Problem 34 - Introductory and Intermediate Algebra for College Students 4th

Question

List all numbers from the given set that are: \(\mathbf{a}\). natural numbers, \(\mathbf{b}\). whole numbers, \(\mathbf{c}\). integers, \(\mathbf{d}\). rational numbers, \(\mathbf{e}\). irrational numbers, \(\mathbf{f}\), real numbers. $$\\{-7,-0 . \overline{6}, 0, \sqrt{49}, \sqrt{50}\\}$$

Step-by-Step Solution

Answer

The natural numbers are \(\sqrt{49}\) or 7. The whole numbers are 0 and \(\sqrt{49}\) or 7. The integers are -7, 0, and \(\sqrt{49}\) or 7. The rational numbers are -7, -0.\overline{6}, 0, and \(\sqrt{49}\). The irrational number is \(\sqrt{50}\). All numbers in the set are real numbers.

Step 1: Identification of the Number Types in Given Set

Firstly, identify and categorize the numbers in the given set. Take one number in the set at a time and check which category it falls into.

Step 1: Natural Numbers

Natural numbers are counting numbers starting from 1. From our set, only one number is a natural number, which is \(\sqrt{49}\) or 7.

Step 2: Whole Numbers

Whole numbers include all natural numbers along with zero. From our set, the whole numbers are 0 and \(\sqrt{49}\), that is 7.

Step 3: Integers

Integers include all whole numbers and their negatives. From our set, the integers are -7, 0, and \(\sqrt{49}\), that is, 7.

Step 4: Rational Numbers

Rational numbers are numbers that can be expressed as a quotient or a fraction of two integers. From our set, rational numbers are -7, -0.\overline{6}, 0, and \(\sqrt{49}\) or 7.

Step 5: Irrational Numbers

Irrational numbers cannot be expressed as a simple fraction, and their decimal representation never ends or repeats. From our set, the only irrational number is \(\sqrt{50}\).

Step 6: Real Numbers

Real numbers include all rational and irrational numbers. Hence, all the numbers in our given set are real numbers: -7, -0.\overline{6}, 0, \(\sqrt{49}\) or 7, and \(\sqrt{50}\).

Key Concepts

Natural Numbers

Natural numbers are the most basic number type and are often referred to as counting numbers. They start from 1 and continue indefinitely (1, 2, 3, 4, ...). These numbers are simple and used for straightforward counting in everyday life.
Because natural numbers do not include zero or any negative values, they are considered to be positive integers only. Thus in our provided set, \(\sqrt{49}\), which equals 7, is the sole natural number.

Whole Numbers

Whole numbers expand upon natural numbers by including zero. This forms the set of non-negative integers (0, 1, 2, 3, ...). The introduction of zero allows for more comprehensive counting and basic arithmetic operations.
Whole numbers are still limited to positive values and zero, meaning they do not include negative numbers or fractions. Thus, from our set, the numbers 0 and \(\sqrt{49}\), which equals 7, are whole numbers.

Integers

Integers form a larger set than whole numbers, incorporating positive numbers, zero, and negative numbers. This set is represented as (... -3, -2, -1, 0, 1, 2, 3 ...).
Integers are comprehensive for calculations involving debts or losses as well as gains and additions. It includes all whole numbers and their negative counterparts. In the given set, -7, 0, and \(\sqrt{49}\) (or 7) are integers.

Rational Numbers

Rational numbers are numbers that can be expressed as a quotient or division of two integers, where the denominator is not zero. For example, 1/2, -3/4, and 5 (which can be written as 5/1) are rational numbers.
These numbers include all integers and fractions with finite or repeating decimals. From the given set, -7, -0.\overline{6}, 0, and \(\sqrt{49}\) (7) are rational, since they can all be expressed as a fraction of integers.

Irrational Numbers

Irrational numbers cannot be expressed as a simple fraction. Their decimal expansions are non-terminating and non-repeating, making them unique compared to rational numbers.
Famous examples include \(\pi\) and \(e\), but any number that cannot be perfectly divided into two integers is classified here. In our set, the only irrational number is \(\sqrt{50}\), because it continues infinitely without repeating.

Real Numbers

Real numbers include all rational and irrational numbers. This is the broadest category that encompasses essentially every number that can be found on the number line.
Real numbers can represent any quantity and include decimals, fractions, and square roots that are not imaginary. Thus, from the given set, -7, -0.\overline{6}, 0, \(\sqrt{49}\), and \(\sqrt{50}\) are real numbers, as they cover both types.