Problem 57 Graph each set of numbers on a n... [FREE SOLUTION]

Chapter 1: Problem 57

Graph each set of numbers on a number line. $$ \left\\{-\pi, 4.25,-1 \frac{1}{2},-0.333 \ldots, \sqrt{2},-\frac{35}{8}, 3\right\\} $$

Short Answer

Expert verified

Plot the numbers on the number line at their approximate locations.

Step by step solution

Understand the Set of Numbers

We have a set of numbers $ \{-\pi, 4.25, -1 \frac{1}{2}, -0.333\ldots, \sqrt{2}, -\frac{35}{8}, 3\} $. Some are integers, others are fractions, decimals, or irrational numbers. We need to represent them all on a number line.

Approximate Non-Integer Values

Approximate the values: $-\pi \approx -3.14$, $4.25$ remains the same, $-1 \frac{1}{2} = -1.5$, $-0.333 \ldots$ is about $-1/3$, $\sqrt{2} \approx 1.414$, $-\frac{35}{8} = -4.375$.

Layout a Number Line

Draw a number line with enough range to represent all numbers from approximately $-4.5$ to $4.5$, since our largest number $4.25$ is close to $5$ and the smallest $-4.375$ is less than $-4$.

Plot Each Number on the Line

Mark each number on the line at the appropriate approximate point:- $-\pi $ at about $-3.14$.- $4.25$.- $-1.5$.- $-0.333\ldots $ near $-0.3$.- $\sqrt{2} $ at about $1.414$.- $-\frac{35}{8} $ clearly placed at $-4.375$.- $3$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction. They are the kind of numbers that include endless, non-repeating decimals. For example, while rational numbers like $ \frac{1}{2} $ can be nicely written as 0.5, irrational numbers such as $ \pi $ or $ \sqrt{2} $ do not have precise decimal forms.
The representation of irrational numbers includes:

Pi ($ \pi $): Approximately 3.14159, but it continues infinitely without repeating.
Square root of 2 ($ \sqrt{2} $): Again, it does not end or repeat, with a decimal approximation around 1.414.

When plotting these numbers on a number line, we rely on their decimal approximations, which brings us naturally to the next concept: approximation.

The Art of Approximation

To plot irrational or complex fractional numbers on a number line, we first need to find their approximate decimal values. This process, known as approximation, allows us to place these numbers accurately in a visual format.
Approximation tips:

For irrational numbers like $ \pi $, use its commonly accepted approximate value, like -3.14 when negative.
Decimals like $ -0.333\ldots $ can be approximated as -0.33 or -1/3, depending on the precision needed.
For fractions like $ -\frac{35}{8} $, perform division to convert them into a decimal, giving approximately -4.375.

These approximations guide us to place numbers accurately, bridging the gap between raw mathematical values and human-friendly visualization.

Plotting Numbers on the Number Line

Plotting numbers on a number line is an essential skill in visualizing numerical relationships. Once you have approximated the numbers, the next step is to comfortably fit them within a drawn number line.
Steps to plot:

First, sketch a horizontal line with incremented markers that cover the expected range of your numbers. For the numbers in our set, ensure your number line runs from approximately -4.5 to 4.5.
Mark each number based on its approximate value:
- -$ \pi $: at about -3.14, slightly to the left of -3.
- 4.25: just beyond 4.
- -1.5: precisely between -1 and -2.
- -0.333\ldots: close to -0.3.
- $ \sqrt{2} $: about 1.414, a little past 1.
- -$ -\frac{35}{8} $: place it at -4.375, between -4 and -5.
- 3: directly on the 3 mark.

These steps ensure that each number is accurately visualized in respective positions on the number line, giving you a clear snapshot of their relative values.

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91影视

Graph each set of numbers on a number line. $$ \left\\{-\pi, 4.25,-1 \frac{1}{2},-0.333 \ldots, \sqrt{2},-\frac{35}{8}, 3\right\\} $$

Short Answer

Step by step solution

Understand the Set of Numbers

Approximate Non-Integer Values

Layout a Number Line

Plot Each Number on the Line

Key Concepts

Understanding Irrational Numbers

The Art of Approximation

Plotting Numbers on the Number Line

One App. One Place for Learning.

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