Goal
Convert a bounded arithmetic expression from infix form to prefix Polish notation and postfix reverse Polish notation, then evaluate it without executing arbitrary code.
Polish Notation and RPN Calculator
Convert a bounded arithmetic expression from infix form to prefix Polish notation and postfix reverse Polish notation, then evaluate it without executing arbitrary code.
Key facts
- What it does
- Convert a bounded arithmetic expression from infix form to prefix Polish notation and postfix reverse Polish notation, then evaluate it without executing arbitrary code.
- Formula
Operator precedence and associativity convert infix tokens to postfix with a stack; the resulting expression tree produces prefix notation and a stack evaluation.- You enter
- Arithmetic expression
- Worked example
- Postfix = 3 4 2 2 ^ * +; prefix = + 3 * 4 ^ 2 2; value = 19.
A clearer path to an answer
From your question to a useful result
This page keeps the calculation transparent: define the goal, enter the matching values, inspect the method, and decide what the result means in your situation.
Inputs
Arithmetic expression
Method
Operator precedence and associativity convert infix tokens to postfix with a stack; the resulting expression tree produces prefix notation and a stack evaluation.
Next step
Calculate, review the assumptions below, then compare a related tool when the decision needs more context.
Polish Notation and RPN Calculator
Convert a bounded arithmetic expression from infix form to prefix Polish notation and postfix reverse Polish notation, then evaluate it without executing arbitrary code.
Open the Polish Notation and RPN Calculator pageMore math tools
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Result
Enter your values above and choose Calculate to see the result here.
Calculation map
Follow the path from input to answer
01
Inputs (1)
- Arithmetic expression Ready
02
Formula
Operator precedence and associativity convert infix tokens to postfix with a stack; the resulting expression tree produces prefix notation and a stack evaluation.
Bounded, transparent calculation
03
Result
- Calculate to preview the result.
Recent runs
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Formula, assumptions, and example
Formula: Operator precedence and associativity convert infix tokens to postfix with a stack; the resulting expression tree produces prefix notation and a stack evaluation.
Infix notation puts operators between operands, while Polish notation puts operators before them and reverse Polish notation puts them after them. This calculator converts a small explicit arithmetic language and shows the representations without using eval or accepting code.
- The expression contains finite numeric literals, parentheses, binary +, −, ×, ÷, exponentiation, and unary signs.
- Exponentiation and unary signs use the declared precedence model and exponentiation is right-associative.
- Implicit multiplication, variables, functions, commas, assignment, and comparisons are rejected.
- Division by zero and non-finite real results are rejected.
- The expression is limited to 240 characters and 40 tokens for predictable browser calculation.
- Prefix and postfix strings describe the parsed expression tree; they are not executable code.
- Real arithmetic follows ordinary finite-number behavior and may round decimal results.
Worked example: Postfix = 3 4 2 2 ^ * +; prefix = + 3 * 4 ^ 2 2; value = 19.
Displayed input contract
- Arithmetic expression
The displayed limits are checked before the handler runs. Model-specific domain checks may also reject impossible or non-finite inputs.
Methodology: This calculator follows the WorldCalculate input, formula, precision, and boundary policy. Read the official methodology.
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Answer-first guide
How to use the Polish Notation and RPN Calculator for a real question
Convert a bounded arithmetic expression from infix form to prefix Polish notation and postfix reverse Polish notation, then evaluate it without executing arbitrary code. Start with one clearly defined goal, enter values in the units shown, and keep the result attached to the assumptions below.
What this answers
This tool is useful when your question includes Polish notation calculator, reverse Polish notation, RPN converter. It returns the outputs declared in the calculator contract rather than a live quote, approval, diagnosis, or professional sign-off.
What you enter
Arithmetic expression. Keep the same time period, unit system, and currency wherever the form requires comparable values.
How to check it
Run the worked example first, compare its output with the page's example, then change one input at a time. This makes an unexpected result easier to trace to a unit, boundary, or assumption.
Three checks before you rely on the answer
- Match the question. Confirm that the result means the quantity you need, not a similar-sounding percentage, balance, rate, or estimate.
- Match the inputs. Use the requested units and period, and read each hint before replacing the example values with your own.
- Read the boundary. Review the assumptions and limits. The expression contains finite numeric literals, parentheses, binary +, −, ×, ÷, exponentiation, and unary signs.
Need a wider view? Browse Math Calculators or compare the related tools below. The WorldCalculate methodology explains how formulas, examples, limits, and revisions are reviewed.
How to use the Polish Notation and RPN Calculator
- Enter Arithmetic expression — Use finite numbers, parentheses, +, −, ×, ÷, and ^. No variables or functions.
- Choose Calculate and read the result panel.
- Use Download PDF or Download Word to save a result sheet.
Formula
Operator precedence and associativity convert infix tokens to postfix with a stack; the resulting expression tree produces prefix notation and a stack evaluation.
Infix notation puts operators between operands, while Polish notation puts operators before them and reverse Polish notation puts them after them. This calculator converts a small explicit arithmetic language and shows the representations without using eval or accepting code.
Worked example
Postfix = 3 4 2 2 ^ * +; prefix = + 3 * 4 ^ 2 2; value = 19.
Assumptions and limits
- The expression contains finite numeric literals, parentheses, binary +, −, ×, ÷, exponentiation, and unary signs.
- Exponentiation and unary signs use the declared precedence model and exponentiation is right-associative.
- Implicit multiplication, variables, functions, commas, assignment, and comparisons are rejected.
- Division by zero and non-finite real results are rejected.
- The expression is limited to 240 characters and 40 tokens for predictable browser calculation.
- Prefix and postfix strings describe the parsed expression tree; they are not executable code.
- Real arithmetic follows ordinary finite-number behavior and may round decimal results.
Who uses this calculator?
- Programming and computer-science students
- Learners practicing expression precedence
- Visitors converting calculator-style expressions
When is it useful?
- Convert an infix expression to RPN/postfix form.
- Compare prefix Polish and postfix representations.
- Check a stack-based evaluation for a small arithmetic expression.
Context and background
The mathematical structure behind the tools
Math calculators move from named quantities to a relation, then to a result that can be checked with substitution, units, or an alternate form.
Arithmetic, algebra, geometry, trigonometry, and number theory provide reusable structures for classroom work and everyday reasoning. Each page narrows that structure to one declared problem.
Research and review
How this guide was researched
Researched by Hassan ALRowaie, Founder and editorial researcher at WorldCalculate.
This guide follows the live calculator's declared inputs, formula, worked example, assumptions, validation boundaries, and source-backed methodology. The review date describes editorial review of the calculator explanation; it is not a promise that external facts or rates remain current.
An expression can have the same value in several notations, but the order of operations becomes easier for a machine when the order is explicit. This page converts a small arithmetic expression into prefix and postfix forms, then evaluates the same tree with a stack.
Infix, prefix, and postfix
Infix notation places an operator between operands, as in 3 + 4. Prefix Polish notation places the operator first: + 3 4. Postfix reverse Polish notation places it last: 3 4 +.
The prefix and postfix forms remove the need for most parentheses because operator position and operand order carry the structure. They are representations of the same expression.
Why precedence matters
In 3 + 4 × 2, multiplication is performed before addition. A converter must preserve that rule when it moves operators. Parentheses and exponentiation make the exact order even more important.
The handler uses an explicit precedence table rather than delegating to JavaScript evaluation. That keeps the supported language reviewable and prevents input text from becoming executable code.
The shunting-yard idea
The conversion scans tokens from left to right. Numbers go to output, while operators wait on a stack until precedence and associativity say they can be emitted. Parentheses hold a group temporarily.
When the scan finishes, remaining operators leave the stack in the order required by the expression. The postfix sequence can then be evaluated left to right with a value stack.
Building prefix notation
After postfix conversion, the page builds an expression tree. A number is a leaf, a binary operator has two children, and a unary sign has one. Reading that tree with the operator first produces prefix notation.
Both representations come from the same tree so a second independent interpretation does not silently disagree with the postfix evaluator.
Stack evaluation
To evaluate postfix, push each number. A binary operator pops the right operand and then the left operand, applies the operator, and pushes the result. A unary sign pops one value.
For 3 4 2 2 ^ * +, the stack first builds 2^2, multiplies by 4, and adds 3. The trace records token progression and stack depth without exposing internal code.
Supported input
Use decimal or scientific-notation numbers, parentheses, plus, minus, multiplication, division, exponentiation, and unary signs. Alternative minus and multiplication glyphs are normalized for convenience.
The parser rejects implicit multiplication, variables, function names, commas, and comparisons. A rejection is preferable to guessing whether 2x means a variable or a missing multiplication sign.
Signs and exponents
Unary minus is distinct from binary subtraction. The expression −3 + 5 begins with a unary sign, while 8−3 uses subtraction. The handler gives exponentiation and unary signs higher binding than multiplication and makes exponentiation right-associative.
For an ambiguous expression such as −2^2, add parentheses when the intended grouping matters. The declared assumptions should travel with a saved result.
Safety and model limits
The calculator never executes the typed expression. It tokenizes a bounded grammar, builds its own operator representation, checks division and finite results, and returns safe text and numbers.
It is not a symbolic algebra system or general compiler. It does not simplify variables, call functions, preserve exact fractions, or prove equivalence outside the supported finite real grammar.
Frequently asked questions
What is the Polish Notation and RPN Calculator?
Convert a bounded arithmetic expression from infix form to prefix Polish notation and postfix reverse Polish notation, then evaluate it without executing arbitrary code.
What is the formula for the Polish Notation and RPN Calculator?
Operator precedence and associativity convert infix tokens to postfix with a stack; the resulting expression tree produces prefix notation and a stack evaluation. Infix notation puts operators between operands, while Polish notation puts operators before them and reverse Polish notation puts them after them. This calculator converts a small explicit arithmetic language and shows the representations without using eval or accepting code.
What do I need to use this calculator?
Enter Arithmetic expression, then choose Calculate.
What are the limits of this calculator?
The expression contains finite numeric literals, parentheses, binary +, −, ×, ÷, exponentiation, and unary signs. Exponentiation and unary signs use the declared precedence model and exponentiation is right-associative. Implicit multiplication, variables, functions, commas, assignment, and comparisons are rejected. Division by zero and non-finite real results are rejected. The expression is limited to 240 characters and 40 tokens for predictable browser calculation. Prefix and postfix strings describe the parsed expression tree; they are not executable code. Real arithmetic follows ordinary finite-number behavior and may round decimal results.
Methodology
This calculator is part of the WorldCalculate library. Its formula, example, assumptions, input bounds, and output formatting follow the official methodology.
Use this calculator as part of a bigger plan
These WorldCalculate collections connect this tool with related questions while keeping each calculation separate and transparent.