How Modulo Fits in the Order of Operations
The standard order of operations — PEMDAS/BODMAS (parentheses, exponents, multiplication/division, addition/subtraction) — usually skips over the modulo operator, but it isn't a special case: % is a form of division, so it sits at the same precedence tier as multiplication and division, above addition and subtraction. This calculator takes a three-term expression, A op1 B op2 C, and evaluates it using that exact rule, showing you which pair of numbers is grouped first and why.
Where % ranks in PEMDAS
In an expression like 17 + 5 % 3, the modulo runs before the addition because it shares the multiplication/division tier: 5 % 3 = 2 first, then 17 + 2 = 19. Without that rule you might expect (17 + 5) % 3 = 22 % 3 = 1 — a completely different, and incorrect, answer. The only way to force that alternate grouping is to add explicit parentheses.
Evaluating a three-term expression
For A op1 B op2 C, compare the precedence of op1 and op2. If op2 outranks op1 (for example + then %), evaluate B op2 C first, then combine that result with A using op1. If op1 outranks op2, or the two operators share a precedence tier (both from {×, ÷, %} or both from {+, −}), evaluate strictly left to right: (A op1 B) op2 C. This calculator applies that rule automatically and displays both intermediate steps.
Common mistakes
- Treating % as the lowest-precedence operator: a % b + c means (a % b) + c, not a % (b + c) — modulo binds exactly as tightly as multiplication and division.
- Assuming mod always returns a nonnegative number: in JavaScript, C, Java, and most calculators, the sign of a % b matches the sign of a (the dividend), so -7 % 3 = -1, not 2.
- Forgetting parentheses when you want a different grouping: if you want (A + B) % C instead of A + (B % C), you must write the parentheses — default precedence always groups % with its neighboring × / ÷ first.
Where this shows up
Order-of-operations questions involving modulo come up constantly in programming and everyday math: hashing (index = key % tableSize), parity checks (n % 2), clock and calendar arithmetic (hour % 12), cyclic array indexing, and checksum digits. Misjudging when % actually fires relative to the surrounding + or − can silently shift an index or a wrapped value by the wrong amount.