Problem 3 - College Algebra and Calculus: An Applied Approach

Question

Determine which numbers in the set are (a) natural numbers, (b) integers, (c) rational numbers, and (d) irrational numbers.$$ \left\\{12,-13,1, \sqrt{4}, \sqrt{6}, \frac{3}{2}\right\\} $$

Step-by-Step Solution

Answer

The natural numbers in the set are \(12\) and \(1\). The integers are \(12\), \(-13\), and \(1\). The rational numbers are \(12\), \(-13\), \(1\), \(\sqrt{4}\)(which is equivalent to \(2\)), and \(\frac{3}{2}\). The only irrational number is \(\sqrt{6}\).

Step 1: Classify as Natural Numbers

Natural numbers are non-negative and non-zero whole numbers. Inspect the set and classify the numbers fitting this criteria, which are: \(12\) and \(1\).

Step 2: Classify as Integers

Integers are whole numbers, both negative and positive, as well as zero. From the set, the integers are: \(12\), \(-13\), and \(1\).

Step 3: Classify as Rational Numbers

Rational numbers are numbers that can be expressed as a fraction where both the numerator and denominator are integers and the denominator is not zero. From the set, the rational numbers are: \(12\), \(-13\), \(1\), and \(\sqrt{4}\), which is equivalent to \(2\), and \(\frac{3}{2}\).

Step 4: Classify as Irrational Numbers

Irrational numbers cannot be expressed as fractions and do not end or repeat. From the set, the only irrational number is: \(\sqrt{6}\).

Key Concepts

Natural Numbers

Natural numbers are the set of positive whole numbers that you learn about early on in math. Essentially, they are the numbers you can count on your fingers starting from one and moving onwards. They do not include zero or any negative numbers. For example, numbers like 1, 2, 3, 4, and so on.

Natural numbers can be found in day-to-day activities such as counting objects, calculating age, or the number of books on a shelf.

• They are represented by symbols like: \( \{1, 2, 3, \ldots \} \).
• They start from 1 and go upwards infinitely.

In the original exercise, the natural numbers from the set are 12 and 1, as they are purely whole and positive.

Integers

Integers cover a broader range of whole numbers, including all the positive numbers (natural numbers), zero, and the negative counterparts.

They form a critical part of number classifications because they include negative values that represent owing money, temperatures below zero, and so forth.

• The integer set includes: \( \{ \ldots -3, -2, -1, 0, 1, 2, 3, \ldots \} \).
• Unlike natural numbers, integers can be negative and zero.

In the exercise, the integers in the set are 12, -13, and 1. These numbers are all whole, without any fractional part.

Rational Numbers

Rational numbers include any number that can be expressed as a quotient or fraction \( \frac{p}{q} \), where both \( p \) and \( q \) are integers and \( q \) is not zero.

These numbers can be positive, negative, represent whole numbers, and even fractions. They are noteworthy because they include many types of numbers that you encounter daily.

• They can be finite decimals (like 0.5 or 2.75).
• They also include repeating decimals, such as 0.333... which is equivalent to \( \frac{1}{3} \).

In the given set, the rational numbers are 12, -13, 1, \( \sqrt{4} \) (which equals 2), and \( \frac{3}{2} \). All these can be written as fractions.

Irrational Numbers

Irrational numbers are numbers that cannot be neatly expressed as a simple fraction of two integers.

These numbers have decimal expansions that go on forever without repeating, making them fascinating yet tricky to grasp fully.

• An example is \( \pi \), known famously for its non-terminating decimal representation of about 3.14159...
• Another example is the square root of non-perfect squares, like \( \sqrt{2} \) or \( \sqrt{6} \).

In the exercise you've tackled, the number \( \sqrt{6} \) is the only example of an irrational number from the set. It does not simplify to an exact fraction, thus bending the rules of neat division.