Problem 93 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether natural numbers, whole numbers, integers, rational numbers, or all real numbers are appropriate for each situation. The number of pets a person has

Step-by-Step Solution

Answer

The most appropriate set of numbers for the given situation is the set of Whole numbers.

Step 1: Understand the different kinds of numbers

Natural numbers are numbers that start from 1 and go to infinity. They are the most basic numbers, and they represent quantities of whole items. Whole numbers include all natural numbers and also 0. Integers include all whole numbers, but also include the negatives of all natural numbers. Rational numbers are any numbers that can be expressed as a fraction of two integers. Real numbers include all rational numbers but also include irrational numbers, which cannot be expressed as a fraction of integers.

Step 2: Analyze the question

In the given situation, the number of pets a person can have is being considered. Can a person have a negative number of pets, or a fractional number of pets? No, this doesn't make sense in the real world. A person can have zero pets, or any whole number of pets, but nothing else.

Step 3: Conclusion

The most appropriate kinds of numbers for expressing the number of pets a person has are the Whole numbers. These include 0 (representing no pets) and all the natural numbers (representing any specific count of pets).

Key Concepts

Natural Numbers

Natural numbers are a simple yet essential part of mathematics. They are the numbers we begin counting with, starting from 1 and continuing infinitely onward.
These are numbers like 1, 2, 3, 4, and so forth.

• Natural numbers do not include zero.
• They also never include negative numbers or decimals.

These numbers are used in everyday scenarios, like counting apples or people. They help us understand quantities in the simplest form. For example, when you count how many apples you have, you use natural numbers. Simply, these are straightforward and practical in many daily contexts.

Whole Numbers

Whole numbers expand on natural numbers by including zero.
They are similar in many ways, encompassing all natural numbers plus the number zero.

• Whole numbers are 0, 1, 2, 3, and so on.
• They do not include negative numbers or fractions.

Whole numbers are often used when zero is a possible option. Think about counting items, like eggs or books, where having none is a possibility. Whole numbers portray this situation perfectly. They provide a complete picture by acknowledging the presence of zero, which natural numbers do not.

Integer Numbers

Integer numbers build upon whole numbers by adding negative values.
They include zero, the natural numbers, and their negative counterparts.

• Examples of integer numbers are -3, -2, -1, 0, 1, 2, 3, etc.
• Integers do not contain fractions or decimals.

Integers are significant in scenarios where negative values might appear, such as temperatures or bank balances. They help us explore and work with numbers when simply having no or positive numbers isn’t enough. With integers, we can express a broader range of values, opening new possibilities in mathematical situations.

Real Numbers

Real numbers include just about every number you can think of.
They encompass all integers, as well as numbers with fractions and decimals.

• Real numbers feature rational numbers (like 3/4 or 0.75) and irrational numbers (like \( \sqrt{2} \)).
• Irrational numbers cannot be exactly expressed as simple fractions.

In essence, real numbers represent the continuum of points on the number line. They are practical for calculations involving both exact and approximate measurements, such as distance or price. Real numbers are crucial for progressing in higher levels of math and science, allowing us to tackle complex problems with ease.