Problem 43 - Beginning and Intermediate Algebra

Question

List all numbers from each set that are the following. (a) natural numbers (b) whole numbers (c) integers (d) rational numbers (e) irrational numbers (f) real numbers \(\left\\{\frac{7}{9},-2 . \overline{3}, \sqrt{3}, 0,-8 \frac{3}{4}, 11,-6, \pi\right\\}\)

Step-by-Step Solution

Answer

Natural: 11. Whole: 0, 11. Integer: 0, 11, -6. Rational: \( \frac{7}{9}, -2 . \overline{3}, 0, -8 \frac{3}{4}, 11, -6 \). Irrational: \( \pi, \sqrt{3} \). Real: \( \frac{7}{9}, -2 . \overline{3}, \sqrt{3}, 0, -8 \frac{3}{4}, 11, -6, \pi \).

Step 1 - Identify Natural Numbers

Natural numbers are positive integers starting from 1. Identify numbers from the given set that are natural numbers: \[ \text{Natural numbers:} \ 11 \]

Step 2 - Identify Whole Numbers

Whole numbers include all natural numbers along with 0. Identify numbers from the set that are whole numbers: \[ \text{Whole numbers:} \ 0, 11 \]

Step 3 - Identify Integers

Integers include all positive and negative whole numbers, including 0. Identify numbers from the set that are integers: \[ \text{Integers:} \ 0, 11, -6 \]

Step 4 - Identify Rational Numbers

Rational numbers can be expressed as a fraction of two integers. Identify numbers from the set that are rational numbers: \[ \text{Rational numbers:} \ \frac{7}{9}, -2 . \overline{3}, 0, -8 \frac{3}{4}, 11, -6 \]

Step 5 - Identify Irrational Numbers

Irrational numbers cannot be expressed as a fraction of two integers. They have non-repeating, non-terminating decimal expansions. Identify numbers from the set that are irrational numbers: \[ \text{Irrational numbers:} \ \pi, \sqrt{3} \]

Step 6 - Identify Real Numbers

Real numbers include all rational and irrational numbers combined. Identify numbers from the set that are real numbers: \[ \text{Real numbers:} \ \frac{7}{9}, -2 . \overline{3}, \sqrt{3}, 0, -8 \frac{3}{4}, 11, -6, \pi \]

Key Concepts

Natural Numbers

Natural numbers are the simplest type of numbers we learn about. These are the positive counting numbers starting from 1. Think about when you're counting objects like apples: 1, 2, 3, and so on. They don’t include zero, decimals, or negative numbers. Examples include:

• 1
• 2
• 3

From the given set: \(\frac{7}{9}, -2 . \overline{3}, \sqrt{3}, 0,-8 \frac{3}{4}, 11,-6,\pi\)\.
The only natural number is 11.

Whole Numbers

Whole numbers expand on natural numbers by including zero. These are the non-negative integers that do not have any decimal or fractional part. For example:

• 0
• 1
• 2

It's important to note that while whole numbers include zero, they do not include negative numbers or fractions.
From the given set: \(\frac{7}{9}, -2 . \overline{3}, \sqrt{3}, 0,-8 \frac{3}{4}, 11,-6, \pi\). The whole numbers are 0 and 11.

Integers

Integers include all whole numbers as well as their negative counterparts. These numbers can be positive, negative, or zero but do not include fractions or decimals. Examples include:

• -3
• -2
• 0
• 1
• 2

From the given set: \(\frac{7}{9}, -2 . \overline{3}, \sqrt{3}, 0,-8 \frac{3}{4}, 11,-6, \pi\). The integers are 0, 11, and -6.

Rational Numbers

Rational numbers can be written as a fraction of two integers, where the denominator is not zero. They include integers, finite decimals, and repeating decimals. For instance:

• \frac{1}{2}
• 0.75
• -2
• 3

From the given set: \(\frac{7}{9}, -2 . \overline{3}, \sqrt{3}, 0,-8 \frac{3}{4}, 11,-6, \pi\), the rational numbers are \(\frac{7}{9}, -2 . \overline{3}, 0, -8 \frac{3}{4}, 11, -6\).

Irrational Numbers

Irrational numbers are numbers that cannot be written as a simple fraction. Their decimal forms are non-repeating and non-terminating. Two famous examples include \(\pi\) and \(\sqrt{2}\). For example:

• \pi
• \sqrt{3}

From the given set: \(\frac{7}{9}, -2 . \overline{3}, \sqrt{3}, 0,-8 \frac{3}{4}, 11,-6, \pi\), the irrational numbers are \(\pi\) and \(\sqrt{3}\).

Real Numbers

Real numbers include all the numbers on the number line. These include both rational and irrational numbers. Hence, they include integers, fractions, finite decimals, repeating decimals, and non-repeating/non-terminating decimals. Examples include:

• -3
• 0
• \frac{1}{2}
• \pi

From the given set: \(\frac{7}{9}, -2 . \overline{3}, \sqrt{3}, 0,-8 \frac{3}{4}, 11,-6, \pi\), all the numbers listed belong to the set of real numbers. So, the real numbers are:\br> \(\frac{7}{9}, -2 . \overline{3}, \sqrt{3}, 0,-8 \frac{3}{4}, 11,-6, \pi\).