Problem 10 - Algebra and Trigonometry

Question

\(9-16\) State whether each inequality is true or false. $$ \sqrt{2}>1.41 $$

Step-by-Step Solution

Answer

True: \( \sqrt{2} > 1.41 \).

Step 1: Evaluate the Square Root

Calculate the approximate value of \( \sqrt{2} \). We know that \( \sqrt{2} \approx 1.414 \). This is a non-terminating decimal that starts with 1.414.

Step 2: Compare Values

Compare the two values: \( 1.414 \) from \( \sqrt{2} \) and \( 1.41 \). Notice that \( 1.414 \) is greater than \( 1.41 \).

Step 3: Conclude the Inequality

Since \( 1.414 > 1.41 \), this means the inequality \( \sqrt{2} > 1.41 \) is true.

Key Concepts

Understanding Square Roots

Square roots are a fundamental concept in mathematics. They represent a number which, when multiplied by itself, results in the original number. For instance, the square root of 4 is 2, because when 2 is multiplied by itself, you get 4. This is denoted as \( \sqrt{4} = 2 \). Square roots of non-perfect squares (like the square root of 2) result in decimal numbers that are irrational, meaning they have non-repeating, non-terminating decimal expansions.

This is why \( \sqrt{2} \approx 1.414 \), and not a simple decimal. It is crucial to understand this nature while dealing with square roots, as rounding or approximating them is a common practice to simplify calculations. Knowing how to approximate a square root to various decimal places can provide precision suitable for the task at hand.

Practicing Approximations

Approximations come into play particularly in real-world scenarios where exact values of irrational numbers (like \( \sqrt{2} \)) are impractical. Approximating the value to a certain number of decimal places allows for ease of use while maintaining necessary precision.

To approximate \( \sqrt{2} \), one can use a method like long division or numerical algorithms to reach an estimate that is accurate enough for the problem. The approximation \( 1.414 \) is commonly used as it provides a reasonable level of accuracy with three decimal places.

• Using calculators or computational tools can quickly provide these approximations, but understanding the process builds stronger mathematical comprehension.
  
• Regular practice with these approximations helps in developing intuition about how close certain values are to their rooted numbers. This is especially useful in exam settings or mathematical reasoning tasks.

Comparing Real Numbers

In mathematics, comparing real numbers involves determining which is larger or smaller. This is an essential skill, as evident when looking at inequalities like \( \sqrt{2} > 1.41 \).

Once we determine the approximated value of \( \sqrt{2} \) as \( 1.414 \), comparing it to the number 1.41 is straightforward. Since each digit of 1.414 is equal or greater than the corresponding digit of 1.41 from left to right, we confirm that 1.414 is indeed larger.

• Comparing real numbers requires careful attention to decimal places. Even a difference of .001 can determine the truth of an inequality.
  
• Practicing with inequalities enhances understanding of number size relationships and stimulates mental arithmetic skills, providing a crucial tool in mathematics. Being comfortable with such comparisons can aid in various analytical tasks and everyday problem solving.