# Division and periodicity


## PostgreSQL-faithful scale

`div(a, b)` without an explicit scale uses PostgreSQL's own `select_div_scale`
rule — `max(16 - 4·qweight, scale_a, scale_b)` where `qweight` is the weight of
the quotient's integer part. The output is **digit-for-digit identical** to
`numeric` division on the database:

| PostgreSQL | `div()` result |
|------------|----------------|
| `1/3` | `0.33333333333333333333` |
| `10/4` | `2.5000000000000000` |
| `100/3` | `33.3333333333333333` |

`pgDivScale(a, b)` exposes the computed scale if you need it for pre-validation.

## Fixed scale override

```ts
div(total, 8, 2);                               // force scale 2 (HALF_UP)
div(a, b, { scale: 6, rounding: "HALF_EVEN" }); // force scale + rounding mode
```

## The `isPeriodic` flag

`isPeriodic(d)` is `true` when the last division produced a non-terminating
expansion that was truncated (e.g. `1/3`). The flag **propagates through
operations** — a downstream computation on a periodic operand stays flagged,
so callers can decide to round for display while keeping the precise value
internally:

```ts
const third = div(1, 3);
isPeriodic(third);              // true
isPeriodic(mul(third, 2));      // true — propagates
isPeriodic(add(third, "1"));    // true — propagates
isPeriodic(div(4, 2));          // false
```