Problem 61 - Prealgebra

Question

Write each fraction or mixed number as a decimal. Use a bar to show a repeating decimal. $$2 \frac{5}{9}$$

Step-by-Step Solution

Answer

The decimal form of \(2 \frac{5}{9}\) is \(2.\overline{5}\).

Step 1: Understand the Fraction

The fraction given is a mixed number, which can be split into a whole number and a proper fraction: \(2\) and \(\frac{5}{9}\). We need to convert \(\frac{5}{9}\) into a decimal.

Step 2: Divide the Fraction

Divide the numerator by the denominator. Divide 5 by 9 to get the decimal equivalent of the fraction \(\frac{5}{9}\). This division gives a repeating decimal of \(0.5555...\), which can be written as \(0.\overline{5}\).

Step 3: Combine the Whole Number and Decimal

Combine the whole number from the mixed number with the decimal of the fraction. The whole number is \(2\), and the decimal for \(\frac{5}{9}\) is \(0.\overline{5}\), so the result is \(2.\overline{5}\).

Key Concepts

Understanding Mixed Numbers

Mixed numbers are a way to represent numbers that contain both a whole part and a fractional part. For example, in the mixed number \(2 \frac{5}{9}\), the number \(2\) is the whole part, and \(\frac{5}{9}\) is the fractional part. Mixed numbers are often used because they align more intuitively with concepts we encounter in daily life, like measurements.
To convert a mixed number into a decimal, one should separately handle the whole and fractional parts. The whole part remains untouched in terms of its decimal representation. The focus is primarily on converting the fractional part into a decimal. Only then do we combine both parts for a complete decimal representation. This process involves basic arithmetic operations such as division, which can help unravel the decimal form of the fraction.

Understanding Repeating Decimals

When you convert some fractions into decimals, sometimes the decimal is non-terminating and one or more digits repeat infinitely. These are known as repeating decimals. For example, with the fraction \(\frac{5}{9}\), dividing 5 by 9 will give you \(0.5555...\); as you continue your division, the 5 repeats indefinitely.
To denote this repeating sequence, a bar is placed over the repeating digit or group of digits. In the previous example, the repeating decimal is written as \(0.\overline{5}\), where the bar indicates that the digit 5 continues without end. Repeating decimals are a fascinating subject because they showcase the complexity and beauty of mathematics—even simple numbers can open the door to endless patterns. Also, recognizing repeating decimals is important to ensure clarity and precision in mathematical calculations.

Decimal Representation of Numbers

Decimal representation is a way of expressing numbers in a base-10 number system, which is widely used due to its simplicity and prevalence in everyday life. Converting fractions or mixed numbers to decimals involves determining how many times the denominator fits into the numerator, which often results in a precise repeating or terminating decimal.

• A terminating decimal is one that ends after a finite number of digits. For instance, \(\frac{1}{4}\) converts to \(0.25\).
• A repeating decimal, on the other hand, keeps repeating a sequence of digits. Like \(\frac{5}{9}\), which becomes \(0.\overline{5}\).

These forms help with better understanding and using numbers in various scenarios like financial computations, where decimals are more commonplace than fractions. Converting mixed numbers to decimals, like with \(2 \frac{5}{9}\), where the fraction becomes \(0.\overline{5}\), and is combined with the whole number to form \(2.\overline{5}\), demonstrates a clear, unified numerical representation.