College Algebra Chapter R: Real Numbers, Properties, and Number Line Distance

Q1. Name the rational numbers from the list below.

Background

Topic: Real Number System

This question tests your understanding of the classification of numbers, specifically identifying rational numbers from a given set.

Key Terms:

• Rational Numbers: Numbers that can be expressed as a fraction $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.

Step-by-Step Guidance

1. Review the list of numbers provided and recall the definition of rational numbers.

2. Identify which numbers can be written as a ratio of two integers (including integers, terminating decimals, and repeating decimals).

3. Exclude any numbers that are non-repeating, non-terminating decimals or cannot be written as a fraction.

Try solving on your own before revealing the answer!

Final Answer:

The rational numbers are those that can be written as fractions or integers from the list.

Q2. Name the irrational numbers from the list below.

Background

Topic: Real Number System

This question tests your ability to identify irrational numbers, which cannot be written as a simple fraction.

Key Terms:

• Irrational Numbers: Numbers that cannot be expressed as a fraction $\frac{a}{b}$, and have non-repeating, non-terminating decimals (e.g., $\sqrt{2}$, $\pi$).

Step-by-Step Guidance

1. Review the list and recall the definition of irrational numbers.

2. Identify numbers that are non-repeating, non-terminating decimals or roots that are not perfect squares.

3. Exclude any numbers that can be written as a fraction or have repeating/terminating decimals.

Try solving on your own before revealing the answer!

Final Answer:

The irrational numbers are those that cannot be written as fractions and have non-repeating, non-terminating decimals.

Q3. Name the real numbers from the list below.

Background

Topic: Real Number System

This question tests your understanding of the broad category of real numbers, which includes both rational and irrational numbers.

Key Terms:

• Real Numbers: All numbers that can be found on the number line, including both rational and irrational numbers.

Step-by-Step Guidance

1. Review the list and recall the definition of real numbers.

2. Include all numbers that are either rational or irrational.

3. Exclude any numbers that are not real (such as imaginary numbers, if present).

Try solving on your own before revealing the answer!

Final Answer:

The real numbers are all numbers from the list that are either rational or irrational.

Q4. Name the property illustrated by the following sentence.

Background

Topic: Properties of Real Numbers

This question tests your knowledge of the basic properties of real numbers, such as commutative, associative, distributive, identity, and inverse properties.

Key Terms:

• Properties of Real Numbers: Rules that apply to arithmetic operations (addition, multiplication, etc.).

Step-by-Step Guidance

1. Read the sentence carefully and identify which property it demonstrates (e.g., changing order, grouping, distributing, etc.).

2. Recall the definitions of the main properties: commutative, associative, distributive, identity, and inverse.

3. Match the sentence to the correct property based on its structure.

Try solving on your own before revealing the answer!

Final Answer:

The property illustrated is one of the basic properties of real numbers (e.g., commutative, associative, etc.).

Q5. Name the property illustrated by the following sentence.

Background

Topic: Properties of Real Numbers

This question is similar to the previous one and tests your ability to recognize another property of real numbers.

Key Terms:

• Properties of Real Numbers: Rules for arithmetic operations.

Step-by-Step Guidance

1. Read the sentence and look for clues about which property is being used (e.g., order, grouping, distributing, etc.).

2. Recall the definitions of the main properties.

3. Identify the property based on the structure of the sentence.

Try solving on your own before revealing the answer!

Final Answer:

The property illustrated is one of the basic properties of real numbers.

Q6. Find the distance between the points on a number line.

Background

Topic: Distance on a Number Line

This question tests your ability to calculate the distance between two points on a number line using the distance formula.

Key Formula:

Distance between points $a$ and $b$:

$d = |a - b|$

Step-by-Step Guidance

1. Identify the coordinates of the two points on the number line.

2. Subtract one coordinate from the other.

3. Take the absolute value of the result to ensure the distance is positive.

Try solving on your own before revealing the answer!

Final Answer:

The distance is the absolute value of the difference between the two points.