Problem 26 - Intermediate Algebra

Question

List two other ways that the fraction \(-\frac{2}{3}\) can be written.

Step-by-Step Solution

Answer

\(-\frac{2}{3}\) can be written as \(-0.\overline{6}\) or \(-\frac{6}{9}\).

Step 1: Converting to Decimal Form

To convert the fraction \(-\frac{2}{3}\) to a decimal, divide the numerator (-2) by the denominator (3). This gives \(-0.666...\), which is an infinite repeating decimal often written as \(-0.\overline{6}\).

Step 2: Expressing as a Mixed Number

Since \(-\frac{2}{3}\) is less than zero but not an integer, expressing it as a mixed number directly gives \(-\frac{2}{3}\), because there's no whole number part. Thus, it cannot be expressed in standard mixed number format without at least one whole part.

Key Concepts

Converting Fractions to Decimals

When converting fractions to decimals, the process is quite straightforward. The fraction is essentially a division problem. The numerator, or the top number, is divided by the denominator, or the bottom number.
This division will yield a decimal number. For example, with the fraction \(-\frac{2}{3}\), you divide \(-2\) by \(3\).
The result is \(-0.666...\), which indicates a repeating decimal, as the digit \(6\) extends infinitely. This method of converting fractions to decimals is very useful. It serves as a bridge between two different number systems: fractions, which represent parts of a whole, and decimals, which give a numerical expression of fractions.

• Start with the fraction \( \frac{a}{b} \).
• Compute \( a \div b \) to get the decimal equivalent.
• If it does not terminate, identify if it is a repeating decimal.

Infinite Repeating Decimals

Infinite repeating decimals occur when the division of the numerator by the denominator in a fraction results in a decimal that continues indefinitely without terminating. Such decimals have a pattern or sequence which repeats itself forever.
In some cases, it might seem challenging to write down an infinite number of digits, so we use a bar notation.
For instance, in the fraction \(-\frac{2}{3}\), the resulting decimal \(-0.666...\) can be written as \(-0.\overline{6}\), where the bar over the \(6\) signifies that this digit repeats endlessly. This concept is essential for understanding the nature of many simple fractions:

• A fraction like \( \frac{1}{3} \) also repeats as \(0.\overline{3} \).
• The presence of repeating decimals indicates that the fraction cannot be expressed as a finite decimal.
• Not all fractions are repeating; some will terminate, like \( \frac{1}{4} = 0.25 \).

Mixed Numbers

Mixed numbers combine whole numbers with fractions. They are crucial for representing numbers that lie between whole numbers. Normally, a mixed number consists of a non-zero whole number and a proper fraction. However, there are cases where a decimal or fraction might not translate neatly into a mixed number.
For example, \(-\frac{2}{3}\) does not have a whole number component, therefore it cannot be directly expressed as a standard mixed number. The fraction is already as simplified as it can get, given its negative orientation and lack of integer whole number.

1. Mixed numbers typically have the form: \(a \frac{b}{c}\) where \(a\) is a whole number.
2. If there is no whole number, as with negative proper fractions, it remains as such.
3. This makes mixed numbers versatile for more than just addition and subtraction of fractions.