Question

Evaluate the following postfix expressions: a. 8 2 + 3 * 16 4 / - = b. 12 25 5 1 / / * 8 7 + - = c. 70 14 4 5 15 3 / * - - / 6 + = d. 3 5 6 * + 13 - 18 2 / + =

Short answer

a. 26, b. 45, c. 8, d. 29.

Step-by-step solution

Understand the Postfix Expression

Postfix expressions, also known as Reverse Polish Notation, are sequences where the operators follow their operands. To solve these, evaluate the expression from left to right, using a stack to hold operands. When an operator is encountered, apply it to the most recent operands.

Solve Expression (a)

Evaluate `8 2 + 3 * 16 4 / -`: 1. Push 8 and 2 to stack. '+' means pop 8 and 2, push 10. 2. Push 3, pop 10 and 3, compute 10 * 3 = 30, push 30. 3. Push 16 and 4, '/' means pop 4 and 16, compute 16 / 4 = 4, push 4. 4. '-' means pop 4 and 30, compute 30 - 4 = 26.

Solve Expression (b)

Evaluate `12 25 5 1 / / * 8 7 + -`: 1. Push 12, 25, 5, and 1. '/' means pop 1 and 5, compute 5 / 1 = 5, push 5. 2. '/' means pop 5 and 25, compute 25 / 5 = 5, push 5. 3. '*' means pop 5 and 12, compute 12 * 5 = 60, push 60. 4. Push 8 and 7. '+' means pop 7 and 8, compute 8 + 7 = 15, push 15. 5. '-' means pop 15 and 60, compute 60 - 15 = 45.

Solve Expression (c)

Evaluate `70 14 4 5 15 3 / * - - / 6 +`: 1. Push 70, 14, 4, 5, 15, and 3. '/' means pop 3 and 15, compute 15 / 3 = 5, push 5. 2. '*' means pop 5 and 5, compute 5 * 5 = 25, push 25. 3. '-' means pop 25 and 4, compute 4 - 25 = -21, push -21. 4. '-' means pop -21 and 14, compute 14 - (-21) = 35, push 35. 5. '/' means pop 35 and 70, compute 70 / 35 = 2, push 2. 6. Push 6, '+' means pop 6 and 2, compute 2 + 6 = 8.

Solve Expression (d)

Evaluate `3 5 6 * + 13 - 18 2 / +`: 1. Push 3, 5, and 6. '*' means pop 6 and 5, compute 5 * 6 = 30, push 30. 2. '+' means pop 30 and 3, compute 3 + 30 = 33, push 33. 3. Push 13, '-' means pop 13 and 33, compute 33 - 13 = 20, push 20. 4. Push 18 and 2. '/' means pop 2 and 18, compute 18 / 2 = 9, push 9. 5. '+' means pop 9 and 20, compute 20 + 9 = 29.

Key concepts

Reverse Polish Notation

Reverse Polish Notation (RPN) is an intuitive way of writing mathematical expressions without parentheses. In RPN, also referred to as postfix notation, the operator comes after its operands. This is different from the more common infix notation, where operators are placed between operands. The biggest benefit of RPN is the removal of the need for parentheses to indicate operation precedence, which makes calculations and computer parsing straightforward. In postfix expressions, this order allows for step-by-step evaluation using a simple scan.

Stack Data Structure

The stack data structure is perfect for evaluating postfix expressions. It works on the Last In, First Out (LIFO) principle. Imagine it as a stack of plates where you can only add or remove the top plate. This makes it particularly useful for handling the operands and operators as they appear in a postfix expression. When evaluating such expressions, the stack is used to temporarily store operands. When an operator is encountered, operands are popped off the stack, the operation is performed, and the result is pushed back onto the stack.

Operand Evaluation

Evaluating operands in a postfix expression involves a couple of straightforward steps:

• Operands are placed onto a stack as they are read.
• When an operator is encountered, the required number of operands are popped from the stack.
• The operator is applied to these operands, and the result is pushed back onto the stack.

This process is repeated until the entire expression has been read, at which point the stack should have exactly one element, which is the result of the expression.

Operator Precedence

In postfix notation, operator precedence is implicitly handled by the order of the operators and operands. This differs significantly from infix notation, which uses parentheses and the inherent precedence of operators to dictate the order of operations. For example, in an expression like `8 2 + 3 * 16 4 / -`, precedence is determined not by the conventional rules (like multiplication before addition) but by the position of operators relative to operands in the sequence, naturally eliminating ambiguity and simplifying calculations.