Polish Notation Converter

Convert between Infix, Prefix (Polish), and Postfix (Reverse Polish) notations.

Enter your mathematical expression and choose the desired conversion type.

Examples

See how to use the converter with these common examples.

Infix to Prefix

infix-to-prefix

Convert a standard infix expression to its prefix equivalent.

Expression: (A + B) * C - D

Infix to Postfix

infix-to-postfix

Convert a standard infix expression to its postfix equivalent.

Expression: (A + B) * C - D

Prefix to Infix

prefix-to-infix

Convert a prefix expression back to its standard infix form.

Expression: * + A B - C D

Postfix to Infix

postfix-to-infix

Convert a postfix expression back to its standard infix form.

Expression: A B + C D - *

Other Titles
Understanding Polish Notation: A Comprehensive Guide
An in-depth look at infix, prefix, and postfix notations, their applications, and conversion methods.

What is Polish Notation?

  • The Basics of Mathematical Notations
  • Prefix vs. Postfix: Key Differences
  • Why Use Different Notations?
In mathematics and computer science, the way we write expressions can significantly impact how they are parsed and evaluated. While we are most familiar with infix notation (e.g., 3 + 4), there are two other important forms: prefix notation (Polish Notation) and postfix notation (Reverse Polish Notation). This calculator helps you seamlessly convert between them.
Infix, Prefix, and Postfix Explained
Infix Notation: The operator is written between the operands (e.g., A + B). This is the standard human-readable format, but it often requires parentheses to enforce the order of operations.
Prefix Notation (Polish Notation): The operator is written before the operands (e.g., + A B). It was introduced by logician Jan Łukasiewicz. A key advantage is that it's unambiguous and never requires parentheses.
Postfix Notation (Reverse Polish Notation - RPN): The operator is written after the operands (e.g., A B +). This notation is highly efficient for computer evaluation, as it can be processed straightforwardly using a stack.

Notation Examples

  • Infix: (5 - 2) * 3
  • Prefix: * - 5 2 3
  • Postfix: 5 2 - 3 *

Step-by-Step Guide to Using the Polish Notation Converter

  • Selecting the Right Conversion
  • Entering Your Expression
  • Interpreting the Results
Using this tool is straightforward. Follow these simple steps to convert your expressions.
Conversion Process
1. Enter Expression: Type or paste your mathematical expression into the 'Expression' input field. You can use variables (A, B, C) or numbers, along with standard operators (+, -, *, /, ^).
2. Choose Conversion Type: From the dropdown menu, select the conversion you want to perform. For example, if you have an infix expression like (3 + 4) * 5, you would choose 'Infix to Prefix' or 'Infix to Postfix'.
3. Calculate: Click the 'Convert' button to process the expression.
4. View Result: The converted expression will appear in the 'Result' section below. You can easily copy it using the copy button.

Example Walkthrough

  • Input Expression: `A * (B + C)`
  • Selected Conversion: `Infix to Postfix`
  • Result: `A B C + *`

Real-World Applications of Polish Notation

  • Compiler Design and Expression Parsing
  • Stack-Based Calculators
  • Data Structures and Algorithms
While infix notation is intuitive for humans, prefix and postfix notations are fundamental in computer science for several reasons.
Use Cases in Computing
Compilers: When a compiler reads your code, it often converts infix expressions into an intermediate representation like an Abstract Syntax Tree (AST), which is closely related to prefix/postfix notations. This makes it easier to evaluate and generate machine code.
Stack-Based Systems: Postfix notation is the foundation of stack-based calculators (like many early HP models) and programming languages like Forth and PostScript. Evaluation is simple: read the expression, push numbers onto a stack, and when an operator is encountered, pop the required operands, perform the operation, and push the result back.
Algorithmic Efficiency: Because they eliminate the need for parentheses and complex precedence rules, these notations simplify algorithms for expression evaluation.

Practical Scenario

  • A spreadsheet program parsing the formula `= (A1+B1)/2` might convert it to `A1 B1 + 2 /` (postfix) for efficient calculation.

Common Misconceptions and Correct Methods

  • Operator Precedence in Conversions
  • Handling Parentheses Correctly
  • Dealing with Variables and Numbers
Converting between notations requires careful attention to rules, especially regarding operator precedence and associativity.
Key Conversion Principles
Misconception: Just reordering is enough. Simply moving operators isn't sufficient. The conversion from infix to postfix/prefix is often done using the Shunting-yard algorithm, which uses a stack to correctly handle the order of operations (e.g., multiplication before addition).
Correct Method (Infix to Postfix): When converting 3 + 4 * 2, the algorithm knows that * has higher precedence. It processes 3, then holds +, processes 4, then *, then 2. The * is applied to 4 and 2 first, resulting in 3 4 2 * +.
Parentheses are king: Parentheses override the default precedence. In (3 + 4) * 2, the parentheses force the + operation to be evaluated first, leading to the postfix 3 4 + 2 *.

Precedence Example

  • Infix: `A + B * C ^ D`
  • Correct Postfix: `A B C D ^ * +` (Order: ^, *, +)
  • Incorrect Postfix: `A B + C * D ^`

Mathematical Derivation and Algorithms

  • The Shunting-Yard Algorithm
  • Prefix to Infix Conversion Logic
  • Postfix to Infix Conversion Logic
The conversions are not arbitrary; they are based on well-defined algorithms that ensure the logical structure of the expression is preserved.
Core Algorithms
Infix to Postfix (Shunting-Yard): This algorithm, invented by Edsger Dijkstra, uses an operator stack and an output queue. It scans the infix expression token by token. Operands are immediately added to the output. Operators are pushed onto the stack, but before pushing, any operators on the stack with higher or equal precedence are popped and added to the output. Parentheses are used to group operations.
Prefix/Postfix to Infix: This conversion is also stack-based. For postfix-to-infix, you read the expression from left to right. When you see an operand, push it onto a stack. When you see an operator, pop the top two operands, combine them with the operator (placing parentheses around the new sub-expression), and push the resulting string back onto the stack. The final item on the stack is the infix expression. A similar process, reading from right to left, works for prefix-to-infix conversion.

Algorithm Snapshot (Postfix to Infix)

  • Expression: `A B + C *`
  • 1. See `A`, push `A`.
  • 2. See `B`, push `B`.
  • 3. See `+`, pop `B`, pop `A`, push `(A + B)`.
  • 4. See `C`, push `C`.
  • 5. See `*`, pop `C`, pop `(A + B)`, push `((A + B) * C)`.