Problem 56 - Algebra 2

Question

Name the sets of numbers to which all of the following numbers belong. Then arrange the numbers in order from least to greatest. $$2.49,2.4 \overline{9}, 2.4,2.49,2 . \overline{9}$$

Step-by-Step Solution

Answer

The numbers belong to rational sets and are ordered: 2.4, 2.49, 2.49, 2.49\overline{9}, 2.\overline{9}.

Step 1: Identifying the Type of Real Numbers

First, we identify the types of numbers. \(2.49\) is a terminating decimal.\(2.49\) overline (repeating) is a repeating decimal, represented as \(2.49999...\). \(2.4\) is a terminating decimal.\(2.49\) is a repeating decimal. \(2.9\) overline means it's a repeating decimal equal to \(3\). All these numbers are rational because they can be expressed as fractions.

Step 2: Converting Repeating Decimals to Fractions

Convert repeating decimals to fractions to simplify comparison. The number \(2.49 \overline{9}\) is approximately \(2rac{49}{99}\) when converted. The number \(2.\overline{9}\) equals \(3\) because repeating nines after two implies the number is effectively rounded up to the next integer.

Step 3: Comparing Decimal Forms

Line up decimal places to compare accurately:- \(2.49\) is already a decimal- \(2.49 \overline{9}\) can be seen as 2.499999...- \(2.4\) is effectively \(2.40\)- \(2.49\) is already a decimal- \(2.\overline{9}\) is \(3.00\) when compared.

Step 4: Arranging from Least to Greatest

Order the numbers based on decimal comparison:1. \(2.4\)2. \(2.49\)3. \(2.49\)4. \(2.49 \overline{9}\)5. \(2.\overline{9}\)

Key Concepts

Decimal Representation

When we talk about decimal representation, we consider the different forms numbers take when they are written with a decimal point. A rational number can be expressed as either a terminating or a repeating decimal.

• **Terminating Decimals**: These decimals have a finite number of digits after the decimal point. For example, the number \(2.49\) ends after the second decimal place.
• **Repeating Decimals**: These numbers have one or more repeating digits after the decimal point. The notation \(2.4\overline{9}\) means the digits "49" repeat indefinitely: i.e., \(2.49999...\).

Understanding these forms is essential because they help us in recognizing patterns and performing calculations or conversions more easily.

Number Comparison

Comparing numbers with decimal points can be a bit tricky, but it's straightforward when you align the decimal places. Here's how you do it:

• **Identify the Decimal Forms**: Start by writing out each number with the same number of decimal places. For instance, compare \(2.40\) to \(2.49\) by adjusting \(2.4\) to \(2.40\).
• **Line Up the Decimals**: Arrange the numbers vertically, lining up their decimal points to help you see which is larger or smaller. Comparing decimal numbers is similar to comparing different values digit by digit.

Through this clear comparison method, it becomes easier to establish which number is greater, less than, or equal in value.

Converting Repeating Decimals

To convert a repeating decimal into a fraction, use a neat little trick involving algebra. Let's take \(2.4\overline{9}\) as an example, which converts to a fraction like so:

• **Setup an Equation**: Let \(x = 2.4999...\).
• **Multiply**: Choose a power of 10 to multiply both sides, here \(10x = 24.999...\).
• **Subtract**: Subtraction gives \(10x - x = 24.999... - 2.4999...\), simplifying to \(9x = 22.5\).
• **Solve**: Finally, solve for \(x\) by dividing both sides by 9, which leads to \(x = \frac{225}{90}\), simplifying to \(\frac{25}{11}\).

This fraction form is useful for precise calculations and understanding the value inherent in a repeating decimal.

Ordering Rational Numbers

Once you have converted and compared the decimals, ordering rational numbers becomes much easier. Simply follow these steps:

• **Convert to Uniform Representation**: Make sure each number is in a comparable decimal or fractional form. This ensures there’s no ambiguity when comparing.
• **Put in Sequence**: Start with the smallest and move to the largest. With our example, begin with the smallest decimal, \(2.4\), and continue until you've reached the highest value, \(2.\overline{9}\).
• **Validate Ordering**: Check that each step accurately follows the comparison logic you initially applied.

Proper ordering allows for precision, whether you're performing mathematical operations or analyzing data sets. It's an essential skill that refines your number sense in significant ways.