Problem 32 - Introductory and Intermediate Algebra for College Students 4th

Question

Express each rational number as a decimal. $$-\frac{7}{6}$$

Step-by-Step Solution

Answer

-1.167

Step 1: Identify the fraction to be converted

The fraction to be converted into a decimal is \(-\frac{7}{6}\). The minus sign indicates that this is a negative number.

Step 2: Perform the division

Since a fraction is simply a form of division, convert \(-\frac{7}{6}\) to its decimal equivalent by dividing the numerator (7) by the denominator (6). This gives approximately -1.167 on carrying out the division.

Step 3: Write the decimal form

The decimal representation of \(-\frac{7}{6}\) is thus -1.167

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction of two integers. The numerator is the top integer, and the denominator is the non-zero bottom integer. For example, \(-\frac{7}{6}\) is a rational number where \(-7\) is the numerator, and \(+6\) is the denominator. This means rational numbers can be written as a simple fraction, which allows them to be easily converted into decimal form.

• All fractions are rational numbers.
• Rational numbers can be positive, negative, or zero.
• They include integers, as every integer \(n\) can be expressed as \(-\frac{n}{1}\).

Understanding rational numbers is important because they form the basis for converting fractions into decimal numbers, allowing us to easily compare and perform arithmetic operations with them.

Fraction to Decimal

Converting a fraction to a decimal involves changing a number from a fraction form (where it has a numerator and a denominator) into a decimal form, which is a base 10 numeral. This process is essential as it provides a visual representation that is often easier to interpret in everyday contexts.

• Any fraction \(\frac{a}{b}\) can be transformed into a decimal through division.
• The decimal form of a fraction is often a more intuitive format.
• Decimals can be terminating, like 0.5, or repeating, like 0.333...

To convert \(-\frac{7}{6}\), notice how the minus sign in the fraction tells us the result will be negative. The conversion process helps us understand how decimal numbers represent the value of fractions in real-world scenarios.

Division Process

The division process in math is critical when converting fractions to decimals. This process involves dividing the numerator (the top portion of the fraction) by the denominator (the bottom portion of the fraction).For the example \(-\frac{7}{6}\), follow these steps:1. **Ignore the Negative Sign Temporarily**: Start by dividing 7 by 6, imagining 7 as the numerator and 6 as the denominator.2. **Division**: Begin dividing 7 by 6 as whole numbers: - 6 fits into 7 once, so write down 1 and a remainder of 1. - Work with the remainder by bringing down a 0 to make 10. - Divide 10 by 6, fitting once again, and note the new remainder, then repeat.3. **Decimal Representation**: Continue dividing until a clear pattern or exact division forms. The result for \(-\frac{7}{6}\) is -1.167, which is a repeating decimal, signaling the division process. Round to desired precision.Remember to maintain the negative sign throughout as the original number was negative. The division transforms a static fraction into a dynamic and useful decimal form essential for deeper mathematical applications.