How Scientific Calculators Parse Mathematical Expressions
A scientific calculator is a crucial tool for engineers, students, and scientists. Unlike standard calculators that evaluate operations strictly in the order they are typed, scientific evaluators must parse strings of symbols and numbers according to strict rules of mathematical precedence. This system, commonly known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which sub-expressions are calculated.
To achieve this, computer programs and calculators use specialized parsing algorithms. The most prominent of these is the Shunting-Yard Algorithm, designed by Edsger Dijkstra. This algorithm translates infix notation—the standard way humans write math like 3 + 4 × 2—into postfix notation, also known as Reverse Polish Notation (RPN). RPN removes the need for parentheses by placing operators after their operands (e.g., 3 4 2 × +). Once in postfix form, a simple stack-based evaluator can resolve the entire expression without ambiguity.
Implicit operations, such as multiplying a number by a constant or a parenthesis block without an explicit multiplication symbol (e.g., 2π or 3(4+5)), are recognized by the tokenizer. The tokenizer inserts a multiplication operator automatically to ensure standard algebra rules are followed.