Problem 36 - Precalculus

Question

Say whether the statement is TRUE or FALSE. (In Exercises \(37-40\), do not use a calculator or table; use instead the approximations \(\sqrt{2} \approx 1.4 \text { and } \pi \approx 3.1 .)\) $$0 . \overline{7}>0.7$$

Step-by-Step Solution

Answer

TRUE

Step 1: Understand the Problem

We need to determine whether the repeating decimal \(0.\overline{7}\) is greater than \(0.7\). The bar above the "7" indicates that the digit repeats indefinitely, meaning the number is \(0.7777...\) (with an infinite sequence of 7s).

Step 2: Compare the Two Numbers

The number \(0.7\) is equivalent to \(0.7000...\), where zeros extend indefinitely. Comparing \(0.7777...\) to \(0.7000...\), we can see that each digit in \(0.7777...\) is greater than the corresponding digit in \(0.7000...\).

Step 3: Conclusion

Since every digit of the repeating number \(0.7777...\) is greater than \(0.7000...\), it follows that \(0.\overline{7} > 0.7\). The statement given in the exercise is TRUE.

Key Concepts

Repeating Decimals

A repeating decimal is a decimal number in which a digit or group of digits repeats infinitely. This repeating section is denoted by placing a bar above the repeating digit(s). For example, in the number \(0.\overline{7}\), the digit "7" repeats endlessly, creating the number sequence \(0.7777...\) that never stops.
Understanding repeating decimals is crucial because they represent fractions as infinite decimal expansions. In many cases, these decimals represent rational numbers, such as fractions where both the numerator and the denominator are integers.
By recognizing repeating patterns, we can easily convert these decimals into their exact fractional equivalents. For instance, \(0.\overline{7}\) can be expressed as the fraction \(\frac{7}{9}\). This conversion can often facilitate easier arithmetic calculations and deeper numerical insights.

• Repeating decimals indicate infinite repetition.
• They often denote rational numbers.
• Understanding them helps in accurate numerical operations.

Number Approximation

Number approximation is the method used to round or estimate a number to make calculations simpler and more intuitive. Using approximations enables you to perform mental math more easily without needing a calculator. In mathematics, certain values are approximated for simplicity, such as \(\pi \approx 3.14\) or \(\sqrt{2} \approx 1.4\).
These approximations are valuable in daily life and various academic tasks where precise figures are not necessary. For instance, when asked to evaluate an inequality like \(0.\overline{7} > 0.7\), understanding the approximation helps us quickly compare and deduce the relationship between numbers.

• Facilitates quicker calculations.
• Often used in practical scenarios for simplicity.
• Helps in making judgments about the size and order of numbers.

Inequality Statements

Inequality statements are expressions that show the relationship between two values, indicating if one is larger, smaller, or equal to another. When comparing decimals, it’s essential to align them properly and compare digit by digit from the left. The inequality \(0.\overline{7} > 0.7\) shows that the repeating decimal is greater than the non-repeating decimal.
Understanding inequality statements involves not just identifying symbols like \(>\), \(<\), and \(=\), but also thoroughly comparing number values. In mathematics, this comparison might involve converting repeating decimals to fractions or aligning decimals accurately.

• Symbols indicate relationships between numbers.
• Requires precise digit-by-digit comparison.
• Helps in understanding the magnitude and relationships of numbers.