Problem 5 - Calculus with Concepts in Calculus

Question

Use the fact that \((\sqrt{2})^{2}=2\) to determine whether \(\sqrt{2}<\) \(1.41, \sqrt{2}=1.41\), or \(\sqrt{2}>1.41\)

Step-by-Step Solution

Answer

\(\sqrt{2} > 1.41\)

Step 1: Square the Given Value

Let's calculate \(1.41^2\) to compare it with 2. We know that huge numbers or decimals may make us a bit more hesitant. So, calculate the square: \(1.41^2 = 1.41 \times 1.41 = 1.9881\).

Step 2: Compare Squared Value with 2

Now, we compare the result from Step 1 with 2. We find that \(1.9881 < 2\). This implies that \((1.41)^2\) is less than 2.

Step 3: Draw Conclusion about \(\sqrt{2}\)

Since \(1.41^2\) is less than 2, and we know \((\sqrt{2})^2 = 2\), it means \(1.41 < \sqrt{2}\). Therefore, \(\sqrt{2}\) must be greater than 1.41.

Key Concepts

Squaring Numbers

Squaring a number means multiplying the number by itself. It's an essential concept in mathematics that simplifies many complex problems. When you square a number, it turns positive, which is important for comparisons. With our original exercise, we squared 1.41, calculating: \[1.41^2 = 1.41 \times 1.41 = 1.9881.\]This process helps sets a foundation for further comparison by eliminating square roots, which can be tricky to compare directly. Squaring makes it simpler as it brings numbers back to the simple arithmetic of multiplication. When you square decimals, just remember to carefully handle the placements correctly, as decimal squaring follows the same rules as whole numbers.

Inequality Comparison

When we compare numbers, inequalities provide a way to express the relationship between them. Inequalities show us three possibilities: a number is either less than, greater than, or equal to another number. In our solution, we compared the squared value of 1.41, which is 1.9881, with 2, the square of \( \sqrt{2} \). Through this comparison:

• Since \(1.9881 < 2\), 1.9881 is less than 2.
• This implies that \(1.41^2\) is less than \(2\). Therefore, 1.41 is less than \(\sqrt{2}\).

If these inequalities seem tricky, remember that we're looking at how one value relates to another. It's all about understanding not just which number is larger, but why it relates the way it does mathematically.

Mathematical Reasoning

Mathematical reasoning involves logical thinking to solve problems, and it often requires connecting different math concepts together. In the exercise, we used reasoning to connect squaring numbers with inequalities. Here's how this reasoning works in our case:

• We know that \((\sqrt{2})^2 = 2\). This is an established fact.
• Then we check if \((1.41)^2 = 1.9881\) is less than, greater than, or equal to 2.
• We find \(1.9881 < 2\), which indicates \(1.41 < \sqrt{2}\).

The key was to logically follow through each step, by combining known facts with calculated outcomes. This way, mathematical reasoning helps in providing a sound conclusion based on borrowed and proven mathematical principles.