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The given infix expression is \(P = 12 / (7 - 3) + 2\).
To convert an infix expression to a postfix expression, we typically use an algorithm based on a stack, considering operator precedence and associativity. Higher precedence operators are evaluated before lower precedence operators, and parentheses dictate the order of operations.
Step-by-step conversion from Infix to Postfix:
Infix Expression: \(12 / (7 - 3) + 2\)
Initialize an empty operator stack and an empty postfix expression list.
1. Scan '12': It's an operand. Add to postfix.
Postfix: 12
Stack: []
2. Scan '/': It's an operator. Push to stack.
Postfix: 12
Stack: [/]
3. Scan '(': It's a left parenthesis. Push to stack.
Postfix: 12
Stack: [/ (]
4. Scan '7': It's an operand. Add to postfix.
Postfix: 12 7
Stack: [/ (]
5. Scan '-': It's an operator. Push to stack.
Postfix: 12 7
Stack: [/ ( -]
6. Scan '3': It's an operand. Add to postfix.
Postfix: 12 7 3
Stack: [/ ( -]
7. Scan ')': It's a right parenthesis. Pop operators from stack and add to postfix until a left parenthesis '(' is encountered. Discard the '('.
Pop '-': Add to postfix.
Postfix: 12 7 3 -
Pop '(': Discard.
Stack: [/]
8. Scan '+': It's an operator. Its precedence is less than '/'. Pop '/' from stack and add to postfix. Then push '+' to stack.
Pop '/': Add to postfix.
Postfix: 12 7 3 - /
Push '+':
Stack: [+]
9. Scan '2': It's an operand. Add to postfix.
Postfix: 12 7 3 - / 2
Stack: [+]
10. End of expression. Pop remaining operators from stack and add to postfix.
Pop '+': Add to postfix.
Postfix: 12 7 3 - / 2 +
Resulting Postfix Expression: \(12 \ 7 \ 3 \ - \ / \ 2 \ +\)
Step-by-step evaluation of the Postfix Expression:
Postfix Expression: \(12 \ 7 \ 3 \ - \ / \ 2 \ +\)
Initialize an empty operand stack.
1. Scan '12': It's an operand. Push to stack.
Stack: [12]
2. Scan '7': It's an operand. Push to stack.
Stack: [12, 7]
3. Scan '3': It's an operand. Push to stack.
Stack: [12, 7, 3]
4. Scan '-': It's an operator. Pop the top two operands (3 and 7), perform the operation \(7 - 3 = 4\). Push the result to stack.
Stack: [12, 4]
5. Scan '/': It's an operator. Pop the top two operands (4 and 12), perform the operation \(12 / 4 = 3\). Push the result to stack.
Stack: [3]
6. Scan '2': It's an operand. Push to stack.
Stack: [3, 2]
7. Scan '+': It's an operator. Pop the top two operands (2 and 3), perform the operation \(3 + 2 = 5\). Push the result to stack.
Stack: [5]
8. End of expression. The final result is the only value remaining in the stack.
Evaluated Result: \(5\)
Contextual Explanation:
Infix, Postfix, and Prefix expressions are different ways to write arithmetic expressions. Infix notation is the most common form, where operators are placed between operands (e.g., \(A + B\)). However, infix expressions require parentheses and operator precedence rules to resolve ambiguity (e.g., \(A + B * C\)).
Postfix (Reverse Polish Notation or RPN) and Prefix (Polish Notation) notations are unambiguous, meaning they do not require parentheses or operator precedence rules to define the order of operations. In postfix, operators follow their operands (e.g., \(A \ B \ +\)), while in prefix, operators precede their operands (e.g., \(+\ A \ B\)).
These notations are particularly important in computer science. Compilers and interpreters often convert infix expressions into postfix or prefix expressions for easier and more efficient evaluation. Postfix expressions can be evaluated using a simple stack-based algorithm, which is straightforward to implement and avoids the complexities of parsing precedence and parentheses inherent in infix expressions. This makes them fundamental concepts for understanding how programming languages process mathematical expressions and for jobs involving compiler design, data structures, and algorithms.
