Proving The Inequality 1/10 < √101 - √99 Without A Calculator

Introduction

In this article, we delve into an intriguing algebraic inequality problem: proving that $\frac{1}{10} < \sqrt{101} - \sqrt{99}$ without resorting to calculators. This type of problem often appears in intermediate algebra textbooks and serves as an excellent exercise in manipulating radicals and applying algebraic inequalities. Many students find these problems challenging because they require a blend of algebraic techniques and a keen understanding of inequalities. The initial approaches, such as direct manipulation or using radical conjugates, might seem promising but can quickly lead to dead ends if not applied carefully. Therefore, a strategic approach is essential to unravel the problem effectively.

When tackling such inequalities involving radicals, it's crucial to consider methods that can simplify the expressions and make them more amenable to comparison. One common technique is to rationalize the denominator or, in this case, rationalize the expression involving the difference of square roots. This method often helps eliminate radicals from the denominator, making the expression easier to manipulate. Another effective strategy is to look for ways to create a common term or factor that can be used to compare both sides of the inequality. Additionally, thinking about the properties of square roots and how they behave under different operations can provide valuable insights. For instance, understanding that the square root function is increasing can help in making comparisons. The goal is to transform the given inequality into a form where the comparison becomes straightforward and evident, thereby proving the original statement without relying on numerical approximations from a calculator. Let's explore a step-by-step approach to solve this problem and uncover the elegant solution it holds.

The Challenge: $\frac{1}{10} < \sqrt{101} - \sqrt{99}$

This inequality challenges us to demonstrate the given relationship without the aid of a calculator. At first glance, the problem may seem daunting due to the presence of square roots and the need to compare them with a simple fraction. The key here is to manipulate the expression algebraically to reveal the underlying relationship. Direct comparison of $\sqrt{101} - \sqrt{99}$ and $\frac{1}{10}$ isn't immediately obvious, so we need to transform one or both sides of the inequality into a more comparable form. A common and effective technique in dealing with differences of square roots is to multiply by the conjugate. This method helps eliminate the square roots in the numerator, potentially leading to a simpler expression that is easier to evaluate.

The initial instinct might be to square both sides to eliminate the square roots, but this can often complicate the inequality further, especially when dealing with subtractions. Instead, focusing on rationalizing the expression $\sqrt{101} - \sqrt{99}$ provides a more direct path to the solution. By multiplying and dividing by the conjugate, we can transform the left-hand side into a fraction, making it easier to compare with the given fraction $\frac{1}{10}$. This approach leverages the difference of squares identity, which is a powerful tool in simplifying expressions involving square roots. Furthermore, understanding the behavior of square roots and their properties is crucial. For instance, we know that the square root function is monotonically increasing, meaning that if $a > b$, then $\sqrt{a} > \sqrt{b}$. This property can be useful in making comparisons and bounding the values of the expressions. The challenge lies in applying these techniques strategically to reveal the inequality and demonstrate its validity without resorting to approximations.

Strategy: Rationalizing the Denominator (Conjugate Multiplication)

To tackle the inequality $\frac{1}{10} < \sqrt{101} - \sqrt{99}$, a pivotal strategy involves rationalizing the expression $\sqrt{101} - \sqrt{99}$. Rationalizing, in this context, means eliminating the square roots from the expression to make it more amenable to comparison. The most effective way to achieve this is by multiplying the expression by its conjugate. The conjugate of $\sqrt{101} - \sqrt{99}$ is $\sqrt{101} + \sqrt{99}$. By multiplying both the numerator and denominator (in this case, we treat $\sqrt{101} - \sqrt{99}$ as a fraction with a denominator of 1) by this conjugate, we can leverage the difference of squares identity, which states that $(a - b)(a + b) = a^2 - b^2$.

This strategy is particularly useful because it transforms the difference of square roots into a simpler form, allowing us to work with integers instead of radicals in the numerator. The difference of squares identity is a cornerstone of algebraic manipulation and provides a direct route to simplifying expressions involving square roots. When we multiply $\sqrt{101} - \sqrt{99}$ by its conjugate, we get $(\sqrt{101} - \sqrt{99})(\sqrt{101} + \sqrt{99}) = (\sqrt{101})^2 - (\sqrt{99})^2 = 101 - 99 = 2$. This simplification is crucial because it transforms the complex expression into a simple integer, making it easier to compare with the fraction $\frac{1}{10}$. Moreover, by multiplying both the numerator and denominator by the conjugate, we are essentially multiplying by 1, which does not change the value of the original expression, only its form. This technique is a fundamental tool in algebra and is frequently used to simplify and solve problems involving radicals and complex numbers. The next step involves using this simplified expression to prove the given inequality.

Step-by-Step Solution

Let's embark on a step-by-step solution to demonstrate that $\frac{1}{10} < \sqrt{101} - \sqrt{99}$.

1. Multiply by the conjugate: Start by multiplying and dividing the right-hand side ($\sqrt{101} - \sqrt{99}$) by its conjugate ($\sqrt{101} + \sqrt{99}$): ${ \sqrt{101} - \sqrt{99} = (\sqrt{101} - \sqrt{99}) \cdot \frac{\sqrt{101} + \sqrt{99}}{\sqrt{101} + \sqrt{99}} }$

2. Apply the difference of squares: Use the difference of squares identity, $(a - b)(a + b) = a^2 - b^2$, to simplify the numerator: ${ = \frac{(\sqrt{101})^2 - (\sqrt{99})^2}{\sqrt{101} + \sqrt{99}} = \frac{101 - 99}{\sqrt{101} + \sqrt{99}} = \frac{2}{\sqrt{101} + \sqrt{99}} }$

3. Establish the inequality: Now we need to show that $\frac{1}{10} < \frac{2}{\sqrt{101} + \sqrt{99}}$. To do this, we can take the reciprocal of both sides, remembering to flip the inequality sign: ${ 10 > \frac{\sqrt{101} + \sqrt{99}}{2} }$

4. Multiply by 2: Multiply both sides by 2: ${ 20 > \sqrt{101} + \sqrt{99} }$

5. Square both sides: Square both sides of the inequality. Since both sides are positive, squaring preserves the inequality: ${ 20^2 > (\sqrt{101} + \sqrt{99})^2 400 > 101 + 2\sqrt{101}\sqrt{99} + 99 400 > 200 + 2\sqrt{101 \cdot 99} }$

6. Simplify further: Subtract 200 from both sides: ${ 200 > 2\sqrt{101 \cdot 99} }$ Divide both sides by 2: ${ 100 > \sqrt{101 \cdot 99} }$

7. Square both sides again: Square both sides again to eliminate the square root: ${ 100^2 > 101 \cdot 99 10000 > 9999 }$

8. Conclusion: The final inequality, $10000 > 9999$, is clearly true. Since all our steps are reversible, we have successfully shown that $\frac{1}{10} < \sqrt{101} - \sqrt{99}$. This step-by-step approach highlights the power of algebraic manipulation, particularly the use of conjugates and the difference of squares identity, in proving inequalities involving radicals.

Alternative Approaches and Why They Might Fail

While rationalizing the denominator is a highly effective strategy, it's worth exploring alternative approaches and understanding why they might not be as successful. One common initial thought is to square both sides of the original inequality, $\frac{1}{10} < \sqrt{101} - \sqrt{99}$, to eliminate the square roots. However, this method can quickly lead to complications.

If we square both sides directly, we get: ${ \left(\frac{1}{10}\right)^2 < (\sqrt{101} - \sqrt{99})^2 }$ ${ \frac{1}{100} < 101 - 2\sqrt{101 \cdot 99} + 99 }$ ${ \frac{1}{100} < 200 - 2\sqrt{9999} }$

While this step eliminates the immediate square roots, it introduces a mixed term involving a square root ($2\sqrt{9999}$), making it challenging to simplify further and compare directly. We would still need to isolate the square root term and potentially square again, which adds more steps and complexity to the process. This approach doesn't inherently fail, but it's less direct and more prone to errors compared to the conjugate method.

Another approach might involve trying to approximate the square roots individually. However, without a calculator, accurately approximating $\sqrt{101}$ and $\sqrt{99}$ to a degree that allows for a clear comparison is difficult. This method relies on estimation, which can be imprecise and may not provide a rigorous proof. Furthermore, it deviates from the algebraic spirit of the problem, which seeks an exact demonstration rather than an approximation.

These alternative approaches illustrate the importance of choosing the right strategy when tackling algebraic problems. While some methods might eventually lead to a solution, they can be significantly more complex and time-consuming. The conjugate multiplication method provides a clear, efficient path to solving the inequality by leveraging algebraic identities and simplifying the expression in a controlled manner. Understanding why certain approaches are more effective than others is crucial for developing problem-solving skills in mathematics.

Conclusion

In conclusion, we have successfully demonstrated that $\frac{1}{10} < \sqrt{101} - \sqrt{99}$ without using a calculator. The key to solving this inequality lies in the strategic application of algebraic techniques, particularly rationalizing the expression $\sqrt{101} - \sqrt{99}$ by multiplying it by its conjugate. This method transforms the difference of square roots into a simpler fraction, making the comparison straightforward.

By multiplying and dividing by the conjugate $\sqrt{101} + \sqrt{99}$, we were able to utilize the difference of squares identity to eliminate the square roots in the numerator. This resulted in the expression $\frac{2}{\sqrt{101} + \sqrt{99}}$, which is easier to compare with $\frac{1}{10}$. The subsequent steps involved taking reciprocals, squaring, and simplifying the inequality until we arrived at the clearly true statement $10000 > 9999$. This process showcases the power of algebraic manipulation in solving problems involving inequalities and radicals.

We also explored alternative approaches, such as directly squaring both sides of the inequality, and discussed why they might not be as efficient. Squaring both sides initially introduces a mixed term with a square root, complicating the simplification process. Similarly, approximating the square roots without a calculator can be imprecise and doesn't provide a rigorous proof. These alternative attempts highlight the importance of choosing the most effective strategy to simplify and solve mathematical problems.

The successful solution presented here underscores the elegance and efficiency of using conjugates in dealing with expressions involving differences of square roots. This technique is a valuable tool in algebra and can be applied to a wide range of problems involving radicals and inequalities. By mastering such techniques, students can develop a deeper understanding of algebraic principles and enhance their problem-solving abilities.