Problem 2 - Algebra and Trigonometry

Question

\(1-2\) . List the elements of the given set that are (a) natural numbers (b) integers (c) rational numbers (d) irrational numbers $$ \left\\{1.001,0.333 \ldots,-\pi,-11,11, \frac{13}{15}, \sqrt{16}, 3.14, \frac{15}{3}\right\\} $$

Step-by-Step Solution

Answer

(a) 11, 4; (b) -11, 11, 4; (c) 1.001, 0.333..., -11, 11, 4, 5; (d) -π, 3.14

Step 1: Identify Natural Numbers

Natural numbers are positive integers beginning from 1. From the set \( \{1.001, 0.333\ldots, -\pi, -11, 11, \frac{13}{15}, \sqrt{16}, 3.14, \frac{15}{3}\} \), the elements that qualify are \( 11 \) and \( \sqrt{16} = 4 \).

Step 2: Identify Integers

Integers include all whole numbers, both positive and negative, along with zero. From the set, the integers are \(-11, 11\), and \( \sqrt{16} = 4 \).

Step 3: Identify Rational Numbers

Rational numbers can be expressed as the quotient of two integers. The elements in the set that are rational are \(1.001, 0.333\ldots, -11, 11, \frac{13}{15}, \sqrt{16} = 4,\) and \(\frac{15}{3} = 5\).

Step 4: Identify Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction. From the set, the irrational numbers are \(-\pi\) and \(3.14\) (assuming \(3.14\) represents a non-terminating, non-repeating decimal).

Key Concepts

Natural Numbers

Natural numbers are the most basic and fundamental set of numbers we use in mathematics. They start from 1 and continue upwards indefinitely, with each number being a whole, non-fractional number. These numbers are often used for counting and ordering.

• They are represented as \( \{1, 2, 3, 4, \ldots\} \).
• Notice that zero is not considered a natural number.
• They are positive and do not include negative numbers or fractions.

In our set, the elements that are natural numbers are \(11\) and \(\sqrt{16} = 4\). These numbers fit the criteria by being positive whole numbers. No fractions or decimals are included in this category when considering the typical definition of natural numbers.

Integers

Integers expand upon natural numbers by including negative numbers and zero. They encompass all whole numbers, both positive and negative.

• This set is represented as \( \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\} \).
• Integers do not include fractions or decimals.

From our set, the elements that are integers include \(-11, 11\), and \(\sqrt{16} = 4\). These numbers are whole and do not possess any fractional or decimal components, fitting neatly into our definition of integers.

Rational Numbers

Rational numbers are versatile and encompass a wide range of numbers. A number is considered rational if it can be expressed as the quotient of two integers, where the denominator is not zero.

• Any number that can be written as \( \frac{a}{b} \), where \(a\) and \(b\) are integers and \(b eq 0\), is rational.
• This includes all integers (since any integer \(a\) can be written as \(\frac{a}{1}\)).
• Fractions, decimals that terminate, and repeating decimals are all rational numbers.

In our given set, the rational numbers include \(1.001, 0.333\ldots, -11, 11, \frac{13}{15}, \sqrt{16} = 4\), and \(\frac{15}{3} = 5\). These can all be expressed as a fraction of two integers.

Irrational Numbers

Irrational numbers are fascinating because they cannot be perfectly expressed as a ratio of two integers. They are often characterized by non-terminating and non-repeating decimal expansions.

• Examples include well-known numbers like \(\pi\) and \(\sqrt{2}\).
• Any decimal that does not end or does not cycle through a repeating pattern is irrational.

In our set, the irrational numbers are \(-\pi\) and potentially \(3.14\) if it is interpreted as a representation of a non-terminating, non-repeating decimal. These numbers cannot be neatly expressed as a fraction, fitting the criteria for irrational numbers.