Problem 92 - Contemporary Precalculus

Question

Find the distance between the given numbers. \(\pi\) and \(\sqrt{2}\)

Step-by-Step Solution

Answer

Answer: The approximate distance between the numbers \(\pi\) and \(\sqrt{2}\) is 1.73.

Step 1: Identify the larger and smaller numbers

In this case, we are given two numbers, \(\pi\) and \(\sqrt{2}\). We know that \(\pi \approx 3.14\) and \(\sqrt{2} \approx 1.41\). Since 3.14 is larger than 1.41, we can identify that \(\pi\) is the larger number and \(\sqrt{2}\) is the smaller number.

Step 2: Subtract the smaller number from the larger number

Now that we know which number is larger and which is smaller, we need to find the difference between them. Subtract the smaller number (\(\sqrt{2}\)) from the larger number (\(\pi\)): $$\pi - \sqrt{2}$$

Step 3: Use the approximate values to find the distance

Using the approximate values for \(\pi\) and \(\sqrt{2}\), we can now calculate their difference: $$3.14 - 1.41 \approx 1.73$$ Therefore, the approximate distance between the numbers \(\pi\) and \(\sqrt{2}\) is 1.73.

Key Concepts

Approximation

When dealing with numbers like \( \pi \) and \( \sqrt{2} \), it's often necessary to use approximations because their exact values are not straightforward decimals. Approximations help simplify calculations by using closer, easier-to-handle estimates. This can be especially useful when working on problems that require quick numerical answers.

• \( \pi \) is generally approximated as 3.14, a common choice due to its closeness to the actual value of approximately 3.14159.
• Similarly, \( \sqrt{2} \) which is around 1.414 has an approximation of 1.41 for simplicity in calculations.

Choosing these approximations allows us to still maintain reasonable accuracy while performing mathematical operations such as subtraction or addition. This approach is very practical in many mathematical tasks where estimating is sufficient rather than calculating precise values.

Subtraction

To find the distance between two numbers, a basic arithmetic operation is used: subtraction. Subtraction involves taking one value away from another. In this exercise, we're interested in finding how much larger \( \pi \) is than \( \sqrt{2} \).

To perform the subtraction:

• We start by identifying which number is larger; from the approximations, \( \pi \approx 3.14 \), and \( \sqrt{2} \approx 1.41 \).
• Then, we subtract the smaller number from the larger number, which is simplified as \( 3.14 - 1.41 \).
• After calculating, the difference comes out to be approximately 1.73.

This difference, 1.73, represents the 'distance' or the numerical space between the two numbers when plotted on a number line. This concept is fundamental in understanding how values compare and contrast in magnitude.

Pi and Square Root Calculations

Understanding the nuances between different mathematical constants like \( \pi \) and \( \sqrt{2} \) brings greater clarity to calculations. Both are irrational numbers, meaning their exact decimal representations are infinite and non-recurring.

**Why are they important?**

• \( \pi \) is a crucial constant in geometry, especially when dealing with circles, where it represents the ratio of a circle's circumference to its diameter.
• \( \sqrt{2} \) is essential in geometry as it is the length of the hypotenuse in an isosceles right triangle where the other sides are both 1 unit long.

In mathematical operations, when needing to compute values involving these constants, approximations like 3.14 for \( \pi \) and 1.41 for \( \sqrt{2} \) can be used to provide adequate results without necessitating complex calculations or software. This process eases computation while still delivering valuable insights into the nature of the problem being solved.