Problem 13
Question
Fill in the blanks. Graph each irrational number on the number line: \(\\{\pi, e, \sqrt{2}\\}\)
Step-by-Step Solution
Verified Answer
Plot \\(\sqrt{2}\\) near 1.41, \\(e\\) near 2.72, and \\(\pi\\) near 3.14 on the number line.
1Step 1: Understanding the Irrational Numbers
First, we identify each irrational number and approximate its decimal value. We have three numbers: 1. \(\pi\) or Pi is approximately 3.142. \(e\) or Euler's number is approximately 2.723. \(\sqrt{2}\) or the square root of 2 is approximately 1.41.
2Step 2: Draw the Number Line
Draw a horizontal line and label points at intervals for integers, such as 0, 1, 2, 3, 4, etc. This will serve as your reference to place each irrational number in the correct position.
3Step 3: Locate and Mark Approximate Position
Now, locate and mark approximate positions for each irrational number on the number line:1. \(\sqrt{2}\) is approximately 1.41, so it will be slightly to the right of 1.2. \(e\) is approximately 2.72, so it will be a little to the right of 2, nearly 3.3. \(\pi\) is approximately 3.14, so it will be just a bit to the right of 3.
Key Concepts
Understanding the Number LineDecimal Approximation of Irrational NumbersGraphing Irrational Numbers
Understanding the Number Line
A number line is a straight, horizontal line that is used as a visual aid to represent numbers. It helps us understand the order and distance between numbers by placing them at specific points along the line.
On a number line, positive numbers are placed to the right of zero, while negative numbers are to the left. Each mark on the number line represents an integer, and the space between these marks can be divided into smaller intervals to represent fractions or decimals.
When dealing with irrational numbers such as \(\pi\), \(e\), and \(\sqrt{2}\), which cannot be expressed as simple fractions, the number line becomes essential for estimating their placement. Even though these numbers have non-repeating, non-terminating decimal parts, we approximate their values to graph them accurately on the number line.
On a number line, positive numbers are placed to the right of zero, while negative numbers are to the left. Each mark on the number line represents an integer, and the space between these marks can be divided into smaller intervals to represent fractions or decimals.
When dealing with irrational numbers such as \(\pi\), \(e\), and \(\sqrt{2}\), which cannot be expressed as simple fractions, the number line becomes essential for estimating their placement. Even though these numbers have non-repeating, non-terminating decimal parts, we approximate their values to graph them accurately on the number line.
Decimal Approximation of Irrational Numbers
Decimal approximation is a crucial tool when working with irrational numbers. These numbers do not have exact decimal equivalents because their decimal forms go on forever without repeating. Therefore, we use decimal approximation to estimate their values for practical purposes.
Taking the example of \(\pi\), we often use its approximation, \(\pi \approx 3.142\), to simplify calculations. The number \(e\), another irrational number, is roughly equal to 2.723, and \(\sqrt{2}\) is about 1.41. Knowing these approximations allows us to work with these numbers easily when placing them on a number line.
With these approximations, it becomes much easier to make sense of these numbers' positions in relation to the integers we commonly use.
Taking the example of \(\pi\), we often use its approximation, \(\pi \approx 3.142\), to simplify calculations. The number \(e\), another irrational number, is roughly equal to 2.723, and \(\sqrt{2}\) is about 1.41. Knowing these approximations allows us to work with these numbers easily when placing them on a number line.
- \(\pi\): Approximately 3.142
- \(e\): Approximately 2.723
- \(\sqrt{2}\): Approximately 1.41
Graphing Irrational Numbers
Graphing irrational numbers on a number line involves placing these numbers in their approximate positions relative to known integers. This visualization helps us see where these numbers lie and understand their size and magnitude compared to whole numbers.
To graph \(\pi\), which is approximately 3.14, you would place it just slightly to the right of the 3 on the number line. Similarly, \(e\), which approximates to 2.72, would be positioned just past the 2, closer to the 3 but not quite there.
Finally, \(\sqrt{2}\) approximates to 1.41, meaning it would find its place slightly to the right of 1. This careful placement of irrational numbers on the number line helps in understanding their relationship with more familiar numbers and in performing more advanced calculations.
To graph \(\pi\), which is approximately 3.14, you would place it just slightly to the right of the 3 on the number line. Similarly, \(e\), which approximates to 2.72, would be positioned just past the 2, closer to the 3 but not quite there.
Finally, \(\sqrt{2}\) approximates to 1.41, meaning it would find its place slightly to the right of 1. This careful placement of irrational numbers on the number line helps in understanding their relationship with more familiar numbers and in performing more advanced calculations.
Other exercises in this chapter
Problem 13
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Fill in the blanks. The inverse of an exponential function is called a ____ functions.
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