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Chapter 5: Problem 18

Write each number as a decimal. $$ -\frac{7}{9} $$

Short Answer

Expert verified
\(-0.\overline{7}\)

Step by step solution

01

Convert Fraction to Decimal

To convert the fraction \(-\frac{7}{9}\) into a decimal, we need to perform the division of \(-7\) by \(9\). This means we divide the numerator \(-7\) by the denominator \(9\).
02

Perform Long Division

Set up the long division process with \(-7\) as the dividend and \(9\) as the divisor. Since \(9\) cannot go into \(-7\), we consider \(-7.000...\), adding zeros to find the quotient. Divide \(-70\) by \(9\), which gives \(-7\) with a remainder of \(7\).
03

Repeat the Division

Continue dividing by bringing down another zero, making it \(-70\) again. Dividing \(-70\) by \(9\) again results in \(-7\), repeating the same pattern with a remainder of \(7\) to bring down the next zero. The decimal repeats as \(-0.777...\).
04

State the Result as a Repeating Decimal

Recognize the repeating pattern in the division; the decimal representation of \(-\frac{7}{9}\) is \(-0.\overline{7}\) where \(\overline{7}\) indicates that \(7\) is a repeating decimal.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Long Division
Long division is a technique used to divide two numbers and can find the answer in decimal form when dividing fractions. When dealing with a fraction like \(-\frac{7}{9}\), we need to convert it into decimal by dividing \(-7\) by \(9\). This involves setting up \(-7.000...\) because you can't divide \(9\) into \(-7\) directly without ending up in negatives. Thus, you need to add zeros after the decimal in \(-7\) to continue the division.
Here's a simple step-by-step to follow:
  • Identify the dividend (number to be divided) and divisor (number you're dividing by).
  • Set up the dividend beneath the divisor.
  • Proceed with division. If you encounter a smaller number like \(-7\) versus \(9\), add decimal points and zeros.
  • Keep writing zeros and repeating the division calculation until you see a repeating pattern or reach a zero remainder.
Long division takes time to master, but it becomes intuitive with practice. For fractions giving us decimals, it's a crucial skill.
Repeating Decimals
Repeating decimals occur when, during division, a specific sequence of digits starts repeating indefinitely. An example is what happens with \(-\frac{7}{9}\). You'll notice that as you divide, the digit sequence \(7\) consistently appears. This is why, after performing the long division, the result looks like \(-0.777...\).
Let's clarify repeating decimals:
  • Any decimal that does not end (terminate) but shows a pattern or a series of repeating numbers is a repeating decimal.
  • Such decimals are noted with a bar over the repeating part, called a vinculum, \(-0.\overline{7}\) in this case.
  • Repeating decimals can often represent rational numbers and are a result of dividing integers.
Recognizing and representing repeating decimals helps in understanding the nature of some fractional results in decimal form.
Negative Decimals
Dealing with negative decimals involves understanding how the minus sign in a fraction translates. When we convert a negative fraction such as \(-\frac{7}{9}\) to a decimal, the result is negative as well, which stands as \(-0.\overline{7}\).
Here's how to think about it:
  • Negative decimals occur when the division involves a negative number, which can be in the numerator or divisor.
  • Maintain the negative sign throughout the division process to ensure accuracy in your final result.
  • Negative decimals behave similarly to positive ones; however, they are located on the left side of the number line (less than zero).
Don't let the negative sign intimidate you. It's crucial to keep track of signs for accuracy in math problems.

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Most popular questions from this chapter

Divide. \(0.54 \div 12\)

In 2016 , American farmers received an average of \(\$ 84.20\) per hundred pounds of chicken. What was the average price per pound for chickens? Round to the nearest cent.

Perform the indicated operation. $$ \begin{array}{r} 1000 \\ -\quad 95.71 \\ \hline \end{array} $$

Divide. Round the quotients as indicated. Divide: \(98.83 \div 3.5 .\) Round the quotient to the nearest tenth.

Divide. \(\frac{4.615}{0.071}\)
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