100-100 Divided By 100-100 Is Equal To 2

The Mathematical Fallacy: Why 100-100/100-100 ≠ 2

The statement "100-100 divided by 100-100 is equal to 2" is a classic example of a mathematical fallacy. It appears deceptively simple, yet it highlights a crucial concept in mathematics: the undefined nature of division by zero. Understanding why this equation is incorrect requires a deep dive into the fundamental rules of arithmetic and algebra. This article will not only explain why the statement is false but also explore the broader implications of division by zero and its impact on mathematical operations.

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we dissect the fallacy, let's refresh our understanding of the order of operations, often remembered by the acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). These mnemonics dictate the sequence in which we should perform calculations in a mathematical expression. In this case, there are no parentheses or exponents, so we focus on division and subtraction.

The expression "100-100/100-100" appears to suggest:

1. 100 - 100 = 0
2. 100 - 100 = 0
3. 0 / 0 = 2

This is where the fallacy lies. The division step is fundamentally flawed.

The Problem with Division by Zero

Division is fundamentally defined as the inverse operation of multiplication. When we say 6 / 2 = 3, we're asking "what number, when multiplied by 2, equals 6?". The answer is 3. However, there is no number that, when multiplied by 0, results in a non-zero number. Therefore, division by zero is undefined in standard arithmetic.

Why is division by zero undefined?

Let's consider a few scenarios:

• Scenario 1: x / 0 = y (where y is a non-zero number). There's no number 'x' that satisfies this equation. No matter what value you assign to 'x', multiplying it by 0 will always result in 0, never 'y'.

• Scenario 2: 0 / 0 = y. This is called an indeterminate form. Any number 'y' could satisfy this equation because 0 multiplied by any number equals 0. This lack of a unique solution makes it undefined.

• Scenario 3: Approaching Zero: Consider the expression x/x. As 'x' approaches 0, the value of x/x approaches 1. However, this does not mean 0/0 = 1. This limit only applies as 'x' gets infinitesimally close to zero, not at zero itself.

These scenarios clearly demonstrate the inherent problem with division by zero. It leads to contradictions and inconsistencies within the mathematical system, hence its undefined status.

Correctly Evaluating the Expression

To correctly evaluate "100-100/100-100," we must adhere strictly to the order of operations. However, even with PEMDAS/BODMAS, we still encounter division by zero:

1. 100 - 100 = 0 (Subtraction from left to right)
2. 100 - 100 = 0 (Subtraction from left to right)
3. 0 / 0 (Division) - This is undefined.

The result is undefined, not 2.

Beyond the Simple Example: Implications in Calculus

While the simple example 100-100/100-100 is easily understood, the concept of limits and indeterminate forms becomes much more critical in higher-level mathematics, particularly in calculus. Calculus frequently deals with situations where expressions approach zero, and understanding how to handle such limits is essential.

For instance, consider the function f(x) = (x² - 1) / (x - 1). If we substitute x = 1, we get 0/0, an indeterminate form. However, factoring the numerator, we obtain:

f(x) = (x - 1)(x + 1) / (x - 1)

We can cancel out (x - 1) provided x ≠ 1, resulting in f(x) = x + 1. Now, taking the limit as x approaches 1, we get:

lim (x→1) f(x) = 1 + 1 = 2

This illustrates that even though the direct substitution leads to 0/0, through algebraic manipulation and considering the limit, we can find a meaningful result. This is a fundamental concept in calculus known as L'Hôpital's Rule, a technique used to evaluate indeterminate forms.

Common Mistakes and Misinterpretations

Many misconceptions arise from incorrectly interpreting the order of operations or misunderstanding the implications of division by zero. Some common mistakes include:

• Ignoring the order of operations: Incorrectly performing subtraction before division can lead to erroneous results.
• Assuming 0/0 equals 1 or 0: As explained earlier, 0/0 is undefined, not equal to any specific number.
• Cancelling terms inappropriately: While cancellation is a valid algebraic technique, it's crucial to only cancel common factors, not terms involved in addition or subtraction.

Conclusion: The Importance of Mathematical Rigor

The seemingly simple expression "100-100/100-100 = 2" highlights the importance of mathematical rigor. Understanding the rules of arithmetic, the order of operations, and the undefined nature of division by zero is crucial for accurate calculations and avoiding logical fallacies. While such fallacies can be used for amusement, they underscore the significance of a precise and well-defined mathematical framework. The concept of limits and indeterminate forms, further explored in calculus, provides more sophisticated tools for handling situations that might initially appear to involve division by zero. Always remember that mathematics thrives on precision and a deep understanding of its foundational principles. Ignoring these principles can lead to incorrect conclusions and hinder further progress in understanding more complex mathematical concepts.