Problem 5 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Natural: {50}; Integers: {0, -10, 50}; Rational: {0, -10, 50, \(\frac{22}{7}\), 0.538, \(1.2\overline{3}\), \(-\frac{1}{3}\)}; Irrational: {\(\sqrt{7}\), \(\sqrt[3]{2}\)}.

Step 1: Identify Natural Numbers

Natural numbers are positive integers starting from 1. In the given set \( \{0, -10, 50, \frac{22}{7}, 0.538, \sqrt{7}, 1.2\overline{3}, -\frac{1}{3}, \sqrt[3]{2}\} \), the only natural number is 50.

Step 2: Identify Integers

Integers include all natural numbers, zero, and negative whole numbers. From the set, the elements that are integers are \(0\), \(-10\), and \(50\).

Step 3: Identify Rational Numbers

Rational numbers can be expressed as the quotient of two integers, where the denominator is not zero. From the set, the rational numbers are 0, -10, 50, \(\frac{22}{7}\), 0.538, and \(1.2\overline{3}\), and \(-\frac{1}{3}\).

Step 4: Identify Irrational Numbers

Irrational numbers cannot be expressed as the quotient of two integers. They have non-repeating, non-terminating decimal expansions. From the set, the irrational numbers are \(\sqrt{7}\) and \(\sqrt[3]{2}\).

Key Concepts

Natural Numbers

Natural numbers are the simplest class of numbers we deal with. They start from 1 and go upwards without end. These numbers are often used for counting objects, like 1 apple, 2 bananas, etc. In the set context of our exercise, natural numbers must be positive and whole.

Important points to remember about natural numbers include:

• They start from 1, not 0 (which can sometimes be confusing).
• They do not include decimals or fractions.
• They are always positive.

For the set given in the exercise, the only natural number is 50. This is because all other elements either include a negative sign, a decimal, or are zero.

Integers

Integers are a larger set of numbers that encompass natural numbers, zero, and the negative of natural numbers. In other words, they include both positive numbers like 1, 2, 3, negative numbers like -1, -2, -3, and zero itself.

Key aspects of integers include:

• They include zero in the set.
• They do not include fractions or decimals.
• The set is symmetric around zero.

In the exercise set, the integers are 0, -10, and 50. These are the only numbers here that are whole (without fractions or decimal points), and they include zero along with a positive and a negative number.

Rational Numbers

Rational numbers are versatile, covering numbers that can be expressed as a fraction or a ratio of two integers. Here, the denominator should not be zero. Rational numbers can appear as integers, fractions, and even repeating or terminating decimals.

Characteristics of rational numbers include:

• They can be written in fraction form, for example, \( \frac{3}{4} \).
• Decimals that terminate or repeat can be rational. For example, 0.75 and 1.2\overline{3} are rational.
• All integers are also rational numbers since they can be written with a denominator of 1, such as \( 5 = \frac{5}{1} \).

In the given set, 0, -10, 50, \( \frac{22}{7} \), 0.538, 1.2\overline{3}, and \(-\frac{1}{3} \) are rational, as they all fit the criteria of this number type.

Irrational Numbers

Irrational numbers are intriguing because they cannot be written as simple fractions. Instead, their decimal expansions neither terminate nor repeat. These numbers often arise from computations involving roots and special numbers like \( \pi \).

What you should know about irrational numbers:

• They don't have a repeating or terminating decimal form.
• Examples include numbers like \( \sqrt{2} \) or the famous \( \pi \).
• They cannot be expressed exactly as a fraction of two integers.

For our set, the numbers \( \sqrt{7} \) and \( \sqrt[3]{2} \) are irrational. They demonstrate the classic properties of irrational numbers, meaning they can't be simplified into a neat fraction and their decimal form never stops or repeats.