Problem 49 - Beginning and Intermediate Algebra

Question

Tell which set or sets each number belongs to: natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers. See Example 5. $$ -\sqrt{5} $$

Step-by-Step Solution

Answer

-\sqrt{5} is an irrational and real number.

Step 1: Determine if the Number is a Natural Number

Natural numbers are positive integers starting from 1. Since \(-\sqrt{5}\) is not a positive whole number, it is not a natural number.

Step 2: Determine if the Number is a Whole Number

Whole numbers include all natural numbers and zero. \(-\sqrt{5}\) is negative and not an integer, thus it is not a whole number.

Step 3: Determine if the Number is an Integer

Integers are all whole numbers, including negatives, but they must be whole. \(-\sqrt{5}\) is not whole since it's a square root, making it not an integer.

Step 4: Determine if the Number is a Rational Number

Rational numbers are numbers that can be expressed as a fraction of two integers. The square root of a non-perfect square, \(\sqrt{5}\), is irrational, hence \(-\sqrt{5}\) also is not rational.

Step 5: Determine if the Number is an Irrational Number

Irrational numbers cannot be expressed as fractions of two integers and have non-repeating decimals. Since \(\sqrt{5}\) is not a perfect square, it is irrational, making \(-\sqrt{5}\) also irrational.

Step 6: Determine if the Number is a Real Number

Real numbers include all rational and irrational numbers. Since \(-\sqrt{5}\) is irrational, it falls into the category of real numbers.

Key Concepts

Natural Numbers

Natural numbers are the basic counting numbers that we use in everyday life. Think of them as the numbers you would naturally use to count objects, like counting apples in a basket. They start from 1 and go up endlessly: 1, 2, 3, 4, and so on. It's important to remember that natural numbers are always positive and do not include zero.
They are part of a larger set known as whole numbers when zero is included.

• Start from 1
• Always positive
• Used for counting
• Do not include fractions or decimals

When analyzing numbers like \(-\sqrt{5}\), it becomes clear that since it is neither positive nor a whole number, it cannot be a natural number.

Irrational Numbers

Irrational numbers are interesting as they can't be written as a simple fraction of two integers. They have a unique quality where their decimal expansion goes on forever without repeating. Numbers like \(\pi\) and \(\sqrt{2}\) are common examples of irrational numbers. When dealing with irrational numbers:

• Decimals do not terminate or repeat
• Cannot be expressed as a simple fraction
• Include numbers like \(\sqrt{5}\) and \(\pi\)

For the given number \(-\sqrt{5}\), understanding its nature as irrational comes from recognizing that 5 is not a perfect square. This fact indicates that \(\sqrt{5}\) is irrational, making \(-\sqrt{5}\) irrational as well.

Real Numbers

The set of real numbers is vast and important in mathematics. It includes every number you can think of on the number line. Real numbers encompass both rational numbers (like fractions and integers) and irrational numbers. Essentially, any number that doesn't involve the imaginary unit \(i\) is considered real.
Here's what you need to know:

• Includes both rational and irrational numbers
• Found on a continuous number line
• Excludes imaginary numbers

Since \(-\sqrt{5}\) is an irrational number, its presence in the real numbers set is automatic. This universality makes real numbers fundamental in both everyday calculations and complex mathematics.