Problem 18 - Algebra 2

Question

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

Step-by-Step Solution

Answer

-31 is an integer, a rational number, and a real number.

Step 1: Identify if it's a whole number

The number given is -31. Whole numbers include all natural numbers and zero, but not negative numbers. Therefore, -31 is not a whole number.

Step 2: Check if it's an integer

Integers include all positive and negative whole numbers and zero. Since -31 is a negative whole number, it is an integer.

Step 3: Determine if it's a rational number

A rational number can be expressed as a fraction, where both the numerator and denominator are integers, and the denominator is not zero. -31 can be written as \(-31/1\), so it is a rational number.

Step 4: Establish if it's a real number

Real numbers include all numbers that can be found on the number line, including both rational and irrational numbers. Since -31 is a rational number, it is also a real number.

Key Concepts

Whole Numbers

Whole numbers are the basic numbers we use for counting. They include all natural numbers, which start from 1 and go upwards, and also the number 0. Whole numbers, importantly, do not accommodate any negative numbers or fractions. So numbers like 0, 1, 2, 3, 4, etc., fall under this category. Additionally, whole numbers are described as non-negative integers without any decimal or fractional components. In the case of the exercise, -31 is a negative number, and that's why it doesn't belong to the set of whole numbers.

Integers

Integers broaden the horizon from whole numbers to include both negative and positive numbers, along with zero. An easy way to view them is as the set that consists of numbers like -3, -2, -1, 0, 1, 2, 3, continuing infinitely in both directions. Integers do not consist of fractions or decimal numbers. An integer can be described as a "whole" number, but it can also be a negative number. So, -31 fits perfectly in this set, as it is a whole number with a negative sign attached to it.

Rational Numbers

Rational numbers are numbers that can be expressed as a ratio or fraction \( \frac{a}{b} \), where both \( a \) and \( b \) are integers, and \( b eq 0 \). This set of numbers includes integers, fractions, and finite decimals too. The essence of rational numbers is that they can be pinpointed precisely on the number line. For example, \( -31 \) can be represented as \( \frac{-31}{1} \), which proves it is a rational number. Thus, any integer is also a rational number, because it can be expressed as a fraction with a denominator of 1.

Real Numbers

Real numbers include all possible numbers you can find on the number line. They encompass both rational and irrational numbers (like \( \pi \) and \( \sqrt{2} \)), which means they cover all the possible number sets there are—integers, whole numbers, rational numbers, and others like irrational numbers. Real numbers account for every number you can think of—whether it can be written as a fraction or not, finite or infinite, and positive or negative. Since \( -31 \) is a rational number, it naturally falls into the category of real numbers. Hence, if it exists on the number line, it's a real number.