Understanding Mathematical Order of Operations: MDAS vs. PEMDAS

The proper order of operations in mathematics is a fundamental concept that guides us in solving equations and expressions. This article explores the differences between two popular mnemonics, MDAS and PEMDAS, and explains how to correctly evaluate expressions.

Introduction to MDAS and PEMDAS

Mathematical expressions can often be ambiguous, leading to confusion in their evaluation. Two commonly used mnemonics to remember the order of operations are PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) and MDAS (Division, Multiplication, Addition, Subtraction). While both serve the same purpose, it is important to understand the nuances behind each and how they impact the evaluation of expressions.

MDAS vs. PEMDAS: Clarifying the Rules

PEMDAS is the more commonly used mnemonic among teachers and textbooks, emphasizing a clear sequence of operations: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

MDAS, on the other hand, emphasizes the operations of multiplication and division, stating that they are evaluated from left to right. This approach might seem more intuitive in some contexts, but it can lead to confusion in expressions that are not well-defined.

Evaluating the Expression: 8 ÷ 2(2 2) ?

Let's consider the expression 8 ÷ 2(2 2) to see how these rules apply and evaluate the result step-by-step.

First, solve the parentheses:

2   2  4

The expression now becomes:

8 ÷ 2 × 4

Next, follow the order of operations from left to right for multiplication and division:

- First, perform the division:

8 ÷ 2  4

- Then, multiply the result by 4:

4 × 4  16

Therefore, the correct answer to the expression is 16.

Common Misconceptions and the Importance of Clarity

Expressions like 8 ÷ 2(2 2) can be ambiguous and lead to different interpretations, even among mathematicians. This is why it is crucial to be precise and clear in writing mathematical expressions. Without explicit grouping symbols, expressions like these might be interpreted as:

8 ÷ (2(2 2)) (8 ÷ 2)(2 2) 8 ÷ 2 × (2 2) (8 ÷ 2 × 2) 2

Any of these interpretations can be valid, depending on the intended meaning, which is one reason why explicit parentheses and grouping are critical in mathematics.

It is worth noting that the lack of clarity can often be an unintended oversight, making it a good practice to double-check the expression for any potential ambiguities, especially in written or automated contexts.

The True Nature of Operations: Addition and Subtraction, Multiplication and Division

The true nature of the operations in mathematics is that subtraction and division are similar to addition and multiplication, respectively, in that they are inverse operations. This similarity means that these operations are often performed from left to right, with multiplication and division taking precedence over addition and subtraction.

To remember the correct order, just consider the following:

Subtraction and Addition (left to right) Multiplication and Division (left to right) Powers (highest priority)

Additionally, implicit multiplications are given higher priority than explicit multiplications and divisions. For example:

40/432  40/4 × 5  40/20  2

Written differently, the expression could be:

40/(4×32)

This approach helps avoid any potential misunderstandings and ensures consistent evaluation of expressions.

Conclusion

While mnemonics like PEMDAS and MDAS are helpful for remembering the order of operations, it is important to understand that they are not the only rules governing mathematical expressions. The key is to be precise and clear in the writing of mathematical expressions, ensuring that the intended meaning is unambiguous. Remember that subtraction and division are similar to addition and multiplication, respectively, and implicit multiplications are given higher priority over explicit ones.