Problem 6 - 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. A number like \(\sqrt{5},\) which cannot be written precisely in fraction notation or decimal notation, is an example of a(n)_________.

Step-by-Step Solution

Answer

irrational number

Step 1: Identify the given number type

The given number is \( \sqrt{5} \). Notice that this number involves a square root of a non-perfect square.

Step 2: Classify the number

Determine the type of \( \sqrt{5} \) by checking if it can be written in fraction or decimal notation precisely. Since it cannot be expressed as a fraction of two integers, it is not a rational number.

Step 3: Choose the correct term

A number that cannot be written exactly as a fraction or a terminating or repeating decimal is known as an irrational number.

Step 4: Fill in the blank

The blank should be filled with the term 'irrational number'.

Key Concepts

Square Roots

Square roots are numbers that, when multiplied by themselves, yield the original number. For example, the square root of 9 is 3 because \(3 \times 3 = 9\).
There are two kinds of numbers when it comes to square roots:

• Perfect squares
• Non-perfect squares

Let's dive deeper into these categories.

Non-Perfect Squares

Non-perfect squares are numbers that do not have an exact square root in integer form. Examples include numbers like 2, 3, 5, or 7. Their square roots result in decimal numbers that go on forever without repeating. For example, \(\sqrt{5}\) approximates to 2.236067977..., making it hard to represent it as a fraction or a simple decimal.
These types of numbers are considered irrational numbers because they cannot be expressed as a ratio of two integers or a repeating or terminating decimal.

Classification of Numbers

Numbers can be classified into several categories. Here are some of the key types:

• Natural numbers: These are the simplest numbers used for counting, starting from 1, 2, 3, and so on.
• Whole numbers: These include all natural numbers and zero. For example: 0, 1, 2, 3, etc.
• Integers: These consist of whole numbers and their negative counterparts, such as ... , -3, -2, -1, 0, 1, 2, 3, ...
• Rational numbers: These are numbers that can be expressed as the fraction of two integers, where the denominator is not zero. For example, \(\frac{1}{2}\) and 4 (which can be written as \(\frac{4}{1}\)).
• Irrational numbers: These numbers cannot be written as fractions, and their decimal expansions are non-terminating and non-repeating. \(\pi\) and \(\sqrt{5}\) are examples of irrational numbers.