Polish Notation Converter

Convert infix expressions to prefix and postfix notations

Enter a standard mathematical expression (infix notation) to see its equivalent in Polish (Prefix) and Reverse Polish (Postfix) notations. Supports operators +, -, *, /, ^ and parentheses.

Other Titles
Understanding Polish Notation: A Comprehensive Guide
Dive into the world of prefix and postfix notations, the parent-free ways of writing mathematical expressions that are fundamental to computer science.

What is Polish Notation?

In mathematics, we typically write expressions like '3 + 4'. This is called Infix Notation because the operator (+) is in between the operands (3 and 4). While intuitive for humans, it's ambiguous for computers without strict rules for operator precedence and parentheses.
Polish Notation, also known as Prefix Notation, was introduced by logician Jan Łukasiewicz. It places the operator before its operands. The infix expression '3 + 4' becomes '+ 3 4' in prefix notation.
Reverse Polish Notation (RPN), or Postfix Notation, places the operator after its operands. The same expression '3 + 4' becomes '3 4 +' in postfix notation.
The key advantage of both prefix and postfix notations is that they are unambiguous and do not require parentheses or precedence rules to be evaluated. This makes them ideal for computation, and they are widely used in computer stack-based calculators and programming language parsers.

Notation Examples

  • **Infix:** 5 * (6 + 2)
  • **Prefix (Polish):** * 5 + 6 2
  • **Postfix (Reverse Polish):** 5 6 2 + *
  • To evaluate the postfix example: Push 5, push 6, push 2. See '+', pop 6 and 2, calculate 8, push 8. See '*', pop 5 and 8, calculate 40, push 40. Result is 40.

Step-by-Step Guide to Using the Converter

This tool uses a modified version of the Shunting-yard algorithm to perform the conversions.
How to Convert:

Usage Tips

  • Ensure your expression is well-formed. For example, '3 + * 4' is invalid.
  • Use spaces between tokens for clarity, although the calculator can often handle expressions without them (e.g., '3+4*2' works).
  • Check for mismatched parentheses, as this is a common source of errors.

Real-World Applications of Polish Notation

While it looks abstract, Polish notation is a cornerstone of computer science.
Computing and Programming:
Programming Languages:

In Practice

  • A computer evaluating `(a+b)*c` first converts it to `a b + c *` (postfix).
  • It pushes `a`, pushes `b`, executes `+` (replacing `a` and `b` with their sum), pushes `c`, then executes `*`.
  • This stack-based approach is mechanically simple and efficient for a machine to process.

Common Misconceptions and Correct Methods

Misconception 1: Word Order is Arbitrary
The order is extremely strict. In postfix A B +, the + applies to the two most recent operands, A and B. You cannot write + A B and expect it to be evaluated the same way as postfix.
Misconception 2: It's Just for Math
The concept applies to any logical operation. For instance, the logical statement (A AND B) OR C can be written in prefix as OR (AND A B) C or in postfix as A B AND C OR.

Key Takeaways

  • Infix: Operator is in-between operands.
  • Prefix: Operator is before operands.
  • Postfix: Operator is after operands.

The Shunting-yard Algorithm

The algorithm used for this conversion was developed by Edsger Dijkstra. It parses an infix expression and produces a postfix expression using a main output queue and an operator stack.
How it Works (for Postfix):
For Prefix Notation:
A common way to get the prefix notation is to reverse the infix expression, swap all parentheses, run the postfix algorithm, and then reverse the final result.

Example: 3 + 4 * 2

  • 1. Read 3: Output = [3]
  • 2. Read +: Stack = [+]
  • 3. Read 4: Output = [3, 4]
  • 4. Read *: '*' has higher precedence than '+'. Stack = [+, *]
  • 5. Read 2: Output = [3, 4, 2]
  • 6. End of expression. Pop stack to output.
  • 7. Pop *: Output = [3, 4, 2, *]
  • 8. Pop +: Output = [3, 4, 2, *, +]
  • Result (Postfix): 3 4 2 * +