Problem 23 - Prealgebra

Question

Write each fraction or mixed number as a decimal. Use a bar to show a repeating decimal. $$\frac{1}{9}$$

Step-by-Step Solution

Answer

\( \frac{1}{9} = 0.\overline{1} \)

Step 1: Understanding the Fraction

The given fraction is \( \frac{1}{9} \), which needs to be expressed as a decimal. To do this, we must perform the division \( 1 \div 9 \).

Step 2: Set Up the Long Division

Set up the division of 1 by 9. Since 9 is larger than 1, we know that the result will be a decimal less than 1.

Step 3: Perform the Long Division

Begin by dividing: 9 goes into 10 once (since 9 multiplied by 1 is 9), and place the 1 above the division bar, giving a remainder of 1. Bring down a 0 to make it 10 if needed, and repeat the process.

Step 4: Identify the Pattern

As you continue dividing, you will notice that the remainder continues to be 1, which means the decimal repeats itself indefinitely. Therefore, \( 1 \div 9 = 0.111... \).

Step 5: Write the Decimal With a Bar

Since the digit \(1\) repeats indefinitely, the decimal representation of \( \frac{1}{9} \) is written as \( 0.\overline{1} \), indicating that the 1 is repeating.

Key Concepts

Long Division

Long division is a mathematical method used to solve division problems. It is particularly useful when dealing with fractions converted into decimals. When you have a fraction like \( \frac{1}{9} \), performing long division involves dividing the numerator (1) by the denominator (9). Since 1 is less than 9, you would initially have a decimal number that is less than one.

Here is how you perform long division step-by-step:

• First, arrange the division problem with 1 inside the division bar and 9 outside.
• Since 9 cannot go into 1, you would add a decimal point and bring down a zero to make it 10, allowing you to divide.
• 9 goes into 10, once. The result is written above the division bar after the decimal point, and you would then subtract (9×1 = 9) from 10, leaving a remainder of 1.
• Repeat the process by adding another zero, bringing it down to form 10 again, and continue dividing as before.

By following this method, you'll notice a recurring pattern, leading to a repeating decimal, as demonstrated in divisors such as \( \frac{1}{9} \).

Repeating Decimals

In division, a repeating decimal is a result where a sequence of numbers continuously repeats indefinitely. This pattern emerges when performing the division of certain fractions whose results cannot be expressed as a terminating decimal. When you deal with the fraction \( \frac{1}{9} \), by using long division you'll see how each step results in a leftover that leads to the same series of digits after the decimal point.

When the decimal repeats, a bar is used over the repeating number (or numbers) to show this repetition. In the case of \( \frac{1}{9} \), the decimal equivalent is \( 0.\overline{1} \), indicating that the number 1 repeats forever.

• During division, when recurring remainders appear, they indicate that the same digits will endlessly occur.
• Used bars visually convey the repeating nature to maintain neat and understandable expressions.

Recognizing and properly documenting repeating decimals is a crucial aspect of understanding decimal representations of fractions.

Mixed Numbers

Mixed numbers, on the other hand, are numbers composed of an integer and a proper fraction. They are commonly used in everyday scenarios such as measuring quantities. For instance, a mixed number like 1 ½ refers to one whole and one half of something, which is different from our fraction \( \frac{1}{9} \) since it strictly involves decimal conversion without whole numbers.

Converting mixed numbers into decimals involves first converting the fraction part separately as a decimal and then adding the whole number part. For example, if we wanted to convert 2 ¾ into a decimal:

• Convert the fraction \( \frac{3}{4} \) to the decimal 0.75 using long division or known decimal equivalents.
• Add this decimal to the whole number, so 2 + 0.75 = 2.75.

While this is not needed for \( \frac{1}{9} \) specifically, understanding mixed numbers helps solidify knowledge about decimals and fractions. By recognizing the structure of a mixed number, conversions between fractions, decimals, and mixed numbers become intuitive.