Problem 37 - College Algebra Essentials

Question

List all numbers from the given set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, f. real numbers. $$\left\\{-11,-\frac{5}{6}, 0,0.75, \sqrt{5}, \pi, \sqrt{64}\right\\}$$

Step-by-Step Solution

Answer

a. Natural numbers: \( \sqrt{64} =8 \)\n b. Whole numbers: 0, \( \sqrt{64} =8 \)\n c. Integers: -11, 0, \( \sqrt{64} =8 \)\n d. Rational numbers: -11, -5/6, 0, 0.75, \( \sqrt{64} =8 \)\n e. Irrational numbers: \( \sqrt{5} \), \( \pi \)\n f. Real numbers: -11, -5/6, 0, 0.75, \( \sqrt{5} \), \( \pi \), \( \sqrt{64} =8 \)

Step 1: Identifying Natural Numbers

Natural numbers are positive integers. Thus from the given set, the only natural number is \( \sqrt{64} =8 \).

Step 2: Identifying Whole Numbers

Whole numbers are non-negative integers. Therefore, \(0\) and \( \sqrt{64} =8 \) are the whole numbers from the set.

Step 3: Identifying Integers

Integers are all natural numbers including negative numbers and the number zero. So -11, 0, and \( \sqrt{64} =8 \) are integers.

Step 4: Identifying Rational Numbers

Rational numbers are numbers that can be written as a fraction and this includes integers. So -11, -5/6, 0, 0.75 (which equals to 3/4 when represented as a fraction), and \( \sqrt{64} =8 \) are rational numbers.

Step 5: Identifying Irrational Numbers

Irrational numbers cannot be expressed as fractions and they both non-terminating and non-repeating. \( \sqrt{5} \) and \( \pi \) are irrational numbers in the given set.

Step 6: Identifying Real Numbers

Real numbers are all numbers that can be found on the number line. This includes both rational and irrational numbers. So, all the numbers in the set are real numbers.

Key Concepts

Natural Numbers

The simplest numerical classification we encounter in mathematics is that of natural numbers. These are the numbers we use for counting and order in our daily lives, starting from 1 and going onwards indefinitely—2, 3, 4, and so on. They are the backbone of basic arithmetic, being all the positive integers with no fractional or decimal parts. In the context of the exercise, the value of \( \text{\sqrt{64}} \) simplifies to 8, which is a natural number.

Whole Numbers

Whole numbers can be thought of as the natural numbers with the addition of zero. This gives us a set starting from 0, then 1, 2, 3, and so on. Zero is the aspect that differentiates whole numbers from natural numbers. It represents an absence or a neutral position in the number line. From the exercise, the numbers \(0\) and \(8\) (since \( \text{\sqrt{64}} = 8\)) fall into the category of whole numbers.

Integers

Expanding our horizon further, we include the negatives of natural numbers along with the whole numbers to get integers. Integers encompass the whole spectrum of whole numbers and their negative counterparts, making them a symmetrical set around zero. They can be positive, negative, or zero. Specifically referring to the exercise given, \(-11\), \(0\), and \(8\) are the integers from the set.

Rational Numbers

Rational numbers are the fractions and decimals that represent a whole or a part of something. Technically, they are numbers that can be expressed as the ratio of two integers (hence the term 'rational'). This includes numbers that can be written as a finite or repeating decimal. From the provided set, \(-11\), \(-\frac{5}{6}\), \(0\), \(0.75\), and \(8\) are counted as rational numbers because they can all be represented as a fraction.

Irrational Numbers

Irrational numbers hold a mystique in the numerical world because they cannot be represented as simple fractions. Their decimal expansions are infinite and non-repeating. These numbers often arise from natural processes or geometric calculations, such as the value of \( \text{\pi} \) or the square root of a non-perfect square. In the set from the exercise, \( \text{\sqrt{5}} \) and \( \text{\pi} \) are the numbers that fit into the category of irrational numbers.

Real Numbers

The concept of real numbers encapsulates all the numbers we've discussed so far. It is the complete set that includes all the rational and irrational numbers, forming a continuous line that represents every point in space that a number can occupy. Essentially, every number that has a place on the number line is a real number. Therefore, in the context of the exercise, every number in the set (-11, -\frac{5}{6}, 0, 0.75, \( \text{\sqrt{5}} \), \( \text{\pi} \), 8) constitutes as a real number.