Infix, Postfix and Prefix Expressions/Notations

Mathematical formulas often involve complex expressions that require a clear understanding of the order of operations. To represent these expressions, we use different notations, each with its own advantages and disadvantages. In this article, we will explore three common expression notations: infix, prefix, and postfix.

Infix Expressions

Infix expressions are mathematical expressions where the operator is placed between its operands. This is the most common mathematical notation used by humans. For example, the expression “2 + 3” is an infix expression, where the operator “+” is placed between the operands “2” and “3”.

Infix notation is easy to read and understand for humans, but it can be difficult for computers to evaluate efficiently. This is because the order of operations must be taken into account, and parentheses can be used to override the default order of operations.

Common way of writing Infix expressions:

• Infix notation is the notation that we are most familiar with. For example, the expression “2 + 3” is written in infix notation.
• In infix notation, operators are placed between the operands they operate on. For example, in the expression “2 + 3”, the addition operator “+” is placed between the operands “2” and “3”.
• Parentheses are used in infix notation to specify the order in which operations should be performed. For example, in the expression “(2 + 3) * 4”, the parentheses indicate that the addition operation should be performed before the multiplication operation.

Operator precedence rules:

Infix expressions follow operator precedence rules, which determine the order in which operators are evaluated. For example, multiplication and division have higher precedence than addition and subtraction. This means that in the expression “2 + 3 * 4”, the multiplication operation will be performed before the addition operation.

Here’s the table summarizing the operator precedence rules for common mathematical operators:

Operator	Precedence
Parentheses ()	Highest
Exponents ^	High
Multiplication *	Medium
Division /	Medium
Addition +	Low
Subtraction –	Low

Evaluating Infix Expressions

Evaluating infix expressions requires additional processing to handle the order of operations and parentheses. First convert the infix expression to postfix notation. This can be done using a stack or a recursive algorithm. Then evaluate the postfix expression.

Advantages of Infix Expressions

• More natural and easier to read and understand for humans.
• Widely used and supported by most programming languages and calculators.

Disadvantages Infix Expressions

• Requires parentheses to specify the order of operations.
• Can be difficult to parse and evaluate efficiently.

Prefix Expressions (Polish Notation)

Prefix expressions are also known as Polish notation, are a mathematical notation where the operator precedes its operands. This differs from the more common infix notation, where the operator is placed between its operands.

In prefix notation, the operator is written first, followed by its operands. For example, the infix expression “a + b” would be written as “+ a b” in prefix notation.

Evaluating Prefix Expressions

Evaluating prefix expressions can be useful in certain scenarios, such as when dealing with expressions that have a large number of nested parentheses or when using a stack-based programming language.

Advantages of Prefix Expressions

• No need for parentheses, as the operator always precedes its operands.
• Easier to parse and evaluate using a stack-based algorithm.
• Can be more efficient in certain situations, such as when dealing with expressions that have a large number of nested parentheses.

Disadvantages of Prefix Expressions

• Can be difficult to read and understand for humans.
• Not as commonly used as infix notation.

Postfix Expressions (Reverse Polish Notation)

Postfix expressions are also known as Reverse Polish Notation (RPN), are a mathematical notation where the operator follows its operands. This differs from the more common infix notation, where the operator is placed between its operands.

In postfix notation, operands are written first, followed by the operator. For example, the infix expression “5 + 2” would be written as “5 2 +” in postfix notation.

Evaluating Postfix Expressions (Reverse Polish Notation)

Evaluating postfix expressions can be useful in certain scenarios, such as when dealing with expressions that have a large number of nested parentheses or when using a stack-based programming language.

Advantages of Postfix Notation

• Also eliminates the need for parentheses.
• Easier to read and understand for humans.
• More commonly used than prefix notation.

Disadvantages of Postfix Expressions

• Requires a stack-based algorithm for evaluation.
• Can be less efficient than prefix notation in certain situations.

Comparison of Infix, Prefix and Postfix Expressions

Let’s compare infix, prefix, and postfix notations across various criteria:

spect	Infix Notation	Prefix Notation (Polish Notation)	Postfix Notation (Reverse Polish Notation)
Readability	Human-readable	Less human-readable, requires familiarity	Less human-readable, requires familiarity
Operator Placement	Between operands	Before operands	After operands
Parentheses Requirement	Often required	Not required	Not required
Operator Precedence Tracking	Required, parentheses determine precedence	Not required, operators have fixed precedence	Not required, operators have fixed precedence
Evaluation Method	Left-to-right	Right-to-left	Left-to-right
Ambiguity Handling	May require explicit use of parentheses	Ambiguity rare, straightforward evaluation	Ambiguity rare, straightforward evaluation
Unary Operator Handling	Requires careful placement	Simplified handling due to explicit notation	Simplified handling due to explicit notation
Computer Efficiency	Less efficient due to precedence tracking and parentheses handling	More efficient due to fixed precedence and absence of parentheses	More efficient due to fixed precedence and absence of parentheses
Usage	Common in everyday arithmetic and mathematical notation	Common in computer science and programming languages	Common in computer science and programming languages

Frequently Asked Questions on Infix, Postfix and Prefix Expressions

1. What is an infix expression?

An infix expression is a mathematical expression in which the operators are placed between the operands.

2. What is a postfix expression?

A postfix expression is a mathematical expression in which the operators are placed after the operands.

3. What is a prefix expression?

A prefix expression is a mathematical expression in which the operators are placed before the operands.

4. How are infix expressions evaluated?

Infix expressions are evaluated by following the rules of operator precedence and using parentheses to specify the order of operations.

5. How are postfix expressions evaluated?

Postfix expressions are evaluated by scanning the expression from left to right and using a stack to keep track of operands and operators.

6. How are prefix expressions evaluated?

Prefix expressions are evaluated by scanning the expression from right to left and using a stack to keep track of operands and operators.

7. What are the advantages of postfix and prefix expressions over infix expressions?

Postfix and prefix expressions eliminate the need for parentheses to specify the order of operations and make it easier to evaluate expressions using algorithms.

8. Can infix expressions be converted to postfix or prefix expressions?

Yes, infix expressions can be converted to postfix or prefix expressions using algorithms such as the shunting yard algorithm for postfix conversion and the reverse polish notation for prefix conversion.

9. Are postfix and prefix expressions more efficient than infix expressions in terms of evaluation?

Postfix and prefix expressions are more efficient for evaluation as they are easier to parse and evaluate using algorithms that do not require parentheses or multiple passes through the expression.

10. When should I use infix, postfix, or prefix expressions in my programming code?

Infix expressions are commonly used for human readability, while postfix and prefix expressions are preferred for efficient evaluation in programming languages and mathematical computations.