Problem 21 - Elementary and Intermediate Algebra

Question

Graph each rational number on the number line. $$ -4.3 $$

Step-by-Step Solution

Answer

-4.3 is slightly to the left of -4 on the number line.

Step 1: Understand the Given Rational Number

The given rational number is -4.3. This represents a value that is slightly less than -4 on the number line.

Step 2: Identify the Position

To graph -4.3 on the number line, understand that -4.3 is between -4 and -5. Since it's a decimal, it will be closer to -4.

Step 3: Sketch the Number Line

Draw a number line and mark integers. Label points starting from -5 to 0, making sure each point is evenly spaced.

Step 4: Locate -4.3 on the Number Line

Identify the segment between -4 and -5. Since -4.3 is closer to -4, place a point slightly to the left of -4 and label it as -4.3.

Key Concepts

Graphing Rational Numbers on a Number Line

Graphing rational numbers on a number line is a fundamental skill in mathematics. It helps to visualize where numbers lie in relation to each other. To begin, always comprehend the given rational number. In our example, we are working with -4.3.

Understanding the number's position is crucial. Since -4.3 is negative and has a decimal part, it lies between two whole numbers, specifically -4 and -5. The decimal part (.3) indicates that it’s closer to -4.

Here’s a simple way to graph this on a number line:

• Draw a number line and mark the integers, starting from -5 up to 0.
• Now, locate the segment between -4 and -5.
• The number -4.3 lies a bit to the left of -4 since it’s closer to it than to -5.

By following these steps, you can accurately place any rational number on a number line.

Understanding Rational Numbers

Rational numbers are numbers that can be expressed as a fraction of two integers, where the denominator is not zero. For example, -4.3 can be written as \(-43/10\).

Rational numbers include:

• Whole numbers (e.g., 0, 1, 2, 3)
• Fractions (e.g., 1/2, -3/4)
• Decimals that terminate or repeat (e.g., 0.5, -0.333...)

Unlike irrational numbers, rational numbers have a clear, repeating pattern or a finite end. This makes them easier to work with in mathematics.
Understanding and identifying rational numbers is key to mastering more complex math topics.

Decimal Representation of Rational Numbers

Decimals are a way to represent fractions or parts of a whole. For rational numbers, decimals can either terminate (end) or repeat.

In our example, -4.3 is a terminating decimal, meaning it ends at one decimal place. Terminating decimals are straightforward and easy to locate on a number line.

Repeating decimals, on the other hand, have a digit or group of digits that repeat infinitely. For instance, 1/3 is 0.333... This endless repetition can be represented with a bar notation (e.g., \(\bar{0.3}\)).

To convert a repeating decimal to a fraction, recognize the pattern and write an equation to solve. For example:

• Let x = 0.666...
• Multiply by 10: 10x = 6.666...
• Subtract: 10x - x = 6.666... - 0.666...
• Solve for x: 9x = 6
• Thus, x = 6/9, which simplifies to 2/3.

Understanding decimal representation helps in graphing and working with rational numbers more efficiently.