Problem 4 - 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.$$ \left\\{3,-1, \frac{1}{3}, \frac{6}{3},-\frac{1}{2} \sqrt{2},-7.5\right\\} $$

Step-by-Step Solution

Answer

(a) The natural numbers in the set are: \(3\). (b) The integers in the set are: \(3, -1\). (c) The rational numbers in the set are: \(3, -1, \frac{1}{3}, \frac{6}{3}, -7.5\). (d) The irrational numbers in the set are: \(-\frac{1}{2} \sqrt{2}\).

Step 1: Identify the Natural Numbers

Natural numbers, also called counting numbers, are the numbers we use when counting objects: \(1, 2, 3, ...\). Looking at the set, \(3\) is the only natural number.

Step 2: Identify the Integers

Integers include natural numbers, their negatives, and zero. In the set, \(3\) and \(-1\) are integers.

Step 3: Identify Rational Numbers

Rational numbers are any numbers that can be expressed as a fraction of two integers where the denominator is not zero. In the set, \(3, -1, \frac{1}{3}, \frac{6}{3}, \) and \(-7.5\) (which is also \(-\frac{15}{2}\)) are rational numbers.

Step 4: Identify Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a fraction of two integers. These are typically non-repeating and non-terminating decimals. In the set, \(-\frac{1}{2} \sqrt{2}\) is an irrational number.

Key Concepts

Natural Numbers

Natural numbers, often referred to as the counting numbers, are the simplest form of numbers that we use in everyday life. They help us count objects and are the first numbers you typically learn as a child.

• Examples of natural numbers are: 1, 2, 3, and so on.
• They do not include zero or negative numbers.

In the context of the original exercise, the number 3 is a natural number. It is a part of this set because it fits the definitions and characteristics described above. Natural numbers are very straightforward and help in basic arithmetic operations such as addition and multiplication. Whenever you think of numbers in their purest counting sense, you're thinking of natural numbers.

Integers

Integers expand the set of natural numbers by including their negative counterparts and zero. They include:

• Positive integers (like 1, 2, 3)
• Negative integers (like -1, -2, -3)
• The number zero (0)

In the given set, both 3 and -1 are integers. This is because integers are complete in terms of addition, subtraction, and multiplication. They can represent whole numbers, whether positive, negative, or zero. In practical terms, integers allow us to model real-world scenarios involving something being added, lost, or absent entirely (zero).

Rational Numbers

Rational numbers are a bit broader category. They encompass numbers that can be written as a fraction, where both the numerator and denominator are integers and the denominator isn’t zero.

• Rational numbers include integers since numbers like 3 can be written as \(\frac{3}{1}\).
• They also include decimals that are either terminating, like 0.75, or repeating, such as 0.333...

From our exercise, the numbers 3, -1, \(\frac{1}{3}\), \(\frac{6}{3}\) (which simplifies to 2), and -7.5 (which is the same as \(-\frac{15}{2}\)) are all rational. These can all be expressed as fractions, making them rational numbers. Rational numbers are versatile as they are used in various computations involving divisions and ratios, providing more flexibility beyond just whole numbers.

Irrational Numbers

Irrational numbers are quite special. They cannot be precisely written as a simple fraction of two integers, and they appear as non-repeating, non-terminating decimals. This means the decimal form goes on forever without a recurring pattern.

• The most famous examples include \(\pi\) (Pi) and \(\sqrt{2}\) (the square root of two).

In the given set, \(-\frac{1}{2} \sqrt{2}\) is the irrational number. This is because \(\sqrt{2}\) cannot be exactly expressed as a fraction, and multiplying it by another rational number just maintains its irrational nature. Irrational numbers are very important in mathematics and science, especially in areas involving continuous growth and geometric computations. They represent real-world quantities that can't be neatly captured by integers or fractions and thus play a critical role in understanding the complex parts of mathematics.