Book cover for Precalculus: A Right Triangle Approach

Precalculus: A Right Triangle Approach

Judith A. Beecher, Marvin L. Bittinger, David J. Ellenbogen, Judith A. Penna

ISBN #9780321783967

5th Edition

5,905 Questions

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25,202 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section provides foundational knowledge of the real-number system by classifying numbers into various types such as natural, whole, integers, rational, and irrational numbers. It emphasizes the importance of interval notation for expressing number sets, reviews the core properties of real numbers used to manipulate algebraic expressions, and explains the concept of absolute value as the distance from zero, which underpins many real-world applications.

Learning Objectives

1

Identify and classify various types of real numbers (natural numbers, whole numbers, integers, rational numbers, and irrational numbers).

2

Use interval notation to represent sets of real numbers and accurately graph them on a number line.

3

Explain and apply key properties of real numbers including commutative, associative, distributive, and identity properties.

4

Determine and interpret the absolute value of real numbers and calculate distances between points on the number line.

Key Concepts

CONCEPT

DEFINITION

Natural Numbers

The set of positive integers used for counting (e.g., 1, 2, 3, ...).

Whole Numbers

The set of natural numbers including zero (e.g., 0, 1, 2, 3, ...).

Integers

All whole numbers and their opposites, including negative numbers (e.g., ..., -3, -2, -1, 0, 1, 2, 3, ...).

Rational Numbers

Numbers that can be expressed as a fraction p/q, where p and q are integers and q ≠ 0; their decimal expansions terminate or repeat.

Irrational Numbers

Numbers that cannot be written as a simple fraction; their decimal forms are non-terminating and non-repeating (e.g., √2, π).

Real Numbers

The set of all rational and irrational numbers combined, often represented by points on a number line.

Interval Notation

A concise way to denote subsets of real numbers using parentheses for excluding endpoints and brackets for including endpoints (e.g., (a, b), [a, b]).

Absolute Value

The distance of a number from zero on the number line, always nonnegative; denoted by |a|.

Example Problems

Example 1

In Exercises 1-10, consider the numbers $$ \begin{array}{l} \frac{2}{3}, 6, \sqrt{3},-2.45, \sqrt[6]{26}, 18 . \overline{4},-11, \sqrt[3]{27} \\ 5 \frac{1}{6}, 7.151551555 \ldots,-\sqrt{35}, \sqrt[5]{3},-\frac{8}{7}, 0, \sqrt{16}. \end{array} $$ Which are rational numbers?

Example 2

Consider the numbers $$ \begin{array}{l} \frac{2}{3}, 6, \sqrt{3},-2.45, \sqrt[6]{26}, 18 . \overline{4},-11, \sqrt[3]{27} \\ 5 \frac{1}{6}, 7.151551555 \ldots,-\sqrt{35}, \sqrt[5]{3},-\frac{8}{7}, 0, \sqrt{16}. \end{array} $$ Which are natural numbers?

Example 3

Consider the numbers $$ \begin{array}{l} \frac{2}{3}, 6, \sqrt{3},-2.45, \sqrt[6]{26}, 18 . \overline{4},-11, \sqrt[3]{27} \\ 5 \frac{1}{6}, 7.151551555 \ldots,-\sqrt{35}, \sqrt[5]{3},-\frac{8}{7}, 0, \sqrt{16}. \end{array} $$ Which are irrational numbers?

Example 4

Consider the numbers $$ \begin{array}{l} \frac{2}{3}, 6, \sqrt{3},-2.45, \sqrt[6]{26}, 18 . \overline{4},-11, \sqrt[3]{27} \\ 5 \frac{1}{6}, 7.151551555 \ldots,-\sqrt{35}, \sqrt[5]{3},-\frac{8}{7}, 0, \sqrt{16}. \end{array} $$ Which are integers?

Example 5

Consider the numbers $$ \begin{array}{l} \frac{2}{3}, 6, \sqrt{3},-2.45, \sqrt[6]{26}, 18 . \overline{4},-11, \sqrt[3]{27} \\ 5 \frac{1}{6}, 7.151551555 \ldots,-\sqrt{35}, \sqrt[5]{3},-\frac{8}{7}, 0, \sqrt{16}. \end{array} $$ Which are whole numbers?
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Step-by-Step Explanations

QUESTION

How do you write the set of all real numbers between a and b, excluding a and b?

STEP-BY-STEP ANSWER:

Step 1: Identify the endpoints a and b.
Step 2: Since both endpoints are not included, use parentheses around a and b.
Step 3: Write the interval as (a, b).
Final Answer: (a, b) represents all real numbers greater than a and less than b.

Interval Notation

QUESTION

How do you calculate the distance between two points on the number line, say a and b?

STEP-BY-STEP ANSWER:

Step 1: Recognize that the distance between two real numbers is given by the absolute value of their difference.
Step 2: Write the distance as |a - b| or |b - a|.
Step 3: Compute the absolute value to find the numerical distance.
Final Answer: The distance between a and b is |a - b|.

Absolute Value and Distance

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Common Mistakes

  • Incorrectly including endpoints in an open interval by using brackets instead of parentheses.
  • Confusing integers with natural or whole numbers, especially regarding the inclusion of negative numbers or zero.
  • Misapplying the absolute value definition and neglecting the sign of the original number.
  • Mixing up the properties of real numbers, such as confusing the commutative property with the associative property.