Problem 34 - Introductory Algebra for College Students

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. $$\\{-7,-0 . \overline{6}, 0, \sqrt{49}, \sqrt{50}\\}$$

Step-by-Step Solution

Answer

**Natural numbers:** \(\{0, \sqrt{49}\}\), **Whole Numbers:** \(\{\sqrt{49}, 0\}\), **Integers:** \(\{-7, 0, \sqrt{49}\}\), **Rational numbers:** \(\{-7,-0 . \overline{6}, 0, \sqrt{49}\}\), **Irrational numbers:** \(\{\sqrt{50}\}\), **Real numbers:** \(-7,-0 . \overline{6}, 0, \sqrt{49}, \sqrt{50\).

Step 1: Identify the Natural Numbers

Natural numbers are all non-negative counting numbers. From our provided set \(\{-7,-0 . \overline{6}, 0, \sqrt{49}, \sqrt{50}\}\), the numbers that fall into this category are 0, \(\sqrt{49}\).

Step 2: Identify the Whole Numbers

Whole numbers are natural numbers, including zero. From our set, \(\sqrt{49}\) and 0 are whole numbers.

Step 3: Identify the Integers

Integers are whole numbers that include both positive and negative numbers. So from our set, -7, 0, and \(\sqrt{49}\) are integers.

Step 4: Identify the Rational Numbers

Rational numbers are numbers that can be expressed as fractions (ratios) of integers. From the given set, -7, -0.\(\overline{6}\), 0, and \(\sqrt{49}\) are rational.

Step 5: Identify the Irrational Numbers

Irrational numbers are numbers which cannot be expressed as fractions of integers, but they are real numbers. From the provided set, only \(\sqrt{50}\) is irrational as its decimal representation is non-repeating and non-terminating.

Step 6: Identify the Real Numbers

Real numbers encompass natural, whole, integer, rational, and irrational numbers. Every number in our set \(-7,-0 . \overline{6}, 0, \sqrt{49}, \sqrt{50}\) is a real number.

Key Concepts

Natural Numbers

Natural numbers are often referred to as counting numbers. They are the set of positive numbers beginning from 1 and go on infinitely, like 1, 2, 3, 4, and so on. It’s essential to note that natural numbers do not include zero, negatives, or any form of decimal or fraction.

However, in some mathematical contexts, zero may be counted among the natural numbers. That can be a source of confusion, but for the problem's sake, we are looking for simple counting numbers.

From the original set \( \{-7,-0 . \overline{6}, 0, \sqrt{49}, \sqrt{50}\} \), \(\sqrt{49} \ = 7 \) is the natural number as it falls within this straightforward counting category.

Whole Numbers

Whole numbers expand the category of natural numbers by including zero. They start from 0 and move upwards in the counting sequence, so they are

• 0
• 1
• 2
• 3
• and so on...

They do not include decimals or fractions. In the given set, zero and \(\sqrt{49} = 7\) are indeed whole numbers because they follow the rules of being either zero or positive whole number counts.

Integers

Integers are an extended family of whole numbers. They include all those positive numbers, zero, and their negative counterparts too. Integers are numbers without end in both the positive and negative directions. Examples are -2, -1, 0, 1, 2, continuing onward. Due to their expansive range, integers do not include fractions or decimals.

In the original set \( \{-7, -0.\overline{6}, 0, \sqrt{49}, \sqrt{50}\} \), the integers present are

• -7
• 0
• \(\sqrt{49} = 7\)

Given both positive and negative values, these reflect the integer properties.

Rational Numbers

Rational numbers are very versatile. They can be expressed in the form of a fraction \( \frac{a}{b} \), where \(a\) and \(b\) are integers and \(b eq 0\). This means rational numbers can be fractions, integers, positive or negative, and, importantly, they can have repeating or terminating decimals.

The set \( \{-7, -0.\overline{6}, 0, \sqrt{49}, \sqrt{50} \} \)
provides several rational numbers:

• -7 (it can be seen as \( \frac{-7}{1} \))
• -0.\overline{6} (which equals \( \frac{-2}{3} \) when expressed as a fraction)
• 0 (which is \( \frac{0}{1} \))
• \(\sqrt{49} = 7 \)

Each of these fits the typical criteria of rational numbers.

Irrational Numbers

Irrational numbers are the opposite of rational numbers. They cannot be expressed as a simple fraction and their decimal representations are neither terminating nor repeating. This makes them a fascinating set of numbers that exist between the rational ones on the number line.

A classic example of an irrational number is \( \pi \). In the given set \( \{-7, -0.\overline{6}, 0, \sqrt{49}, \sqrt{50} \} \), \(\sqrt{50}\) is irrational. While \(\sqrt{50}\) can be approximated as about 7.071, it continues in a non-repeating, non-terminating fashion, solidifying its place as irrational.