## Encyclopedia > Reverse Polish Notation

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# 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
4 3, 4 Push operand
* 12 Multiplication
3 12, 3 Push operand
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.

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