Polish Notation Calculator

What is Polish Notation (Prefix) and Reverse Polish Notation (RPN)?

In mathematics and computer science, mathematical expressions can be written in three primary syntactic notations:

• Infix Notation: The standard everyday format where binary operators are placed between operands (for example, $3 + 4$ or $(3 + 4) \times 2$). It relies on operator precedence rules (PEMDAS / BODMAS) and parentheses to resolve ambiguity.
• Prefix Notation (Polish Notation / PN): Invented in 1924 by Polish logician Jan Łukasiewicz, operators are placed before their operands (for example, $+ \; 3 \; 4$ or $\times \; + \; 3 \; 4 \; 2$). Prefix notation is completely parenthesis-free and unambiguous.
• Postfix Notation (Reverse Polish Notation / RPN): Operators are placed after their operands (for example, $3 \; 4 \; +$ or $3 \; 4 \; + \; 2 \; \times$). Developed by Charles Hamblin in the 1950s, RPN is widely used in stack-based virtual machines, compilers, and Hewlett-Packard scientific calculators.

How the Shunting-Yard Algorithm Works

Invented by Edsger Dijkstra, the Shunting-yard algorithm parses infix expressions into postfix or prefix representations using an operator stack:

1. Numbers and variables are immediately passed to the output queue.
2. Operators are pushed onto an operator stack after popping higher or equal precedence operators to the output.
3. Left parentheses $($ are pushed onto the operator stack.
4. Right parentheses $)$ trigger popping of all operators from the stack to the output until the matching $($ is removed.
5. At the end of the input, all remaining stack operators are popped to the output.

Evaluating Reverse Polish Notation (RPN) with a Stack

RPN expressions are evaluated linearly in $O(n)$ time using a last-in, first-out (LIFO) stack:

• When a number is encountered, push it onto the stack.
• When a binary operator ($+$, $-$, $\times$, $/$, \textasciicircum) is encountered, pop the top two operands ($b$ then $a$), compute $a \text{ op } b$, and push the result back onto the stack.
• Once all tokens are processed, the single value remaining on the stack is the final answer.

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