Order of Operations: Why Calculators Get It Wrong

Have you ever encountered a situation where your calculator seems to be giving you the wrong answer? It's a common experience, and it often stems from a misunderstanding of the order of operations in mathematics. In this article, we'll delve into the world of PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), explore why it's crucial for accurate calculations, and reveal why calculators might sometimes get it wrong.

What is the Order of Operations?

The order of operations, also known as PEMDAS or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction), is a set of rules that dictate the sequence in which mathematical operations should be performed. It ensures that everyone arrives at the same answer for a given equation, regardless of how they approach it.

PEMDAS in Action

Let's break down PEMDAS:

1. Parentheses (or Brackets): Perform any operations enclosed in parentheses or brackets first.
2. Exponents (or Orders): Next, calculate any exponents or roots.
3. Multiplication and Division: Perform multiplication and division from left to right.
4. Addition and Subtraction: Finally, perform addition and subtraction from left to right.

Why Calculators Get It Wrong

Calculators, especially basic ones, might not always follow the order of operations correctly. This can lead to incorrect results. For instance, if you input the expression 2 + 3 * 4 into a calculator that doesn't adhere to PEMDAS, it might perform the calculations sequentially, resulting in 20 instead of the correct answer, 14.

Here's the breakdown:

• Calculator without PEMDAS: 2 + 3 * 4 = 5 * 4 = 20
• Calculator with PEMDAS: 2 + 3 * 4 = 2 + 12 = 14

Examples of Common Mistakes

Here are a few common mistakes people make when dealing with order of operations:

• Forgetting Parentheses: If you're working with a complex expression involving multiple operations, forgetting to use parentheses can lead to incorrect results.
• Misinterpreting Multiplication and Division: Remember that multiplication and division have equal priority and should be performed from left to right.
• Ignoring Exponents: Exponents must be calculated before any multiplication or division.

History of the Order of Operations

The concept of order of operations has roots in the development of algebra. Ancient mathematicians recognized the need for consistent rules to avoid ambiguity in calculations. While the modern PEMDAS acronym is relatively recent, the underlying principles have been around for centuries.

Conclusion

Understanding the order of operations is essential for accurate mathematical calculations. While calculators can be helpful tools, it's crucial to be aware of their limitations and to double-check your results. By adhering to PEMDAS, you can ensure that you're performing calculations correctly and avoiding common errors.