Problem 60 - Introductory and Intermediate Algebra for College Students 4th

Question

Insert either \(<\) or \(>\) in the shaded area between each pair of numbers to make a true statement. $$0.6\quad\square\quad0 . \overline{6}$$

Step-by-Step Solution

Answer

The correct comparison is 0.6 \(<\) \(0 . \overline{6}\)

Step 1: Analyzing Decimal Number 0.6

The number 0.6 has only one digit after the decimal point. Therefore, it is less than numbers with any digit greater than 6 at the hundredths place or beyond.

Step 2: Analyzing the Repeating Decimal \(0 . \overline{6}\)

This number is essentially the same as 0.6666 repeated indefinitely. The repetition begins at the tenths place, making this number larger than 0.6. Remember that an overline on a decimal digit indicates that digit is recurring.

Step 3: Comparing the Numbers

Now that we have analyzed both numbers and understood that \(0 . \overline{6}\) is greater than 0.6, the correct symbol to insert in the shaded area is the 'greater than' symbol \(>\)

Key Concepts

Repeating Decimals

Repeating decimals are a type of decimal number where one or more digits repeat infinitely. This kind of decimal is often represented with a line, called a vinculum, over the repeating digits. For instance, the number \(0.\overline{6}\) indicates that the digit 6 repeats infinitely, making it equal to \(0.6666...\), continuing without end.
Understanding repeating decimals is crucial as they often arise from divisions that do not resolve into a terminating decimal. When encountering repeating decimals:

• Recognize the pattern: Identify which digits repeat and denote them with an overline.
• Convert to a fraction: Sometimes, it's easier to deal with fractions when comparing or computing with repeating decimals.

Repeating decimals can appear strange at first, but their infinite precision is a valuable concept in understanding the behavior of real numbers.

Decimal Numbers

Decimal numbers are numbers that contain a decimal point, which separates the whole part from the fractional part. Each place value after the decimal point represents a fractional power of ten, such as tenths, hundredths, thousandths, etc.
Here’s what you should know about decimal numbers:

• Tenths, Hundredths, Thousandths: These are the positions that come after the decimal point, where each step to the right divides by ten.
• Precise Representation: Decimals allow more precise measurements than whole numbers since they can denote very small amounts.
• Comparison: When comparing decimal numbers, like \(0.6\) and \(0.66\), begin from the left and look for the first difference.

Decimals are essential in daily life and science, offering clarity and precision in financial calculations, measurements, and data interpretation.

Inequalities in Mathematics

Inequalities in mathematics describe a relationship between two values when they are not equal. They are expressed using symbols like \(<\), \(>\), \(\leq\), and \(\geq\).
Understanding inequalities is crucial because they help us compare different quantities and express conditions:

• Greater Than (>): Indicates the first number is larger than the second number.
• Less Than (<): Indicates the first number is smaller than the second number.
• Applications: Inequalities are used in various fields for problem-solving, such as comparing prices, lengths, and quantities.

A classic application is in comparing decimal numbers, just as in the exercise where \(0.6\) is compared to \(0.\overline{6}\), leading us to conclude that \(0.\overline{6}\) is indeed greater than \(0.6\). Approaching inequalities with clarity can aid in not only solving equations but also in understanding data and trends across different situations.