## Introduction: The Mathematical Puzzle

In the world of mathematics and computer science, expressing and evaluating mathematical expressions is akin to solving a puzzle. Traditional methods, like infix notation, often require us to navigate through complex rules of precedence and parentheses. However, Reverse Polish Notation (RPN) emerges as a simple and efficient alternative, where every operation follows its operands directly. But **how does RPN achieve this efficiency?** The answer lies in its clever utilization of stacks.

## Understanding RPN: A New Perspective

RPN, also known as postfix notation, flips the script on traditional mathematical notation. Instead of placing operators between operands, as seen in infix notation, RPN puts operators after their operands. This eliminates the need for parentheses to denote the order of operations, making expressions clearer and easier to evaluate.

## The Stack: A Pillar of RPN Evaluation

At the heart of RPN lies the stack data structure. Think of a stack as a pile of plates: you can only add or remove plates from the top. Similarly, in RPN evaluation, we use a stack to keep track of numbers and intermediate results.

Suggested: Implementing Stack Using Linked List vs. Array

## Step-by-Step Evaluation: A Visual Journey

Let’s walk through the evaluation of an RPN expression step-by-step:

- Start with an empty stack.
- Scan each token (operand or operator) from left to right.
- If the token is an operand, push it onto the stack.
- If the token is an operator, pop the necessary operands from the stack, perform the operation, and push the result back onto the stack.
- Continue this process until the entire expression is evaluated, leaving the final result on the stack.

## Illustrating with an Example

Consider the RPN expression: “3 4 5 *”

- Push “3” onto the stack.
- Push “4” onto the stack.
- Encounter “*”: Pop “4” and “3”, perform multiplication, and push “12” onto the stack.
- Push “5” onto the stack.
- Encounter “*”: Pop “5” and “12”, perform multiplication, and push “60” onto the stack.
- Expression evaluation is complete, with “60” remaining on the stack as the final result.

## Comparing RPN with Other Notations

While infix notation is familiar, it often requires complex rules and parentheses. Postfix notation shares similarities with RPN but lacks the efficiency of stack-based evaluation. RPN’s simplicity and reliance on stacks make it a clear winner in terms of computational efficiency and ease of understanding.

## Conclusion: Unleashing the Potential of RPN

Reverse Polish Notation (RPN) offers a fresh perspective on expressing and evaluating mathematical expressions. By leveraging the stack data structure, RPN simplifies the evaluation process, eliminating the need for complex precedence rules and parentheses. Its efficiency and clarity make it a valuable tool in various fields, from computer science to mathematics. So, next time you encounter a mathematical puzzle, consider embracing the simplicity and power of RPN.