Understanding the Order of Operations and Symbols in Mathematical Expressions

Mathematical expressions can sometimes be confusing, especially when it comes to the order of operations and the correct interpretation of symbols. This article will explore these concepts using a series of examples and explain why the BODMAS and GEMS rules are crucial for determining the correct solution.

Example: 88 ÷ 8 × 8 - 8

Let's begin with the example of 88 ÷ 8 × 8 - 8. The use of different operations and the sequence in which they are executed is critical.

Solving Using BODMAS

According to the BODMAS rule, we first perform division: 88 ÷ 8 11 Then, multiplication: 11 × 8 88 Finally, subtraction: 88 - 8 80

The correct answer, following BODMAS, is 80.

Solving Using GEMS

To solve the same expression using the GEMS rule:

No Grouping (G) operations. No Exponents (E) operations. Multiplication and division from left to right: First, perform division: 88 ÷ 8 11 Then, multiplication: 11 × 8 88 Addition and subtraction from left to right: Subtraction: 88 - 8 80

The answer following GEMS is also 80.

Alternative Interpretations

However, the expression can also be interpreted in different ways due to the ambiguity of the symbols used. Let's consider another example using these different interpretations:

Example: 8 8/8 8×8 - 8

Selecting the symbols differently, we can interpret the expression in multiple ways. Here are some examples:

Solving Using Division and Multiplication

First, division: 8 ÷ 8 1 Next, multiplication: 1 × 8 8 Now, addition: 8 1 9 Finally, subtraction: 9 - 8 1

This interpretation results in the answer 1.

Solving Using a Programming Language

Some programming languages, such as APL (TryAPL) and Smalltalk, interpret the expression differently. For example, in APL:

The expression 88 ÷ 88 × 8 - 8 is interpreted as 9. In Smalltalk, the same expression would result in 72.

The reason for these differences lies in the way these programming languages prioritize the operations.

Conclusion

Understanding the order of operations and the interpretation of symbols is crucial in solving mathematical expressions. Guidelines like BODMAS and GEMS provide a standardized approach to ensure consistent results. However, it's essential to be aware of the different ways interpretations can vary, especially with unconventional symbols or programming languages.

By familiarizing yourself with these rules and being mindful of the context, you can avoid confusion and ensure accurate solutions. Happy calculating!