Problem 42 - 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 \(\left\\{-5.3,-5,-\sqrt{3},-1,-\frac{1}{9}, 0,0 . \overline{27}, 1.2,3, \sqrt{11}\right\\}\)

Step-by-Step Solution

Answer

(a) {3}, (b) {0, 3}, (c) {-5, -1, 0, 3}, (d) {-5, -1, -1/9, 0, 0.27, 1.2, 3}, (e) {-\sqrt{3}, \sqrt{11}}, (f) {-5.3, -5, -\sqrt{3}, -1, -\frac{1}{9}, 0, 0.27, 1.2, 3, \sqrt{11}}

Step 1: Identify Natural Numbers

Natural numbers are positive integers starting from 1. From the given set, identify natural numbers: \ \ \( \{ 3 \} \) is the only natural number.

Step 2: Identify Whole Numbers

Whole numbers include all natural numbers and zero. From the given set, identify whole numbers: \ \ \( \{ 0, 3 \} \) are the whole numbers.

Step 3: Identify Integers

Integers include all whole numbers and their negative counterparts. From the given set, identify integers: \ \ \( \{-5, -1, 0, 3 \} \) are the integers.

Step 4: Identify Rational Numbers

Rational numbers can be expressed as a ratio of two integers. From the given set, identify rational numbers: \ \ \( \{-5, -1, -\frac{1}{9}, 0, 0.\overline{27}, 1.2, 3 \} \) are the rational numbers.

Step 5: Identify Irrational Numbers

Irrational numbers cannot be expressed as a ratio of two integers. From the given set, identify irrational numbers: \ \ \( \{-\sqrt{3}, \sqrt{11} \} \) are the irrational numbers.

Step 6: Identify Real Numbers

Real numbers include both rational and irrational numbers. From the given set, identify real numbers: \ \ \( \{-5.3, -5, -\sqrt{3}, -1, -\frac{1}{9}, 0, 0.\overline{27}, 1.2, 3, \sqrt{11} \} \) are the real numbers.

Key Concepts

Natural Numbers

Natural numbers are the simplest set of numbers, which include all positive integers starting from 1. These are the numbers we often use for counting. Think of them as the numbers you'd use to count objects: 1, 2, 3, and so on. Natural numbers do not include zero or any negative numbers. For example, from the given set \(\text{-5.3, -5, -\sqrt{3}, -1, -\frac{1}{9}, 0, 0.\overline{27}, 1.2, 3, \sqrt{11}}\), the only natural number is 3.

Whole Numbers

Whole numbers expand on natural numbers by including zero. They are still positive, but now you can also have 0 in your count. This set does not include any negative numbers or fractions. For instance, the whole numbers from our given set are \(\text{0, 3}\). So, whole numbers start from 0 and include all natural numbers: 0, 1, 2, 3, and so on.

Integers

Integers take whole numbers a step further by including their negative counterparts. This set includes positive numbers, zero, and negative numbers but still no fractions or decimal numbers. In our given set, the integers are \(\text{-5, -1, 0, 3}\). So, integers include \(\text{... -3, -2, -1, 0, 1, 2, 3, ...}\).

Rational Numbers

Rational numbers are numbers that can be expressed as a ratio of two integers (a fraction) where the denominator is not zero. This means they can be written as \(\frac{p}{q}\) where \(\text{p and q are integers and q≠0}\). This set includes fractions, whole numbers, and repeating or terminating decimals. From the given set \(\text{-5, -1, -\frac{1}{9}, 0, 0.\overline{27}, 1.2, 3}\) are the rational numbers. Even 0.\overline{27} is rational because it's a repeating decimal.

Irrational Numbers

Irrational numbers are the numbers that cannot be expressed as a simple fraction. They have non-repeating, non-terminating decimal representations. This means their decimal form goes on forever without repeating. Some familiar examples include \(\text{\sqrt{2}}\) and \(\text{\pi}}\). From the given set \(\text{-\sqrt{3}, \sqrt{11}}\) are irrational. These numbers cannot be written as exact fractions.

Real Numbers

Real numbers include all numbers that can be found on the number line. This set encompasses both rational and irrational numbers. Essentially, any number that is not imaginary falls into this category. From the given set, all numbers except none are real numbers: \(\text{-5.3, -5, -\sqrt{3}, -1, -\frac{1}{9}, 0, 0.\overline{27}, 1.2, 3, \sqrt{11}}\). Thus, real numbers can be positive, negative, fractions, or irrational.