Standard Order of Operations
The Standard Order of Operations (SOO) is the conventional rule set used to evaluate arithmetic expressions written in linear notation. It is commonly taught through mnemonics such as PEMDAS, BEDMAS, BIDMAS, or BODMAS.
Within Metopedia's Canonical Order project, SOO is treated as a legacy convention: useful, widely adopted, and often efficient, but sometimes insufficiently explicit when signs, powers, roots, and grouping interact.
Common sequence
A common classroom version of the Standard Order is:
- parentheses or brackets
- exponents and roots
- multiplication and division
- addition and subtraction
Multiplication and division are normally evaluated left to right, as are addition and subtraction.
Strengths
The Standard Order gives a compact convention for ordinary expressions. It prevents expressions such as 2+3\times4 from requiring parentheses around every multiplication. It is also deeply integrated into education, calculators, programming languages, and mathematical typesetting.
Canonical critique
The Canonical Order critique is not that SOO fails in every case. It is that SOO often depends on convention where a stricter grammar would require explicit structure. The disputed areas include unary negatives, the base of an exponent, radical notation, and fractional exponents involving signed quantities.
| Issue | SOO handling | COO critique |
|---|---|---|
| Unary negative | Often resolved by precedence convention. | The sign should be explicitly separated from or included in the base. |
| Roots | Treated by radical notation and exponent notation. | Roots should be written as fractional exponents. |
| Parentheses | Treated as first operation. | Parentheses also define the object acted on by later operations. |
| Ambiguous input | Often settled by convention or calculator syntax. | Ambiguous expressions should be rewritten. |
Compatibility
COO usually agrees with SOO for ordinary arithmetic. The proposed change appears mainly in how expressions are written, taught, and disambiguated. For this reason, COO can be described as a notation and interpretation reform rather than a rejection of arithmetic itself.