Polish Notation Calculator - Infix, Prefix, and Postfix

A polish notation calculator is a tool that converts arithmetic expressions between the three common notations - infix (3 + 4), prefix (+ 3 4, also called Polish), and postfix (3 4 +, also called Reverse Polish or RPN) - and evaluates any of them on a stack-based engine. You paste an expression, pick a mode, and read the converted form or the numeric result, with the token list and a stack trace shown alongside. Practical contexts include computer science coursework on expression parsing, exam preparation for stack-based calculators like vintage HP RPN models, compiler design exercises that use Dijkstra's shunting-yard algorithm, and quick verification when hand-converting expressions during a math or logic class. The polish notation calculator is the single page that handles all three of these notations, so the rest of this guide focuses on when each form is the right tool for the job.

• • Compiler and parser coursework: Convert infix to postfix and confirm the output of a hand-written shunting-yard implementation.
• • RPN calculator practice: Evaluate 3 4 + 7 2 - * style expressions used on stack-based calculators.
• • Reverse-engineering older calculators: Translate postfix keystrokes from vintage HP calculators into infix to understand what they did.
• • Logic and discrete-math teaching: Demonstrate prefix and postfix variants of propositional logic formulas without precedence rules.

Polish notation was introduced in the 1920s by the Polish logician Jan Łukasiewicz as a way to write logical formulas without brackets; the reverse (postfix) form later became popular for stack-based calculators and compilers because each operator is applied to the values already on the stack. According to the Stanford Encyclopedia of Philosophy - Jan Łukasiewicz, Łukasiewicz designed what he called "parenthesis-free notation" in 1924 to keep propositional logic from being obscured by grouping symbols, which is why the prefix form still carries his name today.

When your expression mixes ^ with very large or very small numbers, Exponential Notation Calculator handles the scientific-notation formatting once the conversion is finished.

The calculator first splits the input into tokens, then runs the matching algorithm: Dijkstra's shunting-yard for infix-to-postfix, a modified shunting-yard with strict-greater precedence for infix-to-prefix, and a simple stack walk for the two evaluation modes. Every step records the current output queue and operator stack so you can see why each token went where it did.

• expression: Numbers, single letters, and the five operators + - * / ^ with optional parentheses; tokens are separated by spaces in prefix or postfix modes.
• mode: One of infix-to-postfix, infix-to-prefix, postfix-to-infix, prefix-to-infix, evaluate-postfix, evaluate-prefix.
• operator precedence: Exponentiation (^) binds tightest, then * and /, then + and -; left parentheses are a stack marker only.

The same shunting-yard engine drives both infix-to-postfix and infix-to-prefix; for prefix the tokens are first reversed, parentheses are swapped, and the pop condition changes from greater-or-equal precedence to strictly greater precedence. According to Wikipedia, this small change in the comparison operator is what differentiates the prefix and postfix outputs from the same infix source, which is why the calculator exposes it as a single mode toggle rather than two separate engines.

According to GeeksforGeeks - Shunting Yard Algorithm, the shunting-yard algorithm, invented by Edsger W. Dijkstra in 1961, parses expressions with operator precedence and parentheses by walking each token once and dispatching it to one of two structures: an output queue for the eventual RPN string, and an operator stack that defers higher-precedence work until a lower-precedence operator or a right bracket triggers a flush.

To sanity-check the ^ step in any converted expression, Exponent Calculator can evaluate the base and exponent separately before re-running the polish notation engine.

Four ideas cover almost everything the calculator does; learning them once is enough to predict the result of any mode.

Once these four ideas are clear, the calculator's six modes are just different ways of combining them: infix to postfix is shunting-yard; infix to prefix is reversed shunting-yard with strict-greater precedence; postfix evaluation is a single stack walk; postfix-to-infix is the same walk with the result written as a string and re-bracketed at every operator.

Boolean expressions follow the same prefix rule for connectives like AND, OR, and NOT, so Boolean Algebra Calculator lets you test the same stack-based evaluation on logical formulas once the arithmetic versions feel natural.

Run any of the six modes with the same two controls.

1. 1 Choose a mode: Pick infix-to-postfix, infix-to-prefix, postfix-to-infix, prefix-to-infix, evaluate-postfix, or evaluate-prefix from the conversion mode dropdown.
2. 2 Type the expression: Enter numbers, single-letter variables, and the operators + - * / ^; for prefix and postfix modes, leave a single space between every token.
3. 3 Read the result: Watch the result, token list, and stack trace update in real time as you type. The result shows the converted expression or the numeric evaluation.
4. 4 Verify the trace: If a value is wrong, scan the stack trace to find the line where the output queue and operator stack diverged from what you expected.
5. 5 Switch to round-trip check: After converting to postfix, paste the output back into the expression box in evaluate-postfix mode to confirm the numeric value matches the original infix expression.

Practical example: type 3+2-7*1 in infix-to-prefix mode to see - + 3 2 * 7 1, then switch to evaluate-prefix and paste the prefix back to confirm the value is -2, matching the original infix expression evaluated left-to-right with standard precedence.

When the polish notation engine returns a numeric value for a single expression, System of Equations Calculator extends that numeric verification to a set of constraints at once, so the same evaluated result can be checked against a 2x2 or 3x3 linear system.

The calculator is useful well beyond a quick numeric answer.

• • Verifies shunting-yard homework: Compare your hand-converted output against the calculator's result when you are learning Dijkstra's algorithm in a compilers or data structures class.
• • Decodes RPN keystrokes: Paste a stack of values and operators from an older HP calculator and the calculator returns the infix expression they represent, including the brackets the RPN form skips.
• • Catches precedence mistakes: When an infix expression gives a different value than the prefix or postfix form, the trace shows exactly which operator fired in the wrong order.
• • Works with letters or numbers: Single-letter variables such as a, b, or x are accepted everywhere; they are treated as placeholder operands so you can test the notation on algebraic expressions.
• • Self-contained reference: The same page shows the formula, the variable definitions, the worked example, and the step-by-step trace, so you do not need to flip between a tutorial and a tool.

For scientific-notation constants that show up once the polish notation value is computed, Scientific Notation Equation Calculator performs the cross-checks with the right number of significant figures.

A few conventions and limits affect every result the calculator produces.

• • The calculator does not support unary minus; write -3 as 0-3 or wrap a negative number in parentheses such as 0-3 inside a larger expression.
• • Exponentiation is right-associative as in standard math, but the modified shunting-yard for prefix still applies a strict-greater precedence rule, which can differ from textbook examples for chains of ^; verify multi-step exponent expressions with the evaluate modes.
• • Floating-point results from ^, /, and large numbers are rounded to six decimal places to keep the displayed value readable; use the raw token list if you need the exact form for an exam or homework.

These limits match the rules used by the source calculators, and keeping the same rules means the prefix, postfix, and infix forms always agree on the value they should produce.

According to GeeksforGeeks - Convert Infix Expression to Prefix Expression, the infix-to-prefix variant of the shunting-yard algorithm reverses the token stream, swaps the roles of left and right parentheses, and only flushes an operator from the stack when the incoming operator's precedence is strictly greater, which is what reverses the associativity of the original scan.