Polish Notation Converter

Infix notation is the standard way most people write arithmetic expressions, where the operator is placed between the operands — for example, A + B or (A * B) + C. It requires parentheses and operator precedence rules to determine the order of operations, making it natural for humans to read but more complex for computers to parse. See also our Scientific Notation Calculator.

Polish notation, also called prefix notation, was invented by the Polish mathematician Jan Łukasiewicz. In this notation, operators are placed before their operands — for example, + A B instead of A + B. It eliminates the need for parentheses entirely, since the order of operations is unambiguous from the position of the operators.

Reverse Polish Notation (RPN), or postfix notation, places operators after their operands — for example, A B + instead of A + B. It is widely used in stack-based calculators and programming language compilers because expressions can be evaluated left-to-right using a simple stack without any need for parentheses or precedence rules.

To convert infix to prefix, first reverse the infix expression (swapping parentheses), then apply the infix-to-postfix algorithm, and finally reverse the resulting postfix expression. Alternatively, you can use a recursive descent approach by identifying the root operator at the lowest precedence level and recursively converting left and right sub-expressions. You might also find our use the Round to Nearest Whole Number Calculator useful.

The postfix (Reverse Polish Notation) of A+B is simply A B +. The two operands A and B are written first, followed by the + operator. To evaluate it, push A onto a stack, push B onto the stack, then pop both values, apply +, and push the result back.

Neither is objectively better — they serve different purposes. Infix notation is more natural and readable for humans. Polish and Reverse Polish notations are preferred in computing contexts because they are unambiguous without parentheses and can be evaluated efficiently using a stack, making them ideal for calculators, compilers, and expression parsers.

RPN is used in many scientific calculators (like classic HP models) because it allows continuous chain calculations without needing parentheses or an equals key. Users enter operands as they appear and apply operators immediately, which reduces keystrokes and closely mirrors how a stack-based processor evaluates expressions internally.

This converter supports the five most common arithmetic operators: addition (+), subtraction (-), multiplication (*), division (/), and exponentiation (^). It respects standard operator precedence (^ highest, then * and /, then + and -) and handles parentheses for explicit grouping in infix expressions.