Problem 15 - Algebra and Trigonometry

Question

\(9-16\) State whether each inequality is true or false. $$ 1.1>1 . \overline{1} $$

Step-by-Step Solution

Answer

The inequality is false.

Step 1: Understand the Inequality

We need to determine if the statement \( 1.1 > 1 . \overline{1} \) is true or false. The number \( 1 . \overline{1} \) represents a repeating decimal, meaning it is \( 1.1111\ldots \) where the digit '1' repeats indefinitely.

Step 2: Compare the Numbers

To compare \( 1.1 \) and \( 1 . \overline{1} \), first observe the first decimal place. For \( 1.1 \), the first decimal place is '1', and for \( 1 . \overline{1} \), it is also '1'.However, in the second decimal place, \( 1 . \overline{1} \) has another '1', whereas \( 1.1 \) has an implicit '0' since no other digit is shown beyond the first decimal place. Thus, numerically, \( 1.1 = 1.10 \), which is less than \( 1.111... \).

Step 3: Conclusion

Since \( 1.1 = 1.10 \) and \( 1 . \overline{1} = 1.1111\ldots \), we see more repeating decimals in \( 1 . \overline{1} \), making \( 1 . \overline{1} \) larger than \( 1.1 \). Therefore, the inequality \( 1.1 > 1 . \overline{1} \) is false.

Key Concepts

Repeating Decimals

Repeating decimals are numbers that have digits repeating indefinitely. They are often talked about in the context of fractions because many fractions, when turned into decimals, create a pattern that repeats forever. For example, the fraction \( \frac{1}{3} \) equals \( 0.3333\ldots \), which is the repeating decimal \( 0.\overline{3} \). Another example is \( 1.\overline{1} \), which represents \( 1.1111\ldots \), meaning the digit '1' repeats indefinitely. Understanding repeating decimals is crucial in mathematics as they help in estimating values and making comparisons. These decimals can appear intimidating at first, but their core idea is simply that a certain sequence of digits go on forever without any change. To identify a repeating decimal, look for a digit or a group of digits that continue in a repetitive sequence. We express them using a bar over the repeating part, like \( 0.\overline{4} = 0.4444\ldots \). Recognizing this will aid in solving problems involving decimal comparisons and inequalities.

Decimal Comparison

Comparing decimals involves evaluating their place values, much like comparing whole numbers. Let's dive into how we deal with decimal comparisons.

• Start by looking at the digits in the same place value, beginning with the digits immediately after the decimal point.
• If two decimals share the same digit in a place, move to the next decimal place. The decimal with the larger digit at the first differing place value is greater.
• For decimals that stop, while others repeat, like \(1.1\) and \(1.\overline{1}\), assume the stopping decimal has zeros filling the rest of the places.

In our exercise, comparing \(1.1\) and \(1.\overline{1}\), we begin with the first decimal place. Both numbers have a '1' after the point, but to compare fully, note that \(1.1\) is equivalent to \(1.10\). Moving to the next decimal place, \(1.\overline{1} = 1.1111\ldots\) clearly is greater since it has another '1' where \(1.10\) finishes off. This simple technique helps us determine their order in value.

Inequality Analysis

Inequality analysis involves determining the relationship between two numbers, establishing whether one is greater, lesser, or equal to the other. This concept extends naturally from comparing numbers. Here, inequalities are particularly helpful:

• "Greater than" (>) and "less than" (<) signs help us set up and analyze relationships.
• When numbers appear close in value, like our case with \(1.1\) and \(1.\overline{1}\), every decimal place must be considered.
• Always ensure to fully expand repeating decimals to accurately compare.

In our example, the inequality \(1.1 > 1.\overline{1}\) was set up to check their relation. Through analysis, we see \(1.10 < 1.1111\ldots\), showing that \(1.1 > 1.\overline{1}\) is false. Understanding and analyzing inequalities like this helps in both mathematical and real-world problem-solving by allowing precise comparisons.