Exploring Mathematical Paradoxes: How 100-100/100-100 Can Equal 1/2

Mathematics is often seen as a pure and unambiguous discipline, yet certain expressions can challenge our understanding and lead us into the intriguing world of mathematical paradoxes. One such example is the expression 100 - 100 / 100 - 100, which is undefined due to division by zero. However, through creative manipulation and algebraic transformations, we can explore how such an expression could be made to equal 1/2.

Understanding the Initial Expression

The expression 100 - 100 / 100 - 100 can be simplified as follows:

100 - 100 / 100 - 100 0 / 0

0 / 0 is an undefined expression in mathematics because division by zero is not defined.

However, this does not mean we can never make an expression that looks similar to equal 1/2. Let's explore different approaches to achieve this.

Using Limits to Approach the Desired Result

One way to conceptually approach this is through limits. Instead of directly substituting 100 in both the numerator and the denominator, we can consider values that are very close to 100.

100 - 99 / 100 - 101 can be rewritten as:

lim_{x to 100} (x - 99 / x - 101) 1 / -1 -1

Although this does not equal 1/2, it demonstrates how limits can create different values.

Adjusting the Numbers to Achieve the Desired Result

Another approach is to adjust the numbers in the expression to ensure it can yield the desired result. We can rewrite the expression as follows:

100 - 99 / 100 - 99 (100 - 99) / (100 - 99) 1 / 1 1

While this does not equal 1/2, it shows how altering the numbers can produce different outcomes.

Creating a New Equation to Equal 1/2

To create an expression that directly equals 1/2, we can consider simpler fractions:

1/2 2/4 3/6

We can then use these fractions to create expressions that equal 1/2 using different numbers:

50 - 50 / 100 - 100 0 / 0 undefined

This expression remains undefined because division by zero is not defined.

In summary, while we cannot directly make the expression 100 - 100 / 100 - 100 equal to 1/2 due to its undefined nature, we can explore limits, adjust numbers, and create new equations to produce similar results.

Conclusion

Mathematics is a rich field with many fascinating paradoxes and contradictions, such as the one illustrated above. By understanding the nuances of undefined expressions and algebraic manipulation, we can delve deeper into the beauty of mathematics and explore the myriad ways in which seemingly simple expressions can surprise and challenge our intuitive understanding.