12 divided by 4 3

Guri Bee

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20-10-2024

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The Ambiguity of 12 Divided by 4 3: Understanding Order of Operations

The expression "12 divided by 4 3" has been a source of debate and confusion, sparking numerous discussions on platforms like Github. Let's delve into this seemingly simple arithmetic problem and understand why it's not as straightforward as it appears.

The Issue: Order of Operations

The root of the confusion lies in the order of operations, which dictates the sequence in which mathematical operations are performed. A commonly used mnemonic for this is PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). However, the ambiguity arises because multiplication and division (and similarly, addition and subtraction) are performed from left to right.

Two Interpretations

1. Interpretation 1: (12 / 4) * 3

   Following PEMDAS, we first perform the division: 12 / 4 = 3. Then we multiply: 3 * 3 = 9.

   This interpretation yields the answer 9.

2. Interpretation 2: 12 / (4 * 3)

   This interpretation assumes the multiplication is performed before the division: 4 * 3 = 12. Finally, we divide: 12 / 12 = 1.

   This interpretation yields the answer 1.

Which Interpretation is Correct?

The truth is, neither interpretation is inherently wrong. The problem lies in the ambiguous notation. Without explicit parentheses, it's unclear which operation should be performed first.

The Importance of Clarity

This example highlights the importance of using clear and unambiguous notation in mathematical expressions. In most programming languages and mathematical contexts, the first interpretation (12 / 4 * 3 = 9) is generally assumed. However, to avoid confusion, using parentheses to explicitly define the intended order of operations is always recommended.

Example in Programming

In languages like Python, the expression 12 / 4 * 3 would evaluate to 9, following the left-to-right order of operations for division and multiplication. However, using parentheses, you can control the order:

>>> 12 / (4 * 3) 
1.0
>>> (12 / 4) * 3
9.0

Conclusion

While the expression "12 divided by 4 3" may appear straightforward, it's a classic example of how ambiguity in mathematical notation can lead to different interpretations. To avoid such confusion, always strive for clarity by using parentheses or explicitly stating the order of operations in your calculations.