How do programming languages specify operator precedence and associativity for multiplication, division, addition, and subtraction?

1. How precedence and associativity are defined in language specs

Languages usually encode precedence and associativity either explicitly in operator tables or implicitly in the grammar that defines expressions; C and C++ publish precedence tables showing multiplicative (, /, %) above additive (+, -) and list left‑to‑right associativity for those groups ^{[4] [5]}, while Java’s formal specification derives precedence from the grammar rather than a separate table even though practitioners use a table for readability ^[6]; JavaScript’s documentation describes the same two‑step model: first group by precedence, then by associativity for adjacent operators with equal precedence ^[3].

**2. The common rule for +, -, , and /**

Across mainstream imperative languages the multiplicative operators ( and /) are assigned a higher precedence than the additive operators (+ and -), so expressions like 6 + 3 4 / 2 are parsed with the multiplications/divisions performed before additions/subtractions ^{[2] [7]}; when operators share precedence — for example * and / or + and - — associativity decides grouping, and most languages make these arithmetic groups left‑associative so 4 / 3 / 2 is parsed as (4 / 3) / 2 ^{[7] [3]}.

3. What “left‑associative” actually means and why it matters

Left‑associativity means adjacent same‑precedence binary operators are grouped left‑to‑right: a - b + c becomes (a - b) + c unless parentheses say otherwise, and this grouping rule is only needed when precedence ties occur ^{[1] [8]}. Language documents emphasize that associativity governs parsing/grouping, not the runtime evaluation order of operands — some languages evaluate operands left‑to‑right regardless of associativity — an important distinction for expressions with side effects ^{[2] [3]}.

4. Exceptions, special cases, and observable quirks

Not all operators follow the simple left‑associative model: exponentiation in many languages is right‑associative (e.g., a b c is a (b c) in JavaScript), and some languages disallow chaining of certain operators entirely by marking them non‑associative in the grammar ^{[3] [1]}. Operator overloading in C++ changes semantics but does not change precedence rules—overloaded operators inherit the language’s precedence and associativity, which can surprise implementers who expect semantic overloading to alter parsing ^[5].

5. How implementers and authors present the rules to programmers

Practical references — tutorials, textbooks, and language reference pages — present precedence as an ordered table (highest to lowest) and list associativity beside each row so engineers can predict expression grouping, while language standards may simply encode those same relations inside the language grammar; authoritative docs (e.g., MDN for JavaScript, cppreference for C) recommend parentheses for clarity since tables are dense and subtle differences exist across languages ^{[4] [3] [6]}.

6. Limits of the available reporting and what remains language‑specific

The sources consistently describe the pattern for standard arithmetic operators, but they do not enumerate every language’s corner cases or historical design tradeoffs; where specific languages deviate from the common left‑associative multiplicative/additive model, the deviation must be confirmed in that language’s specification or reference because the provided material emphasizes the general rules and illustrative exceptions rather than an exhaustive cross‑language catalogue ^{[1] [6] [3]}.