Reverse Polish Notation

Reverse Polish Notation (RPN, or postfix notation) is an arithmetic formula notation, introduced in 1920 by the Polish mathematician Jan Łukasiewicz[?]. In this notation, the operands precede the operator, thus dispensing with the need for parentheses. For example, the expression 3 * ( 4 + 7) would be written as 3 4 7 + *.

Implementations that use RPN are stack-based; that is, operands are popped from a stack, and calculation results are pushed back onto it. Although this concept may seem obscure at first, RPN has the advantage of being extremely easy for a computer to analyze due to it being a regular grammar. RPN has been used in Hewlett-Packard calculators as well as in some programming languages such as Forth and PostScript. Most Unix systems also comes with a calculator program that uses RPN: dc.

Practical implications

From the practical point of view, in RPN:

Calculations proceed from left to right
There are no brackets or parentheses, as they are unnecessary.
Operands precede operator. They are removed as the operation is evaluated.
When an operation is made, the result becomes an operand itself (for later operators)

Example

For instance, the calculation: ((1 + 2) * 4) + 3 can be written down like this in RPN:

 1 2 + 4 * 3 +

The expression is evaluated in the following way (the Stack is displayed after Operation has taken place):

Input	Stack	Operation
1	1	Push operand
2	1, 2	Push operand
+	3	Addition
4	3, 4	Push operand
*	12	Multiplication
3	12, 3	Push operand
+	15	Addition

The final result, 15, lies on the top of the stack in the end of the calculation.

Examples of RPN Use

Compare to Polish notation.