Problem 34 - Elementary and Intermediate Algebra

Question

Write decimal notation for each number. $$ -\frac{2}{9} $$

Step-by-Step Solution

Answer

-0.\overline{2}

Step 1: Identify the Fraction

The given number is \(-\frac{2}{9}\). Recognize this as a fraction that needs to be converted into decimal notation.

Step 2: Understand the Fraction

Note that \(-\frac{2}{9}\) means \(-2 \div 9\). The fraction bar indicates division.

Step 3: Perform the Division

Divide 2 by 9. Using a calculator or long division: \(-2 \div 9 = -0.222\) (repeating). This means the decimal representation is \(-0.222\ldots\).

Step 4: Express as Decimal Notation

Write down the obtained result as \(-0.222\ldots\), showing the repeating sequence. This is denoted as \(-0.\overline{2}\).

Key Concepts

fraction to decimal conversion

Converting fractions to decimals is an essential skill in math. When you see a fraction, like \(-\frac{2}{9}\), think of it as a division problem. In this case, you're dividing -2 by 9. Using a calculator can speed up this process, but knowing how to do it manually is valuable. Thus, \(-\frac{2}{9}\) translates to -2 ÷ 9.
When performing this division, you find the decimal representation of the fraction. Often, the result is a repeating decimal. Understanding this process helps you in both mathematical exercises and real-world applications, like determining proportions.

repeating decimals

A repeating decimal is a decimal number that has digits continuing infinitely in a repeating pattern. For instance, after dividing -2 by 9, you get a decimal sequence of -0.222..., where the digit 2 repeats indefinitely.
This can be expressed as \(-0.\overline{2}\), where the bar over the digit 2 indicates that it repeats endlessly. Recognizing repeating decimals is crucial in mathematics since they frequently appear in various calculations involving fractions. Repeating decimals need special notation to convey their infinite nature clearly.

division

Division is one of the four basic operations in arithmetic. It involves determining how many times one number (the divisor) is contained within another number (the dividend). In the fraction \(-\frac{2}{9}\), -2 is the dividend, and 9 is the divisor. Performing the division \(-2\div 9\) results in -0.222...
Long division can be used for a step-by-step approach, which involves dividing, multiplying, subtracting, and bringing down the next digit sequentially. Understanding division helps in converting fractions to decimals and tackling more complicated problems in math and science.