Problem 22 - College Algebra and Calculus: An Applied Approach

Question

Use a calculator to order the numbers from least to greatest.\(\frac{26}{15}, \sqrt{3}, 1.73 \overline{20}, \frac{381}{220}, \sqrt{10}-\sqrt{2}\)

Step-by-Step Solution

Answer

The numbers ordered from least to greatest are: \(\sqrt{10}-\sqrt{2}\), \(\frac{381}{220}\), \(\sqrt{3}\), 1.7320 \(repeats\) , \(\frac{26}{15}\).

Step 1: Convert All Numbers to Decimal Format

First, convert all numbers to decimal form using a calculator. The values are as follows: \(\frac{26}{15}\) is approximately 1.73333, \(\sqrt{3}\) is approximately 1.73205, 1.7320 \(repeats\), \(\frac{381}{220}\) is approximately 1.73182, and \(\sqrt{10}-\sqrt{2}\) is approximately 1.132724.

Step 2: Arrange Numbers in Ascending Order

Now that all numbers are in decimal form, arrange them from smallest to largest. Here are the results: \(\sqrt{10}-\sqrt{2}\), \(\frac{381}{220}\), \(\sqrt{3}\), 1.7320 \(repeats\) , \(\frac{26}{15}\).

Key Concepts

Converting Fractions to Decimals

Understanding how to convert fractions to decimals is crucial for ordering numbers in decimal form. A fraction represents a part of a whole and consists of a numerator (top number) and a denominator (bottom number). To convert a fraction to a decimal, you divide the numerator by the denominator. For instance, to convert the fraction \(\frac{26}{15}\), you divide 26 by 15, which is approximately 1.73333. It's important to carry out the division to enough decimal places so that you can make accurate comparisons later on.

For fractions that result in a repeating decimal, you can denote this by placing a line over the repeating digit or group of digits, such as in 1.73\(\overline{20}\), which signifies that the '20' repeats indefinitely. While calculators may truncate repeating decimals, understanding this concept ensures that you recognize the true nature of the decimal you are working with.

Approximating Square Roots

When dealing with square roots, sometimes you need to approximate their values, particularly when they are irrational and cannot be expressed as a neat fraction. The square root of a number is the value that, when multiplied by itself, gives the original number. For example, the square root of 3, notated as \(\sqrt{3}\), is an irrational number. To approximate \(\sqrt{3}\), you could use a calculator or estimate it by knowing that \(\sqrt{4}\) is 2 (since 2 x 2 = 4), thus \(\sqrt{3}\) must be slightly less than 2. With a calculator, you might find \(\sqrt{3}\) to be approximately 1.73205.

Ideally, for consistency in setting out your work, round square roots to the same number of decimal places as the other numbers you are working with. This ensures that you have a uniform basis for comparison.

Comparing Irrational Numbers

Irrational numbers are numbers that cannot be exactly expressed as a fraction of two integers; their decimal representations go on infinitely without repeating. Comparing them can be tricky since they don't terminate or repeat in a predictable pattern. Common examples include \(\sqrt{3}\) and \(\pi\). When you need to compare irrational numbers, approximate their decimal equivalents to several places for accuracy.

In our exercise, to compare \(\sqrt{3}\) and \(1.73\(\overline{20}\)\), we approximate them and then line up their decimals to see which is greater. Since both start with '1.73', you look at the succeeding digits to determine their order. Remember, the more decimal places you use, the more precise the comparison, but don't exceed the precision of the least accurate number.

Arranging Numbers in Ascending Order

To arrange numbers in ascending order means to line them up from the smallest to the largest. After converting all numbers to decimals and approximating where necessary, you can compare them digit by digit starting from the left. In the given exercise, once converted, the numbers in decimal form are placed side by side to examine their size.

The difference might be clear in the first few digits after the decimal point, which makes it easier to determine their order. If numbers share initial digits, as they often do when dealing with approximations and repeating decimals, you must compare subsequent digits. For example, with \(\frac{26}{15}\) as approximately 1.73333 and \(\sqrt{3}\) as approximately 1.73205, the former is greater since the third decimal place of 1.73333 is higher than that of 1.73205. Patience and attention to detail are key in accurately arranging numbers in ascending order.