Problem 48 - Intermediate Algebra

Question

Graph each set on a number line. $$ \left\\{2 \frac{1}{9},-3.821134 \ldots,-\frac{\pi}{2}, \sqrt{15},-0.9, \frac{\sqrt{2}}{2}\right\\} $$

Step-by-Step Solution

Answer

Plot each number on a number line based on their order: \(-3.821134, -\frac{\pi}{2}, -0.9, \frac{\sqrt{2}}{2}, 2\frac{1}{9}, \sqrt{15}\).

Step 1: Analyze Each Number

We start by identifying and analyzing each number in the set. Convert them to approximate decimal values if necessary:- \(2\frac{1}{9}\) is approximately \(2.11\).- \(-3.821134\) remains as it is.- \(-\frac{\pi}{2}\) is approximately \(-1.57\).- \(\sqrt{15}\) is approximately \(3.87\).- \(-0.9\) remains as it is.- \(\frac{\sqrt{2}}{2}\) is approximately \(0.707\).

Step 2: Determine the Order

Arrange the numbers in ascending order based on their values:1. \(-3.821134\)2. \(-\frac{\pi}{2} \approx -1.57\)3. \(-0.9\)4. \(\frac{\sqrt{2}}{2} \approx 0.707\)5. \(2\frac{1}{9} \approx 2.11\)6. \(\sqrt{15} \approx 3.87\)

Step 3: Position on the Number Line

Draw a number line marking positions for each number based on their approximate values:- Place a mark for \(-3.821134\) between \(-4\) and \(-3\).- Place \(-1.57\) for \(-\frac{\pi}{2}\) slightly larger than \(-2\).- Place \(-0.9\) between \(-1\) and 0.- Place \(0.707\) for \(\frac{\sqrt{2}}{2}\) between 0 and 1.- Place \(2.11\) for \(2\frac{1}{9}\) just above 2.- Place \(3.87\) for \(\sqrt{15}\) just below 4.

Step 4: Verify Positions

Ensure that the spacing on the number line is consistent and that each number is correctly represented based on its ordering.

Key Concepts

Ordering Numbers

Ordering numbers involves arranging them from the smallest to the largest. This can include a mix of whole numbers, fractions, decimals, and even irrational numbers. To achieve this, each number must be accurately evaluated and compared. You begin by converting numbers to a uniform format, often as decimal approximations. Once converted, the process is straightforward: you compare their sizes.

• You compare the decimal parts only after checking the integer parts when converted numbers have both parts.
• Negative numbers are always less than positive numbers.
• The more negative a number, the smaller it is on the number line.

In our example, the numbers were arranged as: - -3.821134 - -1.57 - -0.9 - 0.707 - 2.11 - 3.87. Double-checking their order ensures accuracy before graphing.

Decimal Approximation

Decimal approximation makes it easier to handle and compare numbers, especially when dealing with irrational numbers. It involves expressing numbers to a convenient decimal form without drastically altering their value. Approximating is essential in graphs because it aligns differences among numbers more clearly and understandably.

• For example, \(\frac{1}{9} \u2026\) is converted to \(0.11\u2026\).
• \(\frac{\pi}{2}\u2026\) becomes approximately -1.57.
• Using square roots like \(\sqrt{15}\) is approximately 3.87.

Using decimal approximation aids in getting a clearer position on the number line. It ensures each number lands at the correct spot, preventing overlaps or misplacements,
especially when the exact position is difficult to perceive with unusual number formats such as fractions or symbols.

Irrational Numbers

Irrational numbers cannot be expressed as a simple fraction. Their decimal representation is non-repeating and non-terminating, which means it goes on forever without forming a clear pattern. Examples include \(\pi\) and \(\sqrt{2}\).

• An irrational number like -\(\frac{\pi}{2}\) is approximately -1.57.
• \(\frac{\sqrt{2}}{2}\) has a decimal that starts with 0.707...

Understanding these helps when plotting because it allows us to know precisely where to place them on a number line using their decimal forms. Even though they are approximations, they are valuable in depicting their relative size compared to rational numbers.
In exercises that involve graphing, such as placing numbers on a number line, careful approximation ensures they are correctly represented.

Negative Numbers

Negative numbers are placed on the number line to the left of zero. Their order is from the most negative to zero because larger negative numbers have smaller values.

• A larger absolute value signifies a smaller negative number, such as -3.821134 being more negative than -0.9.
• Negative numbers are crucial in showing not just scale, but orientation, indicating values less than zero.

They help us understand real-life quantities that may be less than nothing, like debts or temperatures below zero. While positive numbers exhibit growth or increase, negative values reflect reduction or decrease.
In the exercise, understanding negative numbers aids in positioning them accurately on the number line in comparison to positive numbers and other less negative values.