Problem 36 - College Algebra

Question

List all numbers from the given set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, i. real numbers. \(\\{-7,-0.6,0, \sqrt{49}, \sqrt{50}\\}\)

Step-by-Step Solution

Answer

The natural numbers are \{7\} , the whole numbers are \{0,7\}, the integers are \{-7,0,7\}, the rational numbers are \{-7,-0.6,0,7\}, the irrational number is \{\sqrt{50}\} and the real numbers are \{-7,-0.6,0,7,\sqrt{50}\}

Step 1: Identifying Natural Numbers

Natural numbers are counting numbers. Among the given numbers, only \(\sqrt{49}\) which amounts to 7 is a natural number.

Step 2: Identifying Whole Numbers

Whole numbers include all natural numbers and zero. The given numbers which are whole numbers include 0 and \(\sqrt{49}\), which is again 7.

Step 3: Identifying Integers

Integers include all whole numbers and their negatives. Thus, -7 and 0 and \(\sqrt{49}\), which is 7, are integers.

Step 4: Identifying Rational Numbers

Rational numbers are numbers that can be expressed as a fraction. Thus, the numbers that are rational include -7, -0.6, 0, and \(\sqrt{49}\) (which is 7).

Step 5: Identifying Irrational Numbers

\(\sqrt{50}\) is an irrational number. It cannot be expressed as a ratio of two numbers and its decimal representation is non-repeating.

Step 6: Identifying Real Numbers

Real numbers include all rational and irrational numbers. Thus, all the numbers (-7, -0.6, 0, \(\sqrt{49}\), \(\sqrt{50}\)) in the set are real numbers.

Key Concepts

natural numbers

Natural numbers are the numbers we use for counting. These include numbers like 1, 2, 3, and so on. They are positive integers that start from 1 and go on infinitely. In the context of the exercise provided, natural numbers do not include any negative numbers, decimals, or fractions.

• Example: 1, 2, 3, etc.
• Non-example: -1, 0, 0.5

In the given set, \(\{-7, -0.6, 0, \sqrt{49}, \sqrt{50}\}\), the only natural number is \(\sqrt{49}\), which simplifies to 7.

rational numbers

Rational numbers can be written as a fraction of two integers, where the denominator is not zero. This includes numbers like 1/2, 3/4, or an integer like 5 (which can be written as 5/1).

• Example: 1/2, 3, 0
• Features: Terminating or repeating decimal
• Non-example: \(\sqrt{2}\), π

From the set \(\{-7, -0.6, 0, \sqrt{49}, \sqrt{50}\}\), the rational numbers include -7, -0.6, 0, and \(\sqrt{49}\) (since \(\sqrt{49} = 7\), and 7 can be written as 7/1). All these numbers can be expressed as fractions.

irrational numbers

Irrational numbers cannot be written as a simple fraction. Their decimal form is non-repeating and non-terminating, making them unique in this sense. Examples include numbers such as π or \(\sqrt{2}\).

• Characteristics: Non-terminating, non-repeating decimals
• Non-example: 1/3, 2, 5

In the set \(\{-7, -0.6, 0, \sqrt{49}, \sqrt{50}\}\), the only irrational number is \(\sqrt{50}\). This number cannot be expressed as a fraction of two integers.

real numbers

Real numbers encompass all rational and irrational numbers. Essentially, any number that can be found on a standard number line is a real number. This means real numbers include all integers, fractions, and decimals without exception.

• Includes: Rational numbers, irrational numbers
• Examples: 3, -1.5, \(\sqrt{2}\), π

In the set \(\{-7, -0.6, 0, \sqrt{49}, \sqrt{50}\}\), all numbers (-7, -0.6, 0, \(\sqrt{49}\), \(\sqrt{50}\)) qualify as real numbers. Thus, every number you encounter in these exercises is considered a real number, classified either as rational or irrational.