Problem 10 - Contemporary Precalculus

Question

Use a calculator and list the given numbers in order from smallest to largest. $$\frac{385}{177}, \quad \sqrt{10}, \quad \frac{187}{63}, \quad \pi, \quad \sqrt{\sqrt{85}}, \quad 2.9884$$

Step-by-Step Solution

Answer

Question: Arrange the following numbers from smallest to largest: $$\frac{385}{177}, \quad \sqrt{10}, \quad \frac{187}{63}, \quad \pi, \quad \sqrt{\sqrt{85}}, \quad 2.9884$$ Answer: $$\frac{385}{177}, \quad \sqrt{\sqrt{85}}, \quad \frac{187}{63}, \quad 2.9884, \quad \pi, \quad \sqrt{10}$$

Step 1: Calculate the approximate values

Using a calculator, find the approximate values for each expression: - $$\frac{385}{177} \approx 2.1751$$ - $$\sqrt{10} \approx 3.1623$$ - $$\frac{187}{63} \approx 2.9683$$ - $$\pi \approx 3.1416$$ - $$\sqrt{\sqrt{85}} \approx 2.9339$$ - $$2.9884$$

Step 2: Compare the approximate values

Now that we have the approximate values, compare them to determine their order: - Smallest: $$2.1751 (\frac{385}{177})$$ - Next: $$2.9339 (\sqrt{\sqrt{85}})$$ - Next: $$2.9683 (\frac{187}{63})$$ - Next: $$2.9884$$ - Next: $$3.1416 (\pi)$$ - Largest: $$3.1623 (\sqrt{10})$$

Step 3: List the original expressions in order

Based on the comparison of approximate values, list the original expressions in order from smallest to largest: $$\frac{385}{177}, \quad \sqrt{\sqrt{85}}, \quad \frac{187}{63}, \quad 2.9884, \quad \pi, \quad \sqrt{10}$$

Key Concepts

Fractions

Fractions represent parts of a whole. They are expressed as two numbers, one above the other, separated by a line. The top number is called the numerator, and the bottom number is the denominator.
This method of expressing numbers allows us to calculate the ratio of two figures.
For instance, in the fraction \( \frac{385}{177} \), the numerator is 385 and the denominator is 177.
To compare fractions, it is often helpful to convert them into decimal form. This can be done by dividing the numerator by the denominator. For example:

• \( \frac{385}{177} \approx 2.1751 \)
• \( \frac{187}{63} \approx 2.9683 \)

By finding these decimal approximations, we can easily compare fractions with other decimal numbers.

Square Roots

Square roots are a way of finding a number which, when multiplied by itself, gives the original number.
The square root operation is an essential mathematical function used to simplify expressions and solve equations.
For example, \( \sqrt{10} \approx 3.1623 \). This means 3.1623 multiplied by itself is approximately 10.
The process involves finding a number \( x \) such that \( x^2 = 10 \).
Calculators can help to find square roots quickly, allowing you to compare them with other numerical forms like fractions or integers.
In our problem, we also find the square root of a square root. This is represented as \( \sqrt{\sqrt{85}} \), which means finding the square root of \( \sqrt{85} \). Its approximate value is 2.9339.

Approximations

Approximations are a way to represent numbers when absolute precision is unnecessary. They simplify complex numbers, making calculations and comparisons more manageable.
Approximations are particularly useful in cases involving irrational numbers like \( \pi \).
For example, \( \pi \) is often approximated as 3.1416 for simplicity. This allows us to perform operations without dealing with an infinite number of decimal places.
Ultimately, approximations help us achieve practical accuracy when working with otherwise unwieldy figures. They play a vital role in almost every area of mathematics, not just in comparing numbers, but in everyday applications too.
It's important to remember that while approximations provide a close estimate, they are not exact.
Thus, understanding when to use them and how accurate they need to be is crucial for effective problem-solving.

Mathematical Comparison

Mathematical comparison involves evaluating numbers to determine their order or equality. This can be done using symbols like greater than (\( > \)) or less than (\( < \)).
In number ordering, we often need to compare various forms of numbers such as decimals, fractions, and roots.
In our task, we first convert all given values into a common form (decimals) to facilitate easy comparison:

• \( \frac{385}{177} \approx 2.1751 \) is less than \( \sqrt{10} \approx 3.1623 \)
• Comparing \( 2.9884 \) and \( \pi \approx 3.1416 \) shows \( 2.9884 \) is smaller.

By arranging these approximations in ascending order, we effectively sort the original expressions.
This method highlights the importance of comparing the size of numerical expressions accurately, especially when working with mixed forms.