Question

Convert the following infix expressions to postfix notations. a. (A + B) * (C + D) - E b. A - (B + C) * D + E / F c. ((A + B) / (C - D) + E) * F - G d. A + B * (C + D) - E / F * G + H

Short answer

a. AB+CD+*E-; b. ABC+D*-EF/+; c. AB+CD-/E+F*G-; d. ABCD+*+EF/G*-H+.

Step-by-step solution

Understand the Expression Order of Operations

Evaluate the mathematical order of operations for each expression. Remember the sequence: Parentheses, Exponents, Multiplication/Division (from left to right), Addition/Subtraction (from left to right).

Apply the Conversion Algorithm - Expression a

Convert `(A + B) * (C + D) - E` to postfix: 1. Convert within parentheses: `A B +` and `C D +` 2. Apply multiplication: `(A B +) (C D +) *` 3. Apply subtraction: `A B + C D + * E -` The postfix expression is `A B + C D + * E -`.

Apply the Conversion Algorithm - Expression b

Convert `A - (B + C) * D + E / F` to postfix: 1. Convert within parentheses: `B C +` 2. Apply multiplication: `B C + D *` 3. Apply subtraction and addition: `A B C + D * - E F / +` The postfix expression is `A B C + D * - E F / +`.

Apply the Conversion Algorithm - Expression c

Convert `((A + B) / (C - D) + E) * F - G` to postfix: 1. Convert within parentheses: `A B +`, `C D -` 2. Apply division: `A B + C D - /` 3. Apply addition within outer parentheses: `A B + C D - / E +` 4. Apply multiplication and subtraction: `(A B + C D - / E +) F * G -` The postfix expression is `A B + C D - / E + F * G -`.

Apply the Conversion Algorithm - Expression d

Convert `A + B * (C + D) - E / F * G + H` to postfix: 1. Convert within parentheses: `C D +` 2. Apply multiplication: `B C D + *` 3. Apply addition and other operations in order from left to right: `A B C D + * + E F / G * - H +` The postfix expression is `A B C D + * + E F / G * - H +`.

Key concepts

Infix to Postfix Conversion

Infix expressions are the standard mathematical notation used in everyday calculations. For example, in the expression \( A + B \), the operator \(+\) is between the operands \(A\) and \(B\). Postfix notation, also known as Reverse Polish Notation (RPN), is a way of writing these expressions without parentheses or operator precedence. In postfix notation, the operator follows the operands. Thus, the expression \(A + B\) becomes \(A B +\).

The conversion from infix to postfix involves handling the order of operations explicitly and rearranging the operands and operators accordingly. This method is valuable as it simplifies the process of evaluating expressions in programming and computing, since it omits the need to consider precedence while parsing the expression. To convert an infix expression to postfix, you can follow systematic algorithm steps, examining one operator and operand at a time and ensuring that the final order respects the precedence and associativity of operations.

Order of Operations

When converting expressions from infix to postfix notation, understanding and correctly applying the order of operations is crucial. Mathematicians and computers alike follow a specific sequence of operations, often remembered by the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In the process of conversion, parentheses take the highest priority and dictate the sequence of operations. Within parentheses, the regular order follows. Multiplication and division have equal precedence, so they are evaluated from left to right. Similarly, addition and subtraction are handled last in the sequence, respecting their order of appearance.

Correctly respecting the order ensures that operations are performed as intended when rearranged into postfix notation. Overlooking these rules can lead to incorrect evaluations and logical errors.

Mathematical Expressions

Mathematical expressions are at the heart of problems involving infix and postfix notations. These expressions typically consist of elements like operands (e.g., numbers or variables), operators (e.g., \(+\), \(*\), \(/\), \(-\)), and sometimes functions or special symbols. Understanding these components is essential in translating an expression from one notation to another.

In mathematical notation, clarity is key. Complex expressions such as \((A + B) * (C + D) - E\) require mindfulness of structure and clarity of components. Each part of an expression needs to be processed in terms of precedence and associative rules to ensure a smooth conversion to postfix. This awareness not only aids in solving mathematical problems but also in fine-tuning computational operations efficiently.

Algorithm Steps

Transforming an infix expression to a postfix one requires specific algorithmic steps, often involving a stack data structure. Here’s a simplified version of common algorithm steps:

• Read the infix expression from left to right.
• Use a stack to keep operators and ensure correct precedence and associativity.
• When encountering operands, add them directly to the output list (i.e., postfix expression).
• For operators, pop all operators from the stack that have greater or equal precedence than the upcoming operator, append them to the output, then push the new operator onto the stack.
• At the end of the expression, pop all operators left in the stack and append them to the output list.

This method functions effectively by mimicking the natural operation of infix expressions while correctly sequencing operations for postfix evaluation. By rigorously applying these steps, you ensure that each part of the expression is handled correctly and results in an easily computable postfix format.