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Chapter 1: Problem 31

Write decimal notation for each number. $$ \frac{2}{3} $$

Short Answer

Expert verified
0.666... or 0.\bar{6}

Step by step solution

01

- Understand the Fraction

Recognize that the given fraction is \(\frac{2}{3}\).
02

- Perform Division

Divide the numerator (2) by the denominator (3) to convert the fraction into decimal form. \(2 \div 3 = 0.666\text{...}\)
03

- Identify Repeating Decimal

Notice that the digits after the decimal repeat. This fraction can be written in decimal notation as \(\frac{2}{3} = 0.666\text{...} = 0.\bar{6}\).

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Key Concepts

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

Fraction to Decimal
When you want to convert a fraction to a decimal, the process involves dividing the numerator by the denominator. This is a key mathematical concept that helps in understanding how parts of a whole can be represented in different forms. For example, in our exercise, we start with the fraction \(\frac{2}{3}\). Here, 2 is the numerator, and 3 is the denominator. By performing the division operation, we can convert this fraction into its decimal equivalent. Dividing 2 by 3 gives us 0.666..., with the number 6 repeating indefinitely. This simple division method works for any fraction, whether the result is a terminating or repeating decimal.
Repeating Decimal
A repeating decimal, also known as a recurring decimal, is a decimal number that has one or more repeating digits after the decimal point. In our example, \(\frac{2}{3} = 0.666...\), the digit 6 repeats. To indicate a repeating decimal, we place a bar over the repeating digit(s), so \(\frac{2}{3}\) becomes \(0.\bar{6}\). This notation helps us understand that the sequence of 6s continues infinitely without changing. Repeating decimals often arise from fractions where the denominator has factors other than 2 or 5. Identifying and writing repeating decimals correctly ensures clarity in mathematical communication and understanding.
Division Method
The division method is used to convert fractions into decimals. Here's how it works in a step-by-step manner:
  • Step 1: Identify the numerator and denominator in the fraction. Here, for \(\frac{2}{3}\), 2 is the numerator, and 3 is the denominator.
  • Step 2: Perform the division operation going step by step. Divide 2 by 3, which cannot be done directly. So, add a decimal point and zeros to the numerator, making it 2.0 or 20 tenths.
  • Step 3: Divide 20 by 3, which goes 6 times (since 3*6=18). Write down 6 after the decimal point.
  • Step 4: Subtract 18 from 20, leaving a remainder of 2. Bring down another 0, making it 20 again, and repeat the division step creating an ongoing process.
The continuous quotient of 6 shows the repeating nature of the division, giving us the repeating decimal \(0.\bar{6}\). This method is fundamental and widely used in mathematics to understand the relationship between fractions and their equivalent decimal forms.

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Most popular questions from this chapter

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Write an equivalent expression without using grouping symbols. $$ -(9 x+1) $$

Simplify. $$ \frac{7^{2}-(-1)^{5}}{3-2 \cdot 3^{2}+5} $$
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