Problem 111 - Algebra 1: Concepts and Skills

Question

Write the fraction as a terminating or repeating decimal. $$ \frac{8}{9} $$

Step-by-Step Solution

Answer

\(\frac{8}{9}\) can be written as a repeating decimal as \(0.8̅\).

Step 1: Identify the Fraction

The given fraction is \(\frac{8}{9}\). We need to convert this into a decimal.

Step 2: Division

Perform the division of the numerator (8) by the denominator (9) either manually or using a calculator. The result is approximately 0.8888888889.

Step 3: Identify Decimal Pattern

Now, from the result of the division, one can notice that the decimal number 0.8888888889 actually repeats the number '8'. Therefore, we can denote this using a bar over the repeating digit. So, \(\frac{8}{9} = 0.8̅\).

Key Concepts

Repeating Decimals

Converting fractions into decimals can result in different types of decimal numbers. One common outcome is a repeating decimal. This happens when the digits start to repeat in a predictable pattern.
For instance, when converting \(\frac{8}{9}\) to a decimal, we get 0.888...; the digit '8' continues indefinitely. To denote this repeating pattern, we use a bar over the repeating digit: \(0.\overline{8}\).
This helps us clearly represent numbers with endless repeating sections in a concise way. Recognizing repeating decimals is important in understanding how some fractions behave when expressed in decimal form.

Division

The process of converting a fraction to a decimal involves division. Specifically, we divide the numerator (the top number) by the denominator (the bottom number). This calculation either directly or mentally allows us to express the fraction as a decimal.
Let's take \(\frac{8}{9}\), for example. When dividing 8 by 9, you get 0.8888..., showing that the division doesn't resolve to a finite number of digits but repeats instead. This division not only helps find the decimal form but also reveals whether the decimal is terminating or repeating.

Numerator and Denominator

In a fraction, the numerator and denominator are key components. The numerator is the number above the line, showing how many parts we have. The denominator is the number below the line, indicating into how many parts the whole is divided.
For \(\frac{8}{9}\), '8' is the numerator and '9' is the denominator. To convert this fraction into a decimal, you divide the numerator by the denominator. This straightforward operation allows you to unlock the decimal representation of the fraction and understand its nature. Understanding these roles is essential as they determine how fractions convert into decimal forms.