Why Order of Operations Trips Us Up

 Just an explanation on why this topic today.  I've been getting those do this problem things on Facebook and there is at least one person putting down a totally different answer indicating they didn't follow the order of operation properly. So today we look at why this happens.

The order of operations, a set of rules dictating the sequence in which calculations are performed, often proves to be a stumbling block for many. This fundamental concept, while seemingly straightforward, can lead to confusion and errors if not understood and applied correctly.   

One of the primary reasons for this confusion lies in the inherent ambiguity of mathematical expressions. Without a standardized order, different interpretations can lead to vastly different results. For instance, consider the expression 2 + 3 × 4. If we calculate from left to right, we get 20. However, following the correct order of operations, we multiply first and then add, yielding 14.

Another common misconception is the belief that multiplication always takes precedence over division, or that addition always precedes subtraction. In reality, multiplication and division, as well as addition and subtraction, are performed on an equal footing, working from left to right. For example, 6 ÷ 2 × 3 is evaluated as (6 ÷ 2) × 3, not 6 ÷ (2 × 3).   

Exponents, often denoted by a superscript number, represent repeated multiplication. For instance, 7^2 means 7 multiplied by itself twice, or 7 × 7. Some individuals may rewrite exponents as repeated multiplication before applying the order of operations. While this approach is technically correct, it can be time-consuming and prone to errors, especially when dealing with larger exponents.   

The order of operations is essential for ensuring consistency and clarity in mathematical calculations. Without a standardized approach, different individuals could arrive at different answers for the same expression, leading to chaos and confusion. By following a specific order, we guarantee that everyone arrives at the same result.   

So how do we overcome these issues. First,  Familiarize yourself with the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction).   Next, regularly practice the order of operations because consistent practice is key. Solve numerous problems involving different operations to solidify your understanding.  To avoid ambiguity, use parentheses to clarify the intended order of operations.   Always break down complex expressions by dividing them into  smaller, more manageable parts.   Double-check your calculations to ensure accuracy. Consider using calculators and computer software to verify results.   

By understanding the underlying principles and practicing regularly, you can overcome the challenges posed by the order of operations and become more proficient in mathematical calculations. Let me know what ou think, I'd love to hear.  Have a great day.