Problem 35 - 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. $$\left\\{-9,-\frac{4}{5}, 0,0.25, \sqrt{3}, 9.2, \sqrt{100}\right\\}$$

Step-by-Step Solution

Answer

a. Natural numbers: \(\sqrt{100}\) (i.e., 10) b. Whole numbers: 0, \(\sqrt{100}\) (i.e., 10) c. Integers: -9, 0, \(\sqrt{100}\) (i.e., 10), d. Rational numbers: -9, -4/5, 0, 0.25, \(\sqrt{100}\) (i.e., 10) e. Irrational numbers: \(\sqrt{3}\), f. All listed numbers are real numbers.

Step 1: Identify Natural Numbers

Natural numbers are non-negative integers, starting from 1. In our set the natural number is \(\sqrt{100}\), which equals 10.

Step 2: Identify Whole numbers

Whole numbers are natural numbers including zero. From our list, we find \(0\) and \(\sqrt{100}\) (or \(10\)) which are whole numbers.

Step 3: Identify Integers

Integers are numbers without fractional or decimal parts, including negative numbers. In our list, \(0\), \(-9\), and \(\sqrt{100}\) are integers.

Step 4: Identify Rational Numbers

Rational numbers can be expressed as a fraction of two integers. This includes integers themselves since they can be expressed as a fraction with denominator 1. From our list, \(0\), \(-9\), \(-\frac{4}{5}\), \(0.25\), and \(\sqrt{100}\) are rational numbers.

Step 5: Identify Irrational Numbers

Irrational numbers cannot be expressed as a ratio of two integers, and their decimal representation is non-repeating and non-terminating. The only irrational number in our list is \(\sqrt{3}\).

Step 6: Identify Real Numbers

Real numbers include all rational and irrational numbers. Therefore, the entire set are real numbers.

Key Concepts

Natural Numbers

Natural numbers are the basic counting numbers that we use in everyday life. They start from 1 and go on indefinitely: 1, 2, 3, and so on.
It's important to note that these numbers do not include zero or any negative numbers.
In our given set, the only number that is a natural number is \( \sqrt{100} \).
This is because \( \sqrt{100} \) equals 10, which is a natural number.
Natural numbers are the simplest and most familiar type of numbers.
Here are a few characteristics of natural numbers:

• They do not include fractions or decimals.
• They represent quantities in counting and ordering.
• They are infinite; there is no largest natural number.

Whole Numbers

Whole numbers are similar to natural numbers, but they also include zero. This means whole numbers are all natural numbers plus zero.
In our example, the whole numbers found in the set are \(0\) and \( \sqrt{100} \) (or 10).
Whole numbers add the concept of nothing—or zero—to our counting system.
Some important points about whole numbers include:

• Like natural numbers, they do not include fractions or decimals.
• They range from 0 to infinity.
• They include every natural number.

Integers

Integers expand on the concept of whole numbers by including negative numbers as well.
They form a set of numbers that includes negatives, zero, and positive whole numbers.
In our number set \(-9\), \(0\), and \( \sqrt{100} \) (or 10) are integers.
Integers are useful for representing situations with gains and losses, or above/below a certain amount.
Key features of integers include:

• They do not include fractional or decimal components.
• They are arranged in an endless sequence, extending in both positive and negative directions.
• They satisfy the concept of additive inverses (e.g., \(5 + (-5) = 0\)).

Rational Numbers

Rational numbers are a more comprehensive system.
They include numbers that can be expressed in the form of a fraction, where both the numerator and the denominator are integers, with the denominator not being zero.
In the given set, the rational numbers are \(0\), \(-9\), \(-\frac{4}{5}\), \(0.25\), and \( \sqrt{100} \).
Rational numbers are prevalent in measuring and calculating proportions.
Key points about rational numbers:

• They include integers since any integer \(a\) can be expressed as \(a/1\).
• They can have repeating or terminating decimal representations.
• They can represent ratios of two quantities.

Irrational Numbers

Irrational numbers are numbers that cannot be written as a simple fraction.
Their decimal expansions are non-repeating and non-terminating.
In our set, \(\sqrt{3}\) is the only irrational number, as it cannot be neatly expressed as a fraction.
Irrational numbers are crucial for understanding precise measurements, like the lengths of diagonals or the value of \(\pi\).
Here are some important attributes:

• They exist between rational numbers on the number line.
• They are often expressed with radical signs, such as \(\sqrt{2}\), \(\pi\).
• The sum or product of an irrational number with a rational number is generally irrational.