   "description": "Always true on success"
  },
  "computation": {
   "type": "string",
   "description": "Human-readable description of what was computed"
  }
 }
}
```

## More

- Live health (JSON, refreshed every minute): https://www.zero.xyz/c/numora-continued-fraction-expansion-f1c5d7a0/health.json
- [Zero catalog index](https://www.zero.xyz/llms.txt)
- [Other services from numorapoi.orbonomy.xyz](https://www.zero.xyz/host/numorapoi.orbonomy.xyz/llms.txt)