Problem 21 - College Algebra and Calculus: An Applied Approach

Question

Use a calculator to order the numbers from least to greatest.\(\frac{7071}{5000}, \frac{584}{413}, \sqrt{2}, \frac{47}{33}, \frac{127}{90}\)

Step-by-Step Solution

Answer

The numbers from least to greatest are: \( \frac{127}{90}, \frac{7071}{5000}, \frac{584}{413}, \sqrt{2}, \frac{47}{33}\)

Step 1: Convert the fraction into decimal

First, convert the fractions into decimal values. We divide the numerator by the denominator using the calculator: \( \frac{7071}{5000}=1.4142 \), \( \frac{584}{413}=1.4142 \), \( \frac{47}{33}=1.4242 \), \( \frac{127}{90}=1.4111 \)

Step 2: Decimal approximation for square root

Next, find the decimal approximation of the square root by using your calculator: \( \sqrt{2}=1.4142 \)

Step 3: Arrange the numbers

Finally, arrange the numbers we got in increasing order. Therefore, the numbers from least to greatest are: \( \frac{127}{90}, \frac{7071}{5000}, \frac{584}{413}, \sqrt{2}, \frac{47}{33}\)

Key Concepts

Decimal Conversion

Understanding decimal conversion is essential when comparing values, especially when dealing with fractions or mixed numbers. To convert a fraction into a decimal, divide the numerator (the top number) by the denominator (the bottom number) using a calculator or long division. For instance, to convert \( \frac{7071}{5000} \) into a decimal, we divide 7071 by 5000, which results in 1.4142. This conversion makes it easier to compare numbers, because decimals align on the place value chart, allowing for a straightforward comparison digit by digit starting from the left.

It's important to remember that some fractions may result in a repeating decimal, which can be noted by a line or dot above the repeating digits. Although repeating decimals may seem complex, they follow a predictable pattern, which facilitates their use in numerical ordering.

Square Root Approximation

When working with square roots, we often encounter irrational numbers—numbers that cannot be perfectly expressed as a fraction or a decimal. The square root of 2, denoted as \(\sqrt{2}\), is such an irrational number. However, for practical purposes, like ordering or basic calculations, we use a decimal approximation. By using a calculator, we can estimate \(\sqrt{2}\) to several decimal places, such as 1.4142.

Although an approximation, it's generally sufficient for most applications. It's crucial to approximate to enough decimal places to ensure accuracy in comparison, particularly when the other numbers in a set are close in value. In mathematics, precision matters, and the approximation of square roots is a useful concept to ensure comparability among numbers that include both rational and irrational values.

Numerical Ordering

Numerical ordering involves arranging numbers in ascending (from least to greatest) or descending (from greatest to least) order. It's a fundamental concept in mathematics that applies across various contexts. To effectively order numbers, especially when they are in different forms (like fractions, square roots, or decimals), they must first be converted into a common form. In this case, we use decimals because they are easy to compare.

Once all numbers are in decimal form, you can compare them by looking at each digit's place value, starting from the leftmost non-zero digit. If two numbers have the same digit in a place value, you move right to the next digit until you find a difference. All numbers are then ordered based on these comparisons. The ability to order numbers aids not just in pure mathematics but also in real-life situations such as financial calculations, data analysis, and prioritizing tasks.