Problem 27 - Introductory and Intermediate Algebra for College Students 4th

Question

Express each rational number as a decimal. $$\frac{9}{11}$$

Step-by-Step Solution

Answer

The decimal form of \(\frac{9}{11}\) is approximately 0.818

Step 1: Perform the Division

Divide the numerator by the denominator, i.e., \(9 ÷ 11\). Use a calculator for accuracy

Step 2: Write down the decimal

The result from Step 1 will be a decimal number. Write it down as the decimal representation of the given fraction.

Key Concepts

Decimal Representation

When we talk about the decimal representation of a number, we are giving that number in its decimal form rather than as a fraction or a whole number. Decimal numbers are expressed using the base-10 number system, which includes digits from 0 to 9 and a decimal point. For example, the fraction \(\frac{9}{11}\) can be expressed as a decimal.To convert a fraction to its decimal form, you divide the numerator (the top number) by the denominator (the bottom number). This division gives us a decimal value, which might either terminate (end after a few digits) or repeat (continue infinitely in a repetitive pattern).A fraction like \(\frac{9}{11}\) will have a repeating decimal because it doesn't neatly divide into a finite number of decimal places. In practical terms, performing the division \(9 \div 11\) gives a repeating decimal, often rounded as necessary for simplicity during calculations or when writing it down.

Fractions to Decimals

Converting fractions to decimals is a crucial concept in mathematics that helps us understand numbers better. It involves a simple process of division — dividing the numerator by the denominator.Here's a step-by-step guide to converting fractions to decimals:

• Identify the Fraction: The fraction \(\frac{9}{11}\) is our starting point here. The numerator is 9 and the denominator is 11.
• Perform the Division: Use long division to divide 9 by 11. This division will not result in a whole number, so be ready to handle a repeating decimal.
• Write Down the Result: Record the result as a decimal. Many times, you will find the quotient repeating, which is essential to note as it affects the answer's precision.

Using a calculator for such conversions is often helpful because these numbers may involve complex repeating patterns. In our example of \(\frac{9}{11}\), the division results in a repeating decimal of 0.8181..., which can be rounded up for easier usage in most real-world scenarios.

Division of Fractions

The division of fractions involves breaking down the relationship between the numerator and the denominator through a straightforward division. It asks us to see how many times the denominator fits into the numerator. While dividing fractions, especially when converting to decimals, can feel daunting at first, understanding the process makes it easier.Here’s how it works:

• Start with the Fraction: For \(\frac{9}{11}\), treat the fraction as a simple division problem, where the numerator (9) is divided by the denominator (11).
• Perform the Division: Use long division or a calculator to find out \(9 \div 11\). This step helps expand your understanding of the fraction's size compared to the whole number 1.
• Identify Patterns: Many divisions will result in decimal forms that either terminate or repeat. Identifying these repeating patterns can help simplify future mathematical operations.

By ensuring that you grasp the division of fractions, you can more comfortably convert fractions into decimals and better understand the decimals' behavior in various mathematical contexts. Practice is key in mastering these conversions, enhancing both speed and accuracy over time.