Problem 5 - Precalculus Mathematics for Calculus

Question

List the elements of the given set that are (a) natural numbers (b) integers (c) rational numbers (d) irrational numbers $$\left\\{0,-10,50, \frac{22}{7}, 0.538, \sqrt{7}, 1.2 \overline{3},-\frac{1}{3}, \sqrt{2}\right\\}$$

Step-by-Step Solution

Answer

(a) 50, (b) 0, -10, 50, (c) 0, -10, 50, \(\frac{22}{7}\), 0.538, 1.2\overline{3}, -\(\frac{1}{3}\), (d) \(\sqrt{7}\), \(\sqrt{2}\).

Step 1: Understanding Natural Numbers

Natural numbers are the set of positive integers starting from 1. Therefore, we will look for positive whole numbers: \(0,-10,50, \frac{22}{7}, 0.538, \sqrt{7}, 1.23\overline{3},-\frac{1}{3}, \sqrt{2}\). The only natural number is 50.

Step 2: Identifying Integers

Integers are numbers without any fractional or decimal part, including negative numbers and zero. From the set \(\{0, -10, 50, \frac{22}{7}, 0.538, \sqrt{7}, 1.23\overline{3}, -\frac{1}{3}, \sqrt{2}\}\), the integers are \(0, -10, 50\).

Step 3: Recognizing Rational Numbers

Rational numbers are numbers that can be expressed as a fraction of two integers. In the set \(\{0, -10, 50, \frac{22}{7}, 0.538, \sqrt{7}, 1.23\overline{3}, -\frac{1}{3}, \sqrt{2}\}\), the rational numbers are \(0, -10, 50, \frac{22}{7}, 0.538, 1.23\overline{3}, -\frac{1}{3}\).

Step 4: Identifying Irrational Numbers

Irrational numbers cannot be expressed as a simple fraction of two integers. They often include non-repeating, non-terminating decimals. From the set \(\{0, -10, 50, \frac{22}{7}, 0.538, \sqrt{7}, 1.23\overline{3}, -\frac{1}{3}, \sqrt{2}\}\), the irrational numbers are \(\sqrt{7}, \sqrt{2}\).

Key Concepts

Natural Numbers

Natural numbers are the simplest type of numbers we first learn to count with. They are the set of all positive integers starting from 1, 2, 3, and so on. Importantly, natural numbers do not include zero or any negative numbers. These numbers are used to count objects and don't have any fractional or decimal parts.

In mathematics, the set of natural numbers is denoted by \( \mathbb{N} \). They are considered building blocks of more complex numbers. When we look at a set of numbers, as in our example set, we seek whole numbers greater than zero.

• Examples: 1, 2, 3, 50, ..., where "..." indicates they continue infinitely.
• In the given set \( \{0, -10, 50, \frac{22}{7}, 0.538, \sqrt{7}, 1.2\overline{3}, -\frac{1}{3}, \sqrt{2}\} \), the natural number is only 50.

Integers

Integers extend our idea of whole numbers to include negative numbers and zero. This set is denoted by \( \mathbb{Z} \) which includes numbers like -2, -1, 0, 1, 2, and so on. These are numbers without any fractional or decimal component.

Integers are very useful as they allow us to represent values that can go below zero, such as temperature or financial debts. When identifying integers within a set, look for numbers that do not have parts after the decimal or fraction.

• Examples: -3, 0, 2, 50, ..., continuing infinitely in both directions.
• In our set \( \{0, -10, 50, \frac{22}{7}, 0.538, \sqrt{7}, 1.2\overline{3}, -\frac{1}{3}, \sqrt{2}\} \), the integers are 0, -10, and 50.

Rational Numbers

Rational numbers offer a broader category that allows fractions and decimals, as long as these decimals are either terminating or repeating. A rational number can always be expressed as the fraction \( \frac{a}{b} \) where both \(a\) and \(b\) are integers and \(b eq 0\).

It's essential to note that all integers are rational numbers since any integer \(n\) can be written as \(\frac{n}{1}\). Rational numbers provide flexibility in mathematics and are used to describe quantities that aren't whole numbers.

• Examples: \( \frac{1}{2}, -\frac{3}{4}, 0.333\overline{3}, -10, 50\).
• In the set \( \{0, -10, 50, \frac{22}{7}, 0.538, \sqrt{7}, 1.2\overline{3}, -\frac{1}{3}, \sqrt{2}\} \), the rational numbers are 0, -10, 50, \(\frac{22}{7}\), 0.538, \(1.2\overline{3}\), and \(-\frac{1}{3}\).

Irrational Numbers

Irrational numbers are numbers that cannot be written as a simple fraction of two integers. This means their decimal expansions are neither terminating nor repeating. They fill the gaps between rational numbers on the number line.

Common examples of irrational numbers include roots of non-perfect squares and constants like \(\pi\). These numbers are significant in mathematics as they complete the real number system, providing a full picture of numbers without any breaks.

• Examples: \(\sqrt{2}, \sqrt{3}, \pi, e\).
• In the provided set \( \{0, -10, 50, \frac{22}{7}, 0.538, \sqrt{7}, 1.2\overline{3}, -\frac{1}{3}, \sqrt{2}\} \), the irrational numbers are \(\sqrt{7}\) and \(\sqrt{2}\).