Understanding Expression Notations: Infix, Prefix, and Postfix Simplified

When dealing with mathematical expressions, you may encounter three different notations: infix, prefix, and postfix. Each serves a purpose, offering unique advantages depending on whether the user is a human or a computer. In this article, we’ll explore these notations in depth, their use cases, and how they’re evaluated.

What Are Infix, Prefix, and Postfix Notations?

Infix Notation: The operator is placed between the operands.
Example: A + B
Prefix Notation: (Polish Notation): The operator is placed before the operands.
Example: + A B
Postfix Notation: (Reverse Polish Notation — RPN): The operator is placed after the operands.
Example: A B +

Why Do We Have These Notations?

Human Readability (Infix Notation): Humans find infix notation natural and intuitive because it aligns with traditional arithmetic taught in schools.
Computer Efficiency (Prefix/Postfix Notations): Computers process prefix and postfix notations more efficiently as they avoid the need for complex rules of operator precedence and parentheses.
Flexibility in Computation: Each notation has unique advantages in different scenarios, such as stack-based evaluations, formal logic, or user interaction.

Infix Notation

Overview

In infix notation, operators are placed between operands. Parentheses are often required to define the order of operations and resolve ambiguity.

Example

Expression: (3 + 4) * 5

Parentheses indicate that 3 + 4 is evaluated first.
The result of 7 is then multiplied by 5 to yield 35.

Use Cases

Daily Arithmetic: Most programming languages (e.g., Python, Js, C++) use infix notation for arithmetic operations.
Readability: Infix is the most human-readable format but relies on precedence rules and parentheses for clarity.

Challenges

Parsing Complexity: Computers must implement precedence and associativity rules to evaluate expressions accurately.
Parentheses Dependency: Incorrect placement of parentheses can lead to errors or ambiguity.

Prefix Notation (Polish Notation)

Overview

In prefix notation, operators precede operands, eliminating the need for parentheses. The order of operations is determined by the position of operators and operands.

Example

Expression: * + 3 4 5 (equivalent to (3 + 4) * 5 in infix)

Start from the left: + 3 4 evaluates to 7.
Multiply 7 by 5 to get 35.

Use Cases

Compilers and Interpreters: Prefix notation simplifies parsing as there’s no need for precedence rules.
Formal Logic: Used in formal systems to represent logical expressions.

Challenges

Readability: Less intuitive for humans due to the unconventional placement of operators.

Postfix Notation (Reverse Polish Notation — RPN)

Overview

In postfix notation, operators follow operands. Like prefix, postfix does not require parentheses, as the order of operations is inherently unambiguous.

Example

Expression: 3 4 + 5 * (equivalent to (3 + 4) * 5 in infix)

Push 3 and 4 onto a stack.
Encounter +: Pop 3 and 4, compute 3 + 4 = 7, and push 7 back onto the stack.
Push 5 onto the stack.
Encounter *: Pop 7 and 5, compute 7 * 5 = 35, and push 35 back onto the stack.
Result: 35

Use Cases

Calculators: Many scientific calculators use postfix notation for efficient computations.
Stack-Based Evaluations: Ideal for systems using stack data structures.

Challenges

Learning Curve: Less intuitive for beginners but highly efficient for computation.

Comparison of Notations

Real-World Use Cases

1. Programming Languages (Infix)

Most programming languages use infix notation for arithmetic operations due to its readability. For instance:

const result = (3 + 4) * 5;
console.log(result); // Output: 35

2. Compilers and Parsers (Prefix)

Compilers often convert infix expressions to prefix (or postfix) for easier evaluation. For example:

Infix: (A + B) * C
Prefix: * + A B C

3. Calculators and Stack-Based Systems (Postfix)

Postfix is widely used in calculators and stack-based systems because of its simplicity in evaluation.

// JavaScript simulation of postfix evaluation
const evaluatePostfix = (expression) => {
const stack = [];
const tokens = expression.split(" ");
for (const token of tokens) {
if (!isNaN(token)) {
stack.push(Number(token));
} else {
const b = stack.pop();
const a = stack.pop();
switch (token) {
case '+':
stack.push(a + b);
break;
case '-':
stack.push(a - b);
break;
case '*':
stack.push(a * b);
break;
case '/':
stack.push(a / b);
break;
}
}
}
return stack[0];
};
const expression = "3 4 + 5 *";
console.log(evaluatePostfix(expression)); // Output: 35

How to Convert Between Notations

Infix to Prefix

Reverse the infix expression.
Replace ( with ) and vice versa.
Convert to postfix.
Reverse the result.

Infix to Postfix

Use a stack to hold operators.
Output operands immediately.
Pop operators from the stack according to precedence.

Example Conversion

Infix: (3 + 4) * 5

Prefix: * + 3 4 5
Postfix: 3 4 + 5 *

Conclusion

Understanding infix, prefix, and postfix notations helps bridge the gap between human readability and computational efficiency. While infix is the default for most arithmetic operations, prefix and postfix notations shine in scenarios requiring stack-based evaluation or formal logic. Whether you’re designing a parser, building a calculator, or diving into formal systems, these notations are indispensable tools.

Let’s Connect! What’s your favorite use case for these notations? Share your thoughts in the comments! Feel free to connect with me on LinkedIn for more articles on computer science and engineering.