Problem 5 - Elementary and Intermediate Algebra

Question

In each of Exercises \(1-8\) fill in the blank using one of the following terms: natural number, whole number, integer, rational number, tern ting irrational. Any number of the form \(a / b,\) where \(a\) and \(b\) are integers, with \(b \neq 0,\) is an example of a(n)__________.

Step-by-Step Solution

Answer

rational number

Step 1: Identify the given form

The problem specifies a number of the form \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b eq 0 \).

Step 2: Definition review

Review the definitions of the given terms: natural number, whole number, integer, rational number, and irrational number.

Step 3: Match the definition

Recognize that a number of the form \( \frac{a}{b} \), where \(a\) and \(b\) are integers and \( b eq 0 \), aligns with the definition of a rational number.

Step 4: Choose the correct term

Based on the definitions, the correct term to fill in the blank is 'rational number'.

Key Concepts

Natural Numbers

Natural numbers are the simplest and most familiar numbers. They are the set of positive numbers, starting from 1 and going upwards. We often use these numbers for counting objects.

Some examples of natural numbers are:

• 1
• 2
• 3
• 4
• 5

It's important to note that natural numbers do not include 0 or any negative numbers. These numbers are denoted by the symbol \(\text{\mathbb{N}}\).

Whole Numbers

Whole numbers expand upon natural numbers by including 0. So, the set of whole numbers starts from 0 and includes all the natural numbers.

Here are few examples of whole numbers:

• 0
• 1
• 2
• 3
• 4

Just like natural numbers, whole numbers do not include negative numbers or any fractions.

To sum up, every natural number is a whole number, but not every whole number is a natural number! These are represented by the symbol \( \text{\mathbb{W}} \).

Integers

Integers include whole numbers and their negative counterparts. This means integers can be positive, negative, or zero. Here are some examples:

• -3
• -2
• -1
• 0
• 1
• 2
• 3

Every natural number and whole number is an integer. However, not all integers are natural or whole numbers (because they can be negative). Integers are represented by \(\text{\mathbb{Z}}\).

Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction of two integers. They have non-repeating and non-terminating decimal expansions. Here are some famous examples of irrational numbers:

• \(\pi\) (pi)
• \(\sqrt{2}\) (square root of 2)

These numbers do not fit neatly into the sets of natural, whole, or integers, or even rational numbers. This makes them a unique and interesting group of numbers.

In summary, irrational numbers are those you cannot write as a simple fraction, unlike rational numbers which you can write in the form \(\frac{a}{b}\).