Postfix Operator: Understanding Its Role and Application in Notation

A comprehensive overview of postfix operators, including their definition, usage, examples, historical context, and comparisons with other operators.

What is a Postfix Operator?

A postfix operator is an operator that is written immediately after its operand(s). In this notation, also called Reverse Polish Notation (RPN), the operator follows the operands. This is in contrast to the more conventional infix notation, where operators are placed between operands. An example of a postfix expression is \(3 , 4 , +\), which denotes the addition of 3 and 4.

Notation and Evaluation

In postfix notation, there is no need for parentheses to indicate the order of operations, which simplifies the evaluation process. The expression \(a , b , OP\) is evaluated as follows:

• Operands First: Identify the operands (e.g., \(a\) and \(b\)).
• Apply the Operator: Apply the operator \(OP\) to the operands \(a\) and \(b\).

For illustration, consider the evaluation of the postfix expression:

$$ 5 \, 6 \, 2 \, + \, * $$

It is equivalent to:

$$ 5 \times (6 + 2) $$

This facilitates a stack-based approach to expression evaluation.

Types of Postfix Operators

• Arithmetic Operators: Used in mathematical expressions (e.g., \( , +, , -, , *, , / \)).
• Increment and Decrement Operators: Common in programming languages like C and C++ (e.g., \(i++\) and \(i–\)).

Historical Context and Applicability

Reverse Polish Notation (RPN) was introduced by the Polish mathematician Jan Łukasiewicz in the early 20th century. It became popular in computing for its efficiency in expression evaluation and the simplification it brings to the parsing process.

Comparing Notations

Infix Notation:

$$a + b$$

Postfix Notation:

$$a \, b \, +$$

Prefix Notation:

$$\+ \, a \, b$$

Practical Examples

• Arithmetic Calculation: Postfix: \(3 , 4 , +\) Evaluation: \(3 + 4\)

• Programming Iteration:

  1int i = 5;
  2int x = i++; // x is assigned 5, then i is incremented to 6
  

Related Terms

• Infix Operator: An operator placed between its operands.
• Prefix Operator: An operator placed before its operands.
• Stack: A data structure used to evaluate postfix expressions.

FAQs

1. What are the advantages of using postfix notation?

• Simplifies the computational process by eliminating the need for parentheses to define the order of operations.

2. How is postfix notation used in computers?

• Used in compilers and calculators for efficient arithmetic computation and expression parsing.

3. Can postfix notation be converted to infix notation?

• Yes, with the help of algorithms or parsing techniques, postfix expressions can be converted back to infix notation.

References

• Łukasiewicz, Jan. “On the principle of contradiction in Aristotle.” (1910).
• Davis, Martin. “The Universal Computer: The Road from Leibniz to Turing.” (2001).

Summary

Postfix operators offer a streamlined, unambiguous approach to expressing and evaluating mathematical and logical expressions. Their application in RPN has significantly impacted computing through more efficient processing and simpler syntax parsing. Understanding postfix notation is foundational for programmers, mathematicians, and those involved in computational fields.