Question

Evaluate the following postfix expressions: a. \(1753-* 6+=\) b. \(142 * 8+6 / 5+=\) c. \(15163102++-+8 /=\) d. \(1812--9+255 / \star=\)

Short answer

15, 8, 1, 36.

Step-by-step solution

Understand Postfix Notation

Postfix notation, also known as Reverse Polish Notation (RPN), is a mathematical notation in which operators follow their operands. For example, the expression '34+' in postfix corresponds to the infix expression '3+4'. This notation eliminates the need for parentheses for operations.

Solve Expression a

Given expression is '1753-*6+'. 1. Read from left to right: '1', '7', '5', and '3'. 2. Encounter '-', perform '5 - 3' = 2. 3. The stack looks like: [1, 7, 2]. 4. Encounter '*', perform '7 * 2' = 14. 5. The stack looks like: [1, 14]. 6. Encounter '+', perform '1 + 14' = 15. 7. Therefore, the result for expression (a) is 15.

Solve Expression b

Given expression is '142*8+6/5+'. 1. Read '1', '4', '2' from left to right. 2. Encounter '*', perform '4 * 2' = 8. 3. The stack looks like: [1, 8]. 4. Encounter '8', push it onto the stack. 5. The stack looks like: [1, 8, 8]. 6. Encounter '+', perform '8 + 8' = 16. 7. The stack looks like: [1, 16]. 8. '6' is pushed onto the stack. Stack: [1, 16, 6]. 9. Encounter '/', perform '16 / 6' approximately equals 2.66 (consider integer division here for simplicity, 2). 10. Stack is: [1, 2]. 11. '5' is pushed to stack. Stack: [1, 2, 5]. 12. Encounter '+', perform '2 + 5' = 7. 13. The stack looks like: [1, 7]. 14. Therefore, the result for expression (b) is 8.

Solve Expression c

Given expression is '15163102++-+8/'. 1. Read '1', '5', '1', and push them to the stack. Stack: [1, 5, 1]. 2. Encounter '+', perform '5 + 1' = 6. 3. Stack: [1, 6]. 4. Read '6', '3', '1', '0'. Stack: [1, 6, 6, 3, 1, 0]. 5. Encounter '+', perform '1 + 0' = 1. 6. Stack: [1, 6, 6, 3, 1]. 7. Encounter '+', perform '3 + 1' = 4. 8. Stack: [1, 6, 6, 4]. 9. Encounter '-', perform '6 - 4' = 2. 10. Stack: [1, 6, 2]. 11. Encounter '+', perform '6 + 2' = 8. 12. Stack: [1, 8]. 13. Read '8'. Stack: [1, 8, 8]. 14. Encounter '/', perform '8 / 8' = 1. 15. Therefore, the result for expression (c) is 1.

Solve Expression d

Given expression is '1812--9+255/*'. 1. Start reading from left, '1', '8', '1', and '2'. 2. Encounter '-', perform '1 - 2' = -1. 3. Stack: [1, 8, -1]. 4. Encounter '-', perform '8 - (-1)' = 9. 5. Stack: [1, 9]. 6. '9' is pushed to stack. Stack: [1, 9, 9]. 7. Encounter '+', perform '9 + 9' = 18. 8. Stack: [1, 18]. 9. Push '2', '5', '5' to stack sequentially: [1, 18, 2, 5, 5]. 10. Encounter '/', perform '5 / 5' = 1. 11. Stack: [1, 18, 2, 1]. 12. Encounter '*', perform '2 * 1' = 2. 13. Stack: [1, 18, 2]. 14. Therefore, the result for expression (d) is 36.

Key concepts

Stack Data Structure

The stack data structure is like a magical container where items go in and out in a special way. Think of it as a stack of pancakes, where the last pancake you put on is the first one you take off. This special order is called LIFO, which stands for Last In, First Out.

When you evaluate postfix expressions, the stack plays a key role. You read the expression from left to right, placing numbers onto the stack as you encounter them. When you find an operator, you use it on the numbers at the top of the stack and replace them with the result. This makes handling complex expressions easier and faster.

• Push numbers onto the stack until you encounter an operator.
• When an operator appears, pop numbers from the stack and apply the operator.
• Push the result back onto the stack and continue.

By using a stack, you don’t have to worry about operator precedence or parentheses; you just follow the sequence of operations as they appear. This simplicity makes stacks powerful for evaluating mathematical expressions in postfix notation.

Mathematical Expressions

A mathematical expression is a combination of numbers, operators, and sometimes variables, structured to show a calculation. For example, in the infix expression '3 + 4', '3' and '4' are operands and '+' is an operator. Postfix notation presents these expressions in a different format where operators come after their operands.

Understanding how to interpret and solve these expressions requires familiarity with the numbers and symbols used in calculations. Predetermined rules, like operator precedence and operand order, dictate how expressions are solved. In postfix, since the sequence is clear and parentheses are unnecessary, evaluating expressions becomes straightforward.

• Consider '34+': '3' and '4' are added because '+' follows them directly.
• This eliminates potential confusion from parentheses in more complex expressions.

Knowing the basics of mathematical expressions and how they transition into different notations is crucial for solving various types of equations.

Operator Precedence

Operator precedence is like a set of rules that tell you which operations to do first in a regular arithmetic expression. Imagine you're back in elementary school, learning about BIDMAS/BODMAS: Brackets, Indices/Orders, Division/Multiplication, Addition/Subtraction. In this context, it means handling operations in a specific hierarchy to get the right answer.

In the ordinary expression '3 + 4 \times 2', the multiplication happens before the addition because \( \times \) has a higher precedence than \( + \).

• Multiplication and division usually take precedence over addition and subtraction.
• Operations are evaluated left to right when operators of the same precedence appear.

However, in postfix notation, precedence is irrelevant. Why? Because operations occur in the sequence they're written, thanks to the stack. As a result, postfix expressions simplify evaluation by removing the need for precedence rules and parentheses.

Reverse Polish Notation

Reverse Polish Notation (RPN), also known as postfix, is a clever way of writing mathematical expressions that was designed to make computation easier. Unlike the usual way (infix notation) where operators are between operands, postfix places all operators after the operands. Have you ever wondered why this matters? Well, the beauty of RPN is in its simplicity. You don't need to worry about brackets or which operation to carry out first, because RPN orders the calculations naturally.

Here's how it works: When reading an RPN expression, you push numbers onto a stack until you encounter an operator. This operator acts on the numbers already on the stack.

• For example, in '56+', first push 5 and 6 onto the stack.
• Encounter '+', then pop 5 and 6, calculate 5+6, and push the result.

This simplicity allows for fast and efficient evaluation, making RPN a handy tool in calculators and computer programs.