# Audit: The viral expression "8 ÷ 2(2+2)" equals 16

- **Generated:** 2026-03-28
- **Reader summary:** [proof.md](proof.md)
- **Proof script:** [proof.py](proof.py)

---

## Claim Specification

| Field | Value |
|---|---|
| Subject | 8 ÷ 2(2+2) |
| Property | Arithmetic value under standard PEMDAS/BODMAS order of operations |
| Operator | == |
| Threshold | 16 |
| Operator note | The expression '8 ÷ 2(2+2)' is famously ambiguous due to the notation '2(2+2)', which could be parsed two ways. (1) Standard PEMDAS/BODMAS: multiplication and division have equal precedence and are evaluated left-to-right, so '2(2+2)' means '× (2+2)' and the full expression parses as '(8 ÷ 2) × (2+2) = 4 × 4 = 16'. (2) Implicit-multiplication-first convention: juxtaposition (writing a coefficient directly against a parenthesis) binds more tightly than explicit division, so the expression parses as '8 ÷ [2 × (2+2)] = 8 ÷ 8 = 1'. This proof adopts interpretation (1) — the standard left-to-right PEMDAS rule, which is the convention used by Python, most scientific calculators, and ISO 80000-2. The adversarial checks document that interpretation (2) is also used in some academic contexts and yields a different result. |

---

## Fact Registry

| ID | Type | Label |
|---|---|---|
| A1 | Type A (Computed) | Left-to-right PEMDAS evaluation of 8 ÷ 2(2+2): (8÷2)×(2+2) |
| A2 | Type A (Computed) | Algebraic rearrangement cross-check using commutativity: 8×(2+2)÷2 |
| A3 | Type A (Computed) | Alternative convention (implicit multiplication higher precedence): 8÷[2×(2+2)] |

---

## Full Evidence Table

### Type A (Computed) Facts

| ID | Fact | Method | Result |
|---|---|---|---|
| A1 | Left-to-right PEMDAS evaluation of 8 ÷ 2(2+2) | (8÷2)×(2+2) = 4×4 — left-to-right PEMDAS | 16.0 |
| A2 | Algebraic rearrangement cross-check using commutativity | 8×(2+2)÷2 = 32÷2 — algebraic rearrangement via commutativity | 16.0 |
| A3 | Alternative convention (implicit multiplication higher precedence) | 8÷[2×(2+2)] = 8÷8 — implicit multiplication convention | 1.0 |

*Note: A3 is computed for adversarial reference only. It establishes that the alternative convention yields 1, not 16.*

---

## Computation Traces

```
=== PRIMARY COMPUTATION: Left-to-right PEMDAS ===
Expression: 8 ÷ 2(2+2)

Step 1 — Parentheses: (2+2) = 4
  Reduced expression: 8 ÷ 2 × 4

  Step 2a — 8 ÷ 2 (leftmost operation): dividend / divisor = 8 / 2 = 4.0000

  Step 2b — 4 × (2+2): step2_div * inner = 4.0 * 4 = 16.0000

Primary result: 16.0

=== CROSS-CHECK: Algebraic rearrangement ===
Using commutativity: a ÷ b × c = a × c ÷ b  (i.e., ÷b = ×(1/b))
So: 8 ÷ 2 × (2+2) = 8 × (2+2) ÷ 2

  Step 1 — 8 × (2+2): a_val * c_val = 8 * 4 = 32

  Step 2 — 32 ÷ 2: numerator / divisor = 32 / 2 = 16.0000

Cross-check result: 16.0

Cross-check passed: 16.0 == 16.0 ✓

=== PYTHON BUILT-IN EVALUATION ===
Python evaluates 8 / 2 * (2 + 2) left-to-right per IEEE 754 / CPython semantics:
  8 / 2 * (2 + 2) = 16.0
Python evaluation agrees with primary result: 16.0 ✓

=== ALTERNATIVE CONVENTION: Implicit multiplication higher precedence ===
Under this convention: 8 ÷ 2(2+2) parses as 8 ÷ [2 × (2+2)]
  Implicit grouping: 2 × (2+2): coeff * inner = 2 * 4 = 8

  8 ÷ [2×(2+2)]: dividend / implicit_product = 8 / 8 = 1.0000

Alternative convention result: 1.0

=== VERDICT ===
  8 ÷ 2(2+2) under PEMDAS == 16: 16.0 == 16 = True
```

---

## Cross-Checks (Rule 6)

Two independent methods were used, both confirming 16:

| Description | Value A | Value B | Agreement |
|---|---|---|---|
| Primary (left-to-right PEMDAS: (8÷2)×4) vs. algebraic rearrangement (8×4÷2) — different computation order | 16.0 | 16.0 | ✓ |
| Primary vs. Python built-in evaluation of `8 / 2 * (2 + 2)` — CPython left-to-right IEEE 754 semantics | 16.0 | 16.0 | ✓ |

**Independence rationale:** The primary method applies left-to-right PEMDAS sequentially (divide first, then multiply). The algebraic rearrangement method applies commutativity of multiplication to change the computation order (multiply first, then divide). These are structurally different derivations: a bug in step-ordering logic in the primary method would not affect the rearrangement, and vice versa.

---

## Adversarial Checks (Rule 5)

**Check 1:** Under the alternative implicit-multiplication-first convention, is the answer 1 rather than 16, making the claim false under that reading?

- **Performed:** Computed 8 ÷ [2 × (2+2)] = 8 ÷ [2 × 4] = 8 ÷ 8 = 1. This confirms the expression IS genuinely ambiguous. The AMS (American Mathematical Society) and PEMDAS as taught in many US schools treat juxtaposition as higher precedence than explicit ÷. Wolfram Alpha returns 16 by default for '8÷2(2+2)' but acknowledges the ambiguity.
- **Finding:** Under the implicit-multiplication-first convention, the answer is 1. This does NOT break the proof because the proof explicitly states it uses the standard left-to-right PEMDAS convention, under which the answer is 16. The controversy exists precisely because both conventions are in real use.
- **Breaks proof:** No

**Check 2:** Does any widely-used authoritative standard (ISO, NIST, etc.) mandate the implicit-multiplication-first convention for this type of expression?

- **Performed:** Searched for ISO 80000-2 (Mathematical signs and symbols) rules on operator precedence. ISO 80000-2:2019 specifies that when × and ÷ appear in sequence without parentheses, left-to-right evaluation applies: 'a × b ÷ c = (a × b) ÷ c'. The standard also recommends using parentheses to avoid ambiguity in expressions like '8 ÷ 2(2+2)', since the juxtaposition notation is inherently unclear. No major international standard mandates implicit-multiplication-first for this form.
- **Finding:** ISO 80000-2 supports left-to-right evaluation (answer: 16) and explicitly recommends parentheses to resolve ambiguity. No authoritative standard mandates the alternative convention. Does not break the proof.
- **Breaks proof:** No

**Check 3:** Could a rounding or floating-point error cause the Python evaluation to differ from the exact integer result of 16?

- **Performed:** Checked: 8, 2, 2, and 2 are exact integers representable as IEEE 754 doubles. 8 / 2 = 4.0 exactly; 2 + 2 = 4 exactly; 4.0 * 4 = 16.0 exactly. All intermediate values are powers of 2 or small integers — no floating-point rounding occurs. Python result 16.0 == 16 is mathematically exact.
- **Finding:** No floating-point error. The result 16.0 equals exactly 16. Does not break the proof.
- **Breaks proof:** No

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*Generated by [proof-engine](https://github.com/yaniv-golan/proof-engine) v0.10.0 on 2026-03-28.*
