🧮 Reverse Polish Notation (RPN) Tutorial

Master the art of postfix notation with interactive examples

📚 What is RPN?

Reverse Polish Notation (RPN) is a mathematical notation where operators follow their operands. Instead of writing 3 + 4, you write 3 4 +.

Why use RPN?

No parentheses needed - the order of operations is completely unambiguous
Easy to evaluate with a simple stack-based algorithm
Used in calculators (HP calculators), programming languages (Forth, PostScript)
Efficient for computers to process

🎯 How RPN Works: The Stack

RPN uses a stack data structure. Think of it like a stack of plates - you can only add to the top (push) or remove from the top (pop).

The Algorithm:

Read tokens left to right
If it's a number: Push it onto the stack
If it's an operator: Pop two numbers, apply the operator, push the result back
Final answer: The last number remaining on the stack

Example: Evaluating `3 5 + 2 *`

Step	Token	Operation	Stack
1	3	Push 3	[3]
2	5	Push 5	[3, 5]
3	+	Pop 5, 3 → Push 8	[8]
4	2	Push 2	[8, 2]
5	*	Pop 2, 8 → Push 16	[16]

Final Answer: 16

🎮 Interactive RPN Evaluator

Try it yourself! Enter an RPN expression (numbers and operators separated by spaces):

🔄 Converting Infix to RPN

Infix notation is what we normally use: (3 + 5) * 2

To convert to RPN, think about the order of operations:

Example: Convert `(5 + 3) × 4 - 6` to RPN

Identify the operations in order: First 5 + 3, then multiply by 4, then subtract 6
Write each operation in RPN as you go:
- 5 + 3 becomes 5 3 +
- Multiply that result by 4: 5 3 + 4 *
- Subtract 6: 5 3 + 4 * 6 -

Answer: 5 3 + 4 * 6 -

Quick Rules for Converting:

Simple operations: a + b → a b +
Parentheses: Process inner parentheses first
Multiple operations: Follow normal order of operations (PEMDAS)
Tip: Build from the inside out, converting each subexpression to RPN

↩️ Converting RPN to Infix

To convert RPN back to standard notation, use the stack but build expressions instead of numbers:

Example: Convert `a b + c d * e / -` to infix

Token	Operation	Stack
a	Push a	[a]
b	Push b	[a, b]
+	Pop b, a → Push (a+b)	[(a+b)]
c	Push c	[(a+b), c]
d	Push d	[(a+b), c, d]
*	Pop d, c → Push (c*d)	[(a+b), (c*d)]
e	Push e	[(a+b), (c*d), e]
/	Pop e, (cd) → Push (cd/e)	[(a+b), (c*d/e)]
-	Pop (cd/e), (a+b) → Push ((a+b)-(cd/e))	[(a+b) - c*d/e]

Answer: (a+b) - c*d/e

⚠️ Common Mistakes to Avoid

Operator order for subtraction/division: In a b -, the result is a - b (not b - a). The second-to-top element is the left operand.
Counting operands: Every binary operator needs exactly 2 operands on the stack
Parentheses in RPN: RPN has NO parentheses! The order is built into the expression itself
Reading left-to-right: Always process tokens strictly from left to right

💪 Practice Problems

Evaluate these:

8 2 / 3 + 5 -
4 5 * 3 2 + / 6 -
12 3 / 2 + 5 * 1 -

Convert to RPN:

(8 - 3) × 2
7 + 4 × (6 - 2)

🎓 You're Ready!

Now that you understand how RPN works, you're ready to tackle more complex problems. Remember:

Use the stack visualization to track your work
Process tokens left to right, always
Numbers push, operators pop-compute-push
Build infix-to-RPN conversions from the inside out

💡 Pro Tip: When stuck on a complex expression, draw out the stack at each step. It makes everything crystal clear!