Unraveling the Dilemma of PEMDAS and BODMAS: Clarifying Mathematical Ambiguity in Equations

Mathematical ambiguity arises when equations are poorly written, leading to differing interpretations and results. This article explores the confusion surrounding PEMDAS and BODMAS, both of which describe the order of operations in mathematics. We will also discuss the importance of clear equation writing and the necessity of explicit parentheses to avoid ambiguity.

PEMDAS vs. BODMAS: Similarities and Differences

PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) and BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) are acronyms used to remember the order of operations in mathematics. Although they differ in terminology, the core rules they describe are the same. The primary difference lies in the terminology used, with BODMAS being more prevalent in the UK and Bquests, while PEMDAS is more common in the U.S.

Order of Operations: PEMDAS and BODMAS

The acronym for both PEMDAS and BODMAS dictates the steps to follow when solving mathematical expressions, with the operations of multiplication and division (as well as addition and subtraction) being performed from left to right in the expression.

Evaluating the Expression 6/23

Let's evaluate the expression 6/23 using the order of operations:

Division: Perform the division from left to right.
6/2 3 Multiplication: Then perform the multiplication.
3*3 9

Therefore, the expression 6/2 is solved first, and then the result is multiplied by 3, leading to a final result of 9.

The Dilemma of Ambiguously Written Expressions

Mathematical ambiguities often arise from poorly written equations, especially those deliberately designed to cause confusion. This issue affects mathematicians, urban planners, and educators, as evidenced by the case of expressions like 105 / 5 1 2 and 6 / 2 1 2.

Interpreting Ambiguous Expressions

An expression like 6/23 can be interpreted in two ways:

1: The full denominator is 2 3. 2: The division is performed as 6/2 first, then the result is multiplied by 3.

For 105 / 5 1 2, the ambiguity leads to two possible interpretations:

105 / (5 1 2) 105 / 8 13.125 (105 / 5) 1 2 21 1 2 24

These examples highlight the critical need for clear and concise notation in mathematical equations.

Striving for Clear and Concise Notation

To avoid such ambiguities, mathematicians, professors, and teachers should prioritize clarity in their equations, even if they are long-winded. Adding parentheses to clarify the order of operations is crucial. For instance, the expression 6 / 2 3 should be rewritten as 6 / (2 3) or (6 / 2) 3 to avoid confusion.

Addressing Criticism

Despite the clarity provided by order of operations and parentheses, the backlash against those emphasizing the need for better notation can be intense. Some critics may argue to relearn PEMDAS or even suggest legal action. It is important to address these critics by explaining the pedagogical principles behind the order of operations and emphasizing the need for clear and unambiguous equations.

For instance, 105 / 5 1 2 can be clarified as:

105 / (5 * (1 2)) 13.125 (105 / 5) 1 2 21 1 2 24

By adding parentheses, we can ensure that there is no confusion and the equation is solved in a consistent manner.

Conclusion

Mathematical clarity is essential in teaching and understanding, and it is crucial to address the ambiguities created by poorly written equations. Practitioners and educators should strive to make equations as clear as possible by using parentheses where necessary, thereby avoiding endless debates on incorrect interpretations.

Remember, the goal is to ensure that the intent of the equation is clear and unambiguous for all readers and solvers.