# NaN and Infinity

Special numeric values in Internet Object represent mathematical concepts of "Not a Number" and infinite values. These are used to handle edge cases in numerical computations, such as division by zero or invalid mathematical operations. These values follow IEEE 754 standards and provide a way to represent undefined or infinite results within numeric operations.

## Syntax

Special numeric values are expressed as literal keywords:

```ebnf
specialValue = nanValue | infinityValue
nanValue = "NaN"
infinityValue = ["+" | "-"] "Inf"
```

## Structural Characters

| Symbol | Name         | Unicode  | Description                       |
| ------ | ------------ | -------- | --------------------------------- |
| `NaN`  | Not a Number | Multiple | Represents invalid numeric result |
| `Inf`  | Infinity     | Multiple | Represents infinite value         |
| `-`    | Minus Sign   | `U+002D` | Indicates negative infinity       |
| `+`    | Plus Sign    | `U+002B` | Optional positive indicator       |

## Valid Forms

```ruby
NaN                  # Not a Number
Inf                  # Positive infinity
-Inf                 # Negative infinity
+Inf                 # Positive infinity (explicit)
```

## Optional Behaviors

### Literal and Alternate Forms

Special values have fixed representations:

```ruby
NaN                  # ✅ Standard form
Inf                  # ✅ Positive infinity
-Inf                 # ✅ Negative infinity
+Inf                 # ✅ Explicit positive infinity
```

### Mathematical Behavior

Special values follow IEEE 754 semantics:

```ruby
# NaN comparisons always return false
NaN == NaN           # false
NaN != NaN           # true

# Infinity comparisons
Inf > 100            # true
-Inf < 0             # true
```

## Invalid Forms

```ruby
nan                  # ❌ Lowercase not allowed
NAN                  # ❌ All caps not allowed
inf                  # ❌ Lowercase not allowed
INF                  # ❌ All caps not allowed
infinity             # ❌ Full word not supported
-NaN                 # ❌ NaN cannot be signed
```

## Preservation of Structure

Internet Object preserves:

* The exact representation of special values
* Negative sign for negative infinity
* Syntactic fidelity (as written, except that an explicit plus sign is not preserved)

However, it does **not** interpret:

* The mathematical operations that led to these values
* Domain-specific handling of special cases
* Comparison or equality semantics

Such semantics are the responsibility of the **schema layer**, **validators**, or **application logic**.

## See Also

* [Number](/the-structure/values/number/number.md) - For standard floating-point numbers
* [Values](/the-structure/values.md)


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