Problem 59 - 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. $$0 . \overline{3} \quad\square\quad 0.3$$

Step-by-Step Solution

Answer

The correct symbol to use is \(>\), as \(0.\overline{3}\) is greater than \(0.3\). Therefore, the true statement is \(0 . \overline{3} > 0.3\).

Step 1: Understand the decimals

First, it's important to understand the two numbers we're comparing. The first, \(0.\overline{3}\), is shorthand for \(0.3333 \ldots\) to infinity. The second number, \(0.3\), is simply \(0.3\) or \(0.3000 \ldots\) to infinity.

Step 2: Compare the decimals

Next, compare the two numbers. Begin at the tenths place, where the first number has a 3 and the second number also has a 3. Since these numbers are equal, progress to the hundredths place. Here, the first number has a 3 (because the three is recurring), while the second number has a 0. Since 3 is greater than 0, \(0.\overline{3}\) is greater than \(0.3\).

Step 3: Insert the right symbol

Finally, the right symbol to insert between the two numbers is \(>\), because \(0.\overline{3}\) is greater than \(0.3\). So, the true statement is \(0 . \overline{3} > 0.3\).

Key Concepts

Repeating Decimals

When we talk about repeating decimals, we're focusing on numbers that have a set of digits after the decimal point that repeat infinitely. A typical example of a repeating decimal is \(0.\overline{3}\), where the "3" repeats endlessly.
This can also be written as \(0.3333\ldots\). It's crucial to note that these numbers have no end. It keeps going forever, unlike a terminating decimal, which has a finite number of digits after the decimal point. Why is understanding repeating decimals important? Because they can sometimes be frequently encountered in mathematics and everyday situations. Recognizing a repeating decimal helps compare it with other numbers, especially when determining which is larger or smaller. For instance, seeing \(0.\overline{3}\) and understanding it means infinite threes offers a better perspective when making comparisons, contrasting it with numbers like \(0.3\). If you examine them closely, repeating decimals often have more cumulative value due to their never-ending nature.

Place Value

Place value is a foundational concept in understanding and working with decimals. It's the system where the position of a digit in a number determines its value. In the number \(0.3\), the "3" is in the tenths place, meaning it represents three-tenths (or \(\frac{3}{10}\)).
Now consider \(0.\overline{3}\). Here, the repeating "3" also starts in the tenths place. Because it repeats endlessly, it fills subsequent placeholders like hundredths, thousandths, etc., indefinitely. So, in essence, it's equivalent to \(\frac{1}{3}\), a common fraction representation of a repeating decimal. These placeholders allow us to accurately determine the decimal's size and compare it with others. By analyzing the place value, we can confidently state that in a comparison like \(0.\overline{3}\) vs. \(0.3\), \(0.\overline{3}\) is larger. This stems from how it keeps adding value beyond what a terminating decimal can offer.

Decimal Notation

Decimal notation is a way to write numbers that include decimal points, making it easy to display fractions and parts of whole numbers. It is a concise and standardized format to express values between whole numbers.
In our example, knowing what \(0.3\) and \(0.\overline{3}\) mean in decimal notation is essential for comparison. Decimal notation shows that \(0.3\) is a single finite digit after the decimal, while \(0.\overline{3}\) has a repeating string of digits. This representation allows us to visualize and calculate values efficiently. Thanks to decimal notation, mathematicians and students alike can perform operations, comparisons, and even visual explorations of numbers in a consistent and easily understood manner. Moreover, it's a critical part of our number system, enabling clearer communication and better understanding in both academic settings and everyday calculations.