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Chapter 5: Problem 63

For Exercises \(59-66,\) insert the appropriate symbol. Choose from \(<,>,\) or \(=\) $$\frac{1}{3} \square 0.3$$

Short Answer

Expert verified
\( \frac{1}{3} > 0.3 \)

Step by step solution

01

- Convert the Fraction to Decimal

Convert the fraction \(\frac{1}{3}\) into a decimal. \(\frac{1}{3} = 0.3333...\) which is a repeating decimal.
02

- Compare Decimals

Compare the decimal equivalent of \(\frac{1}{3}\), which is 0.3333..., with the given number 0.3.
03

- Determine the Appropriate Symbol

Since 0.3333... is greater than 0.3, the appropriate symbol to use is \(>\).

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Key Concepts

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

fractions to decimals
Understanding how to convert fractions to decimals is a crucial skill in mathematics. To convert a fraction to a decimal, you need to divide the numerator (top number) by the denominator (bottom number).
For example, to convert \(\frac{1}{3}\) to a decimal, you divide 1 by 3. This division is not straightforward because it results in a repeating pattern. When you perform long division, 1 divided by 3 equals 0.3333... The '3' after the decimal point repeats indefinitely, which we write as 0.3 with a line over the 3 to indicate that it repeats.

Key points to remember:
  • Divide the numerator by the denominator.
  • A repeating pattern may emerge in the result.
  • Use a line over repeating digits to indicate they continue endlessly.
repeating decimals
Repeating decimals are decimals that have one or more digits that repeat infinitely. When working with repeating decimals, it鈥檚 important to recognize and properly denote them.
For example, \(\frac{1}{3}\) converts to 0.3333... and because the digit '3' repeats endlessly, it is written as 0.3 with a horizontal line above the 3. This line is called a 'vinculum'.

To compare or work with repeating decimals, it's helpful to know this notation. Bullet points include:
  • A repeating decimal repeats a digit or group of digits forever.
  • Use a line above the repeating part to show it continues endlessly.
  • Recognize and convert fractions that result in repeating decimals.
comparing decimals
Comparing decimals involves looking at the value of each digit from left to right. It's like comparing numbers, but instead of whole numbers, you are comparing decimal values.
Take \(\frac{1}{3}\) which converts to 0.3333... and compare it to 0.3:
  • First, look at the digit in the tenths place. Both numbers have a 3.
  • Next, look at the digits in the hundredths place; 0.3333... has another '3' while 0.3 has a '0' (considering additional zero placeholders).
  • Since 3 > 0 in the hundredths place, 0.3333... is greater than 0.3.
So, for \(\frac{1}{3} \) compared to 0.3, the correct symbol is \(>\). Understanding this concept helps you determine which of two decimals is larger or if they are equal.

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