Problem 66 - Algebra 2

Question

Name the sets of numbers to which each number belongs. 31

Step-by-Step Solution

Answer

31 belongs to the sets of natural numbers, whole numbers, integers, rational numbers, and real numbers.

Step 1: Identify Natural Numbers

Natural numbers are positive integers starting from 1, 2, 3, and so on. Since 31 is a positive integer, it belongs to the set of natural numbers.

Step 2: Identify Whole Numbers

Whole numbers include all natural numbers and the number 0. Since 31 is a positive integer, it also belongs to the set of whole numbers.

Step 3: Identify Integers

Integers include all positive and negative whole numbers, including zero. Therefore, 31 is an integer as it is a positive whole number.

Step 4: Identify Rational Numbers

Rational numbers are numbers that can be expressed as a fraction, where both the numerator and denominator are integers. Since 31 can be expressed as \( \frac{31}{1} \), it is a rational number.

Step 5: Identify Real Numbers

Real numbers include all the numbers that can be found on the number line, encompassing both rational and irrational numbers. Since 31 is a rational number, it also belongs to the set of real numbers.

Key Concepts

Natural Numbers

Natural numbers are the foundation of other number sets and are the simplest to understand. They are the numbers we naturally start to learn as young children when counting objects.

• Natural numbers are positive numbers starting from 1.
• They do not include zero, fractions, or negative numbers.
• The sequence of natural numbers is endless: 1, 2, 3, 4, etc.

Understanding natural numbers is crucial for the basics of arithmetic and is often the first step when classifying numbers. For example, the number 31 is a natural number because it is a positive integer greater than zero.

Whole Numbers

Whole numbers extend the concept of natural numbers by including zero. They provide a slightly broader classification which allows us to perform more mathematical operations.

• Whole numbers are all natural numbers, plus zero.
• They appear in the sequence: 0, 1, 2, 3, 4, etc.
• Like natural numbers, they are non-negative and don't include fractions or decimals.

Whole numbers are useful in everyday situations, such as counting items or assessing quantities, whether they are zero or a certain positive amount. For instance, 31 is a whole number since it is a natural number and naturally fits into this category.

Integers

Integers broaden the scope further by including negative numbers along with positive numbers and zero. This set is very important in mathematics.

• Integers consist of natural numbers, their negative counterparts, and zero.
• They form an infinite set of numbers: ..., -3, -2, -1, 0, 1, 2, 3, ...
• Integers do not include fractions or decimals.

Integers help in considering scenarios such as below freezing temperatures or financial loss, where negative values are meaningful. The number 31 is an integer because it meets the criteria of being a whole number, part of the integer set.

Rational Numbers

Rational numbers represent a much larger class of numbers that includes all integers as well as fractions. They are numbers you can express as a simple fraction.

• Rational numbers are numbers that can be written as a fraction of two integers, where the denominator is not zero.
• They include integers, finite decimals, and repeating decimals.
• Examples are: 1/2, 4, and 0.75, which can be expressed as fractions such as 4/1 or 75/100.

Rational numbers can be found on the number line and are essential in both pure and applied calculations. For instance, 31 is a rational number because it can be expressed as the fraction \( \frac{31}{1} \). This property allows it to belong to the rational number category.