Problem 102 - Introductory and Intermediate Algebra for College Students 4th

Question

From here on, each exercise set will contain three review exercises. It is essential to review previously covered topics to improve your understanding of the topics and to help maintain your mastery of the material. If you are not certain how to solve a review exercise, turn to the section and the worked example given in parentheses at the end of each exercise. Consider the set $$\\{-6,-\pi, 0,0, \overline{7}, \sqrt{3}, \sqrt{4}\\}$$ List all numbers from the set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, f. real numbers. (Section 1.3 Example 5 )

Step-by-Step Solution

Answer

a. Natural numbers: 2,7 b. Whole numbers: 0,2,7 c. Integers: -6,0,2,7 d. Rational numbers: -6,0,2,7 e. Irrational numbers: -\(\pi\), \(\sqrt{3}\) f. Real numbers: -6,-\(\pi\),0,2,7,\(\sqrt{3}\)

Step 1: Identify Natural Numbers

Natural numbers are the set of all positive integers. From the given set, the natural numbers are \(\overline{7}\) and \(\sqrt{4}\), which are equivalent to 7 and 2, respectively.

Step 2: Identify Whole Numbers

Whole numbers are the set of all non-negative integers (that is, natural numbers and zero). From the given set, the whole numbers are 0, \(\overline{7}\), and \(\sqrt{4}\), which are equivalent to 0, 7, and 2, respectively.

Step 3: Identify Integers

Integers are the set of all whole numbers and their additive inverses (negative counterparts). From the given set, the integers are -6, 0, \(\overline{7}\), and \(\sqrt{4}\), equivalent to -6, 0, 7, and 2.

Step 4: Identify Rational Numbers

Rational numbers are the set of all numbers that can be expressed as a fraction where denominator is not zero. From the set, the rational numbers are -6, 0, \(\overline{7}\), and \(\sqrt{4}\), which are equivalent to -6, 0, 7, and 2.

Step 5: Identify Irrational Numbers

Irrational numbers are the set of all numbers that cannot be expressed as a fraction, or in other words, their decimal expansion neither terminate nor repeat. From the given set, irrational numbers are -\(\pi\) and \(\sqrt{3}\).

Step 6: Identify Real Numbers

Real numbers are the set of all rational and irrational numbers. So all the numbers in the given set (-6, -\(\pi\), 0, \(\overline{7}\), \(\sqrt{3}\), \(\sqrt{4}\)) are real numbers.

Key Concepts

Real Numbers

Real numbers encompass almost every number we encounter in daily life. They are an amalgamation of both rational and irrational numbers.
This means any number you can place on a number line is a real number. Whether it's negative, positive, a fraction, or a decimal, if it fits on the number line, it counts as a real number.

• This includes both whole numbers like 0 and integers.
• Rational numbers, like 1/2 or 7.5, are also real numbers.
• Irrational numbers, such as \(\pi\) or \(\sqrt{3}\), fall under this category as well.

Real numbers only exclude complex numbers, which involve the imaginary unit \(i = \sqrt{-1}\). Thus, in mathematical problems and real-world applications, you’ll often find yourself working within the realm of real numbers.

Rational Numbers

Rational numbers are numbers you can express as a fraction or a ratio. These numbers can be written as \(\frac{a}{b}\), where both \(a\) and \(b\) are integers and \(b eq 0\).
Importantly, rational numbers have decimal representations that either terminate or repeat a pattern.

• For instance, the number 0.5 is a rational number because it can be expressed as \(\frac{1}{2}\).
• The number 7 is rational, as it’s equivalent to \(\frac{7}{1}\).
• Even numbers like \(-6\) or \(\sqrt{4} = 2\) fit as they simplify to \(\frac{-6}{1}\) or \(\frac{2}{1}\).

Thus, rational numbers are very flexible and include negative numbers, whole numbers, and some decimals.

Irrational Numbers

Irrational numbers present a bit of mystery. They cannot be written as fractions of two integers. Their decimal forms neither terminate nor repeat any pattern, making them unique.
These numbers are often results of roots or involve constants like \(\pi\).

• So numbers like \(\sqrt{3}\) cannot be precisely expressed as a fraction.
• The well-known \(\pi\), approximately 3.14159, continues infinitely without a repeating pattern.

While harder to work with, irrational numbers are integral to higher-level mathematics, giving depth and detail to our understanding of numbers.

Integers

Integers represent the collection of whole numbers and their negatives. Essentially, integers fill the number line without involving fractions or decimals.

• Examples are ..., \(-3, -2, -1, 0, 1, 2, 3,\) and so on.
• Both positive numbers and negative counterparts, including zero, are counted.
• An integer is defined as any number among this seamless series.

They serve as the bedrock for arithmetic operations and offer a complete set of numbers for simple calculations.

Whole Numbers

Whole numbers begin with zero and continue with all the positive integers. Essentially they are the non-negative integers.

• Examples of whole numbers include 0, 1, 2, 3, 4, etc.
• They omit any negative numbers or fractions.
• Whole numbers help in counting and in simple arithmetic tasks where negative values are not considered.

These numbers provide simplicity in understanding and calculating basic mathematical operations without the complications of negatives or decimals.

Natural Numbers

Natural numbers start at 1 and proceed upwards indefinitely. They are the earliest form of counting numbers children typically learn.

• Natural numbers include 1, 2, 3, 4, and so on.
• They do not include zero or any negative numbers.
• In practical terms, these numbers are used for counting objects and simple measurements.

The set of natural numbers is fundamental in mathematics, laying the groundwork for further numerical exploration and calculation.