Problem 6 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

(a) \{11, 4, 5\}; (b) \{-11, 11, 4, 5\}; (c) \{1.001, 0.333..., -11, 11, \frac{13}{15}, 4, 3.14, 5\}; (d) \{-\pi\}.

Step 1: Identify Natural Numbers

Natural numbers are positive integers starting from 1. Analyze the set: - 1.001 is not a natural number as it is not an integer.- 0.333... is not a natural number as it is a decimal.- \(-\pi\) is not a natural number as it is negative and irrational.- \(-11\) is not a natural number as it is negative.- 11 is a natural number.- \(\frac{13}{15}\) is not a natural number as it is a fraction.- \(\sqrt{16} = 4\) is a natural number.- 3.14 is not a natural number.- \(\frac{15}{3} = 5\) is a natural number.So, the natural numbers are: \{11, 4, 5\}.

Step 2: Identify Integers

Integers can be negative, zero, or positive whole numbers. Check the elements:- 1.001 is not an integer.- 0.333... is not an integer.- \(-\pi\) is not an integer.- \(-11\) is an integer.- 11 is an integer.- \(\frac{13}{15}\) is not an integer.- \(\sqrt{16} = 4\) is an integer.- 3.14 is not an integer.- \(\frac{15}{3} = 5\) is an integer.So, the integers are: \{-11, 11, 4, 5\}.

Step 3: Identify Rational Numbers

Rational numbers can be expressed as the quotient of two integers. Check:- 1.001 is rational because it can be expressed as a fraction (1001/1000).- 0.333... is a repeating decimal equivalent to 1/3, so it's rational.- \(-\pi\) is not rational.- \(-11\) is rational as it can be represented as \(-11/1\).- 11 is rational as it can be represented as \(11/1\).- \(\frac{13}{15}\) is rational.- \(\sqrt{16} = 4\) is rational.- 3.14 is rational as it can be expressed as 314/100.- \(\frac{15}{3} = 5\) is rational.So, the rational numbers are: \{1.001, 0.333..., -11, 11, \frac{13}{15}, 4, 3.14, 5\}.

Step 4: Identify Irrational Numbers

Irrational numbers cannot be expressed as a simple fraction. Look for elements that are not rational:- \(-\pi\) is irrational.None of the other numbers fit the criteria for irrational numbers. Thus, the only irrational number in the set is: \{-\pi\}.

Key Concepts

Natural Numbers

Natural numbers are the basic building blocks of mathematics. They are the simplest set of numbers, starting from 1 and increasing indefinitely (1, 2, 3, ...). They do not include negatives, fractions, or zero.

In our given set, natural numbers are purely positive whole numbers. For example:

• 11 is a natural number because it is a positive whole number.
• \(\sqrt{16} = 4\) is another, since 4 is a non-negative whole number.
• \(\frac{15}{3} = 5\) which simplifies to a whole number, also counts as a natural number.

Decimals, fractions, and negative numbers like 1.001, \(0.333\ldots\), and \(-11\) don’t qualify as natural numbers. They either aren't whole or are negative. Thus, from our set, the natural numbers are \{11, 4, 5\}.

Integers

Integers expand upon natural numbers by including negative numbers and zero. They are whole numbers that can be positive, negative, or zero.

Analyzing our set through this lens:

• \(-11\) is an integer due to it being a whole number, albeit negative.
• 11 remains an integer because it doesn’t have any fractional part.
• \(\sqrt{16} = 4\) is an integer as it results in a whole number.
• \(\frac{15}{3} = 5\) simplifies to an integer.

However, numbers with decimal or fractional representations such as 1.001 and \(0.333\ldots\) aren’t integers, since they aren’t whole. That means our integer set consists of \{-11, 11, 4, 5\}.

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction \(\frac{a}{b}\) of two integers, where the denominator \(b\) is not zero.

Within our set:

• 1.001 is rational because it can be expressed as \(1001/1000\).
• \(0.333\ldots\) simplifies to \(1/3\) which is rational.
• \(-11\) can be seen as \(-11/1\), making it rational.
• 11 is rational since it equals \(11/1\).
• \(\frac{13}{15}\) is already a fraction, thus rational.
• \(\sqrt{16} = 4\), equivalent to \(4/1\), is rational.
• 3.14 converts to \(314/100\), also rational.
• \(\frac{15}{3} = 5\), which is \(5/1\).

Rational numbers from the set includes almost all items except those that are mean to have non-repeating decimals or specific non-rational properties. So, they are \{1.001, 0.333\ldots, -11, 11, \frac{13}{15}, 4, 3.14, 5\}.

Irrational Numbers

Irrational numbers are those that can't be neatly expressed as a fraction of two integers. They have non-terminating and non-repeating decimal expansions.

In our set, we find:

• \(-\pi\) is an irrational number. Although often approximated as \(-3.14\), \(-\pi\) in its exact form cannot be expressed as a fraction of two integers, because its decimal form is non-repeating and infinite.

Any other numbers in the set that can be expressed as a finite or repeating fraction are not considered irrational. Therefore, \(-\pi\) is the only irrational number in the given set.