Problem 20 - College Algebra and Calculus: An Applied Approach

Question

Use a calculator to order the numbers from least to greatest.\(\frac{559}{500}, 1.12, \frac{\sqrt{5}}{2}, \frac{115}{99}, \frac{23}{20}\)

Step-by-Step Solution

Answer

Ordered from least to greatest, the numbers are \(\frac{559}{500}\), 1.12, \(\frac{23}{20}\), \(\frac{\sqrt{5}}{2}\), \(\frac{115}{99}\)

Step 1: Convert each number to decimal

Firstly, convert each fraction or square root in decimal form. For the fractions \(\frac{559}{500}\), \(\frac{115}{99}\), and \(\frac{23}{20}\), conduct the indicated division. For \(\frac{\sqrt{5}}{2}\), calculate the square root of 5 and then divide it by 2.

Step 2: Compare the decimals

After converting each number to a decimal, they can be compared. Remember that the longer a decimal, the smaller it is. Write these decimals in the order from least to greatest.

Step 3: Rewrite in original form

Once the numbers have been ordered as decimals, rewrite the sequence using the original representations. This will give the final answer in the form from least to greatest.

Key Concepts

Decimal Conversion

Understanding decimal conversion is essential when you are dealing with numbers in different formats. It's the process of changing a number into its decimal form, which can be especially useful when you are trying to order or compare numbers from least to greatest, like in the exercise provided.

For fractions, decimal conversion involves dividing the numerator by the denominator. For example, to convert the fraction \( \frac{23}{20} \) into a decimal, you divide 23 by 20. In cases involving square roots, such as \( \frac{\sqrt{5}}{2} \) from the exercise, calculate the square root of 5 first, then divide the result by 2 to get its decimal form.

Comparing Numbers

Comparing numbers is a basic numerical skill that involves determining which of two numbers is greater or whether they are equal. When numbers are presented in decimal form, the process becomes more straightforward. You can compare them digit by digit, starting from the decimal point and moving to the right.

Remember, the number with the higher value digit in the farthest left position where the numbers differ is the greater number. It's not always the case that a number with more digits after the decimal is smaller; you need to compare the values of each digit position. In the context of the textbook exercise, once the numbers have been converted to decimals, it becomes much easier to order them.

Fraction to Decimal Conversion

Fraction to decimal conversion might seem challenging at first, but it's quite simple once you get the hang of it. You're essentially taking the fraction's numerator and dividing it by its denominator. For instance, to convert the fraction \( \frac{559}{500} \) into a decimal, you would divide 559 by 500.

Calculators make this process fast and error-free, which is why the exercise recommended using one. The result of this division gives you the decimal form of the fraction, which then can be used in various mathematical operations, such as ordering or comparing.

Interpreting Square Roots

Interpreting square roots is another important concept in mathematics, particularly when they are part of a fraction as seen in the exercise with \( \frac{\sqrt{5}}{2} \). A square root represents a number that, when multiplied by itself, gives the original number under the square root symbol. To interpret \( \sqrt{5} \), you need to understand that it's the number which, when squared, equals 5. Since such roots often result in an irrational number (a never-ending, non-repeating decimal), use a calculator for an approximate decimal value.

Once this value is obtained, you can then proceed with other operations, such as dividing by another number or comparing with other decimals, as the exercise required.