Problem 133 - Introductory Algebra for College Students

Question

Give an example of an integer that is not a natural number. (Section \(1.3 ;\) Example 5 )

Step-by-Step Solution

Answer

-1 is an example of an integer that is not a natural number.

Step 1: Understand the categories of numbers

There are different types of numbers. Key categories for this task are integers and natural numbers. Integers include all whole numbers both positive and negative, and also the number zero. Natural numbers, on the other hand, are positive integers (1, 2, 3, ...) and do not include zero or any negative integers.

Step 2: Identify an integer that is not a natural number

Having understood that natural numbers are positive integers only, one can easily identify an integer that is not a natural number. This could be any negative integer or even the number zero. Let's opt for '-1' as an integer that is not a natural number.

Key Concepts

Integers

Integers are a fundamental concept in mathematics, representing whole numbers without any fractional or decimal components. They encompass a wide range of numbers:

• Positive integers (e.g., 1, 2, 3, ...)
• Negative integers (e.g., -1, -2, -3, ...)
• Zero (0)

The ability to include both positive and negative numbers, along with zero, makes integers crucial for various mathematical operations and real-world applications. Integers are denoted by the symbol \( \mathbb{Z} \). This inclusive set allowing for negatives and zero is what differentiates integers from natural numbers.

Natural Numbers

Natural numbers are a specific subset of integers that consist exclusively of positive whole numbers. They start from 1 and go upwards infinitely (1, 2, 3, ...). Natural numbers are commonly used when counting objects or when operating in mathematical scenarios that require only positive quantities.

It's important to note that natural numbers do not include zero or any negative numbers. There is debate among mathematicians regarding whether zero should be considered a natural number, but typically, in most elementary contexts, natural numbers start from 1. Natural numbers are often represented by the symbol \( \mathbb{N} \).

Negative Numbers

Negative numbers are all the integers less than zero. They are found on the left side of zero on the number line (e.g., -1, -2, -3, ...). Negative numbers have important uses in various fields such as:

• Finance, to indicate debts or losses
• Thermometers, to show temperatures below zero
• Science, to represent quantities like velocity or position going in the opposite direction

Negative numbers enrich the number system by allowing for the expression of loss, deficit, or direction. While included in the set of integers, they are not part of the natural numbers, as natural numbers do not accommodate any negative values.