Reverse Polish notation

Reverse Polish notation (RPN), also known as reverse Łukasiewicz notation, Polish postfix notation or simply postfix notation, is a mathematical notation in which operators follow their operands (e.g.

3

4

+

{\displaystyle 3\ 4\ +}

), in contrast to the more common infix notation (in which operators are placed between operands, e.g.

3

+

4

{\displaystyle 3+4}

), as well as prefix notation (in which operators precede their operands, e.g.

+

3

4

{\displaystyle +\ 3\ 4}

). The notation does not need any parentheses as long as each operator has a fixed number of operands.

Here are some examples to illustrate the differences between the aforementioned types of notation:

The term postfix notation describes the general scheme in mathematics and computer sciences, whereas the term reverse Polish notation typically refers specifically to the method used to enter calculations into hardware or software calculators, which often have additional side effects and implications depending on the actual implementation involving a stack. The description "Polish" refers to the nationality of logician Jan Łukasiewicz, who invented Polish notation in 1924.

The first computer to use postfix notation, though it long remained essentially unknown outside of Germany, was Konrad Zuse's Z3 in 1941 as well as his Z4 in 1945. The reverse Polish scheme was again proposed in 1954 by Arthur Burks, Don Warren, and Jesse Wright and was independently reinvented by Friedrich L. Bauer and Edsger W. Dijkstra in the early 1960s to reduce computer memory access and use the stack to evaluate expressions. The algorithms and notation for this scheme were extended by the philosopher and computer scientist Charles L. Hamblin in the mid-1950s.