Reverse Polish Notation (RPN) Calculator

Reverse Polish notation, often shortened to RPN and also called postfix notation, is a way of writing mathematical expressions where every operator follows its operands instead of sitting between them. Instead of writing 3 + 4, you write 3 4 +. This ordering removes the need for parentheses and lets a simple stack-based process evaluate the expression without worrying about operator precedence rules.

Type numbers and operators separated by spaces, working left to right the same way you would say them aloud. For example, to calculate (3 + 4) times 2, you would type 3 4 + 2 *. The addition happens first because both of its operands, 3 and 4, appear before the plus sign, and the result is then multiplied by 2 once the multiplication sign appears after it.

This calculator supports addition, subtraction, multiplication, division, and exponentiation using the standard +, -, *, /, and ^ symbols, along with the unary functions sqrt for square root, neg for negation, and abs for absolute value. Numbers can include decimals and a leading minus sign for negative values entered directly onto the stack.

RPN maps directly onto how a simple stack machine processes instructions, which is why many classic HP calculators, the Forth programming language, and the Unix dc calculator utility all use it. A stack-based evaluator only needs to read tokens once, left to right, pushing numbers and applying operators as it goes, with no need to track parentheses or memorize operator precedence tables.

If an operator appears without enough numbers already on the stack, or if extra numbers are left over after every operator has been applied, this calculator flags the expression as invalid and explains where the mismatch occurred. A valid RPN expression always leaves exactly one number on the stack once every token has been processed.

Infix notation places an operator between its two operands, such as 3 + 4, and relies on a fixed order of operations plus parentheses to resolve ambiguity in longer expressions like 3 + 4 * 2. RPN instead places operators after their operands, so 3 + 4 * 2 becomes 3 4 2 * +, and the order tokens appear in is the only thing that determines how the expression is evaluated.

This calculator evaluates expressions that are already written in RPN rather than converting standard infix expressions into RPN form. Converting infix to RPN typically uses an algorithm called the shunting-yard algorithm, which reorders tokens based on operator precedence and parentheses before a stack-based evaluator like this one can process them.

Yes. Certain HP scientific and financial calculators still ship with an RPN entry mode that many engineers and finance professionals prefer for chaining calculations without pressing an equals key repeatedly. RPN also appears under the hood in stack-based virtual machines, PostScript page description language, and some programming language interpreters.