Problem 47 - Beginning and Intermediate Algebra

Question

Tell which set or sets each number belongs to: natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers. See Example 5. $$ \frac{2}{3} $$

Step-by-Step Solution

Answer

\( \frac{2}{3} \) is a rational and real number.

Step 1: Identify the Number Type

The given number is \( \frac{2}{3} \), which is a fraction. This immediately tells us that it can be classified as a rational number, as rational numbers consist of any number that can be expressed as a fraction \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b eq 0 \).

Step 2: Determine Membership in Other Sets

Since \( \frac{2}{3} \) is neither a natural number nor a whole number, it cannot be classified under these sets. Natural numbers start at 1 and increase as whole numbers start at 0, both consisting of only integers. An integer is a whole number without a fractional part, eliminating \( \frac{2}{3} \) also from integer classification.

Step 3: Check for Rational and Real Number Inclusion

Since \( \frac{2}{3} \) can be expressed as a fraction with integer numerator and a non-zero integer denominator, it is confirmed as a rational number. All rational numbers are also part of the real number system. Thus, \( \frac{2}{3} \) is a real number as well.

Step 4: Verify Exclusion from Irrational Numbers

An irrational number cannot be expressed as a fraction. Since \( \frac{2}{3} \) can be expressed as a fraction, it is not irrational.

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction where both the numerator and the denominator are integers, and the denominator is not zero. This means any number that can be written in the form \( \frac{a}{b} \) falls under the category of rational numbers. For instance, \( \frac{2}{3} \) is a rational number because it has an integer numerator (2) and an integer denominator (3), with 3 not being zero.

• Rational numbers include all integers since they can be written with a denominator of 1, like \( \frac{7}{1} \).
• They also encompass finite and repeating decimals, such as 0.333... which equals \( \frac{1}{3} \).

Ultimately, the key characteristic of rational numbers is their ability to be represented as a simple fraction.

Real Numbers

Real numbers are the broadest category of numbers we typically use in everyday arithmetic. They include all the rational numbers, like fractions and integers, as well as all irrational numbers, which we will dive into later.

• Real numbers include both the negative and positive numbers, along with zero.
• They can also be visualized on a number line, stretching from negative infinity to positive infinity.

The set of real numbers forms a continuum, ensuring that between any two distinct real numbers, there is always another real number. Thus, \( \frac{2}{3} \) is included in the real numbers because it is a rational number and all rational numbers form part of the real numbers.

Natural Numbers

Natural numbers are the set of positive integers that we generally begin counting from. They start at 1 and continue infinitely (1, 2, 3, 4, ...). These numbers do not include zero, fractions, decimals, or negatives, positioning them as the most basic set of numbers meant for simple counting or ordering.

• Natural numbers are exclusively positive.
• They are often used for counting physical objects and are integral to basic arithmetic operations.

Since \( \frac{2}{3} \) is a fraction and not a positive whole number, it does not qualify as a natural number.

Whole Numbers

Whole numbers represent the set of non-negative integers, beginning from zero and extending upwards (0, 1, 2, 3, 4, ...). They incorporate all natural numbers along with zero, making them slightly more inclusive.

• Whole numbers start at zero.
• They too do not include fractions, decimals, or negative values.

This means \( \frac{2}{3} \) is not a whole number since whole numbers must be complete without any fractional or decimal parts.

Integers

Integers include all whole numbers along with their negative counterparts (..., -3, -2, -1, 0, 1, 2, 3, ...). They are a comprehensive set containing zero, positives, and negatives, yet notably excluding any fractions or decimal numbers.

• Integers can be negative or positive, and also include zero.
• They do not accommodate fractional or decimal numbers.

Consequently, \( \frac{2}{3} \) does not fit into the integer category because it exists as a fraction and not a pure whole number or its negative.

Irrational Numbers

Irrational numbers are those numbers that cannot be penned as a simple fraction or ratio of two integers. They are the numbers whose decimal forms are non-terminating and non-repeating. Examples include \( \sqrt{2} \), \( \pi \), and \( e \). These numbers are not able to be expressed in the simple form \( \frac{a}{b} \), where both \( a \) and \( b \) are integers.

• An irrational decimal never ends or settles into a permanent repeating pattern.
• They fit into the real number category, forming part of the continuum.

Since \( \frac{2}{3} \) can indeed be simplified into a fraction, it cannot be considered an irrational number.