Problem 27 - Introductory Algebra for College Students

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 0.8181.

Step 1: Understand the problem

The exercise requires converting the fractional number \(\frac{9}{11}\) into decimal form. This can be done through the method of long division.

Step 2: Begin the long division process

To start, divide 9 by 11. Since 9 is less than 11, the result is 0 with a remainder of 9.

Step 3: Continue long division

Now, append a decimal point and a zero to the right side of the 9 in the numerator, making it 90. Then, divide 90 by 11. The result is 8 with a remainder of 2. Therefore, the first decimal digit after the decimal point is 8.

Step 4: Repeat step 3

We can repeat step 3 with the remainder by appending a zero to the right side of the 2 and dividing 20 by 11. The result is 1 with a remainder of 9. The second decimal digit after the decimal point is 1.

Step 5: Continue the long division process

If you continue this process, you will find that the digits repeat in a cycle. Thus, the decimal form of \(\frac{9}{11}\) is 0.8181 (rounded to four decimal places), and will continue in this pattern indefinitely.

Key Concepts

Decimal Conversion

When you talk about converting a fraction to a decimal, you're essentially transforming the number from one format to another. Every fraction represents a division operation where the numerator is divided by the denominator. This division can yield a decimal that either terminates or repeats. Here's a quick breakdown:

• **Terminating Decimal:** Ends after a few digits. For example, \(rac{1}{4} = 0.25\).
• **Repeating Decimal:** Has one or more repeating digits or block of digits, like \(rac{9}{11} = 0.818181...\).

For fractions like \(rac{9}{11}\), the process begins by dividing the numerator by the denominator using long division, which we'll explore in the next section. Once the quotient repeats, you can identify the repeating sequence.

Long Division

Long division is a traditional method used to divide larger numbers. It’s very helpful in converting fractions to decimals. Here’s how it works, using the fraction \(\frac{9}{11}\) as an example:1. **Setup:** Write 9 (the dividend) inside the division bracket and 11 (the divisor) outside. Since 9 is less than 11, you know the whole number part is zero.2. **Add a Decimal:** Append a decimal point to the 9 and then add a zero, making it 90.3. **Divide:** Determine how many times 11 fits into 90. It goes 8 times, resulting in 88. The remainder is 2. 4. **Proceed:** Append another zero to the remainder, making it 20. Now, see how many times 11 fits into 20, which is 1 time with a remainder of 9.This iterative process continues, repeatedly dividing and appending zeros to the remainder. The long division will help find each next digit of the decimal until a pattern appears or the remainder ends at zero.

Repeating Decimals

Some fractions, when converted to decimals, don’t terminate. They go on forever but repeat in a predictable pattern. These are known as repeating decimals. In the fraction \(\frac{9}{11}\), the decimal is 0.818181..., with '81' being the repeating part.

• **Identifying Repeating Decimals:** Look for when the remainder you've been dividing returns to the original number in your process. Once it repeats, the digits will cycle.
• **Representing:** Repeating decimals are denoted by placing a bar over the repeating digits, so \(\frac{9}{11} = 0.\overline{81}\).
• **Understanding Cycles:** Just like with \(\frac{9}{11}\), many fractions will repeat within a few steps, enabling you to predict the outcome.

Harnessing long division to find these repeating patterns helps in converting and representing rational numbers accurately.