RPNs

RPNs, or reverse Polish notation, are a form of postfix mathematical notation in which operators follow their operands. In RPN expressions, elements are processed with a stack: operands are pushed onto the stack until an operator appears, at which point the operator consumes the required number of operands from the top of the stack and pushes the result. RPN is widely used in calculators and in some programming languages.

Origin and history: The concept traces to early Polish notation developed by Jan Łukasiewicz, which influenced later postfix forms. In computing, RPN gained popularity in the 1970s with Hewlett-Packard's handheld calculators, which used RPN for efficient input. The approach later influenced stack-based languages such as Forth and PostScript.

How it works: Push operands onto a stack; on binary operators, pop two values, apply the operator, and push the result. Example: 3 4 + evaluates to 7. A more complex expression: 5 1 2 + 4 * + 3 - evaluates to 14, because 1+2=3, 3*4=12, 5+12=17, 17-3=14.

Advantages and limitations: RPN minimizes or eliminates the need for parentheses and explicit precedence rules, which can reduce typographical effort and be faster on stack machines. It can be less intuitive to people trained in infix notation, and errors can occur if the stack is not managed correctly; some calculators offer both RPN and infix modes.

Relation to broader notation: RPN is a type of postfix notation and is related to languages like Forth and PostScript, where programs are built as sequences of operands and operators. It remains a niche but enduring approach in certain hardware, software, and educational contexts.