Problem 298 - Intermediate Algebra

Question

In the following exercises, list the (a) whole numbers, (b) integers, \(\odot\) rational numbers, \(@\) irrational numbers, \(\Theta\) real numbers for each set of numbers. $$ -9,-3 \frac{4}{9},-\sqrt{9}, 0.4 \overline{09}, \frac{11}{6}, 7 $$

Step-by-Step Solution

Answer

Whole numbers: 7 Integers: -9, -3, 7 Rational numbers: -9, - 3 \frac{4}{9}, -\frac{3}{3}, 0.4 \bar{09}, \frac{11}{6}, 7 Irrational numbers: None Real numbers: -9, - 3 \frac{4}{9}, -\frac{3}{3}, 0.4 \bar{09}, \frac{11}{6}, 7

Step 1: Identify the whole numbers

Whole numbers are non-negative integers including zero. Look at the set and pick the numbers that fit this definition.

Step 2: Identify the integers

Integers include all whole numbers and their negative counterparts. Select numbers from the set that are whole numbers or negative whole numbers.

Step 3: Identify the rational numbers

Rational numbers can be expressed as a fraction of two integers (where the denominator is not zero). Determine which numbers in the set can be written as a fraction.

Step 4: Identify the irrational numbers

Irrational numbers cannot be written as a simple fraction. They have non-repeating, non-terminating decimal expansions. Pick any numbers that fit this definition.

Step 5: Identify the real numbers

Real numbers include all rational and irrational numbers. List all numbers from the set under this category.

Key Concepts

Understanding Whole Numbers

Whole numbers are fundamental in mathematics. They include all positive numbers and zero, without decimals or fractions. In simpler terms, whole numbers are like counting numbers but with zero added.
Examples of whole numbers are 0, 1, 2, 3, 4, and so on. Because whole numbers can't be negative, numbers like -1 or -2 are not considered whole numbers.
So, from the given set, the whole number is just 7.

Exploring Integers

Integers include both positive and negative whole numbers, along with zero. They do not have fractions or decimals.
For example, numbers like -3, 0, 4, and -10 are all integers. This means any number you can count with or against without a fraction is an integer.
So, from the given set \(-9, -\frac{4}{9}, -\text{sqrt}(9), 0.4 \overline{09}, \frac{11}{6}, 7\), the integers are -9 and 7.

Diving into Rational Numbers

Rational numbers are numbers that can be expressed as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b eq 0\).
Rational numbers can be positive, negative, whole numbers, and even fractions and decimals as long as they can be written as a fraction.
Some examples include \(\frac{1}{2}\), -3, and 0.25. So from our given set, the rational numbers include \(-3\frac{4}{9}, -\text{sqrt}(9), 0.4 \overline{09}, \frac{11}{6}, 7\). In this context -\text{sqrt}(9)= -3.

Unraveling Irrational Numbers

Irrational numbers are the numbers that cannot be written as simple fractions. Their decimal expansions are non-repeating and non-terminating.
Examples of irrational numbers include \( \frac{\text{pi}}{\underline{\phantom{xx}}} \), \(\text{sqrt}(2)\), and \(0.10110111011110...\).
From the set provided \(-9,-3\frac{4}{9},-\text{sqrt}(9), 0.4 \backslash overline{09}, \frac{11}{6}, 7\), none of the numbers fit into the irrational category.

Understanding Real Numbers

Real numbers encompass both rational and irrational numbers. Essentially, any number that exists on the number line is a real number.
This includes whole numbers, integers, rational numbers, and irrational numbers.
So from our initial set, all the numbers \(-9, -3\frac{4}{9}, -\text{sqrt}(9), 0.4 \overline{09}, \frac{11}{6}, 7\), all fall under the category of real numbers.