Problem 14 - Fundamentals of Mathematics

Question

For the following 8 problems, next to each real number, note all collections to which it belongs by writing \(N\) for natural number, \(W\) for whole number, or \(Z\) for integer. Some numbers may belong to more than one collection. -805

Step-by-Step Solution

Answer

-805 belongs to Integers \( Z \).

Step 1: Identify the Given Number

The number given is -805. We need to analyze this number to determine which category or categories it belongs to: Natural numbers, Whole numbers, and Integers.

Step 2: Define Each Number Set

First, we'll define each category: - **Natural numbers (N):** These include all positive integers starting from 1, such as 1, 2, 3, etc. - **Whole numbers (W):** These include all natural numbers and zero, such as 0, 1, 2, 3, etc. - **Integers (Z):** These include all whole numbers and their negative counterparts, such as ..., -3, -2, -1, 0, 1, 2, 3, etc.

Step 3: Categorize the Number -805

Now, we need to categorize -805 according to the definitions: - **Natural numbers (N):** -805 is not a positive integer, so it does not belong here. - **Whole numbers (W):** -805 is not a non-negative integer (0 or positive), so it does not belong here. - **Integers (Z):** -805 is a negative integer, so it belongs in this set.

Key Concepts

Natural Numbers

Natural numbers are the simplest set of numbers used for counting and ordering.
They begin from 1 and continue infinitely.
This group includes numbers like 1, 2, 3, and so on.

• Natural numbers are always positive.
• They do not include 0 or any negative numbers.

These numbers are often used in everyday life, like when counting objects or measuring quantities.
For instance, if you are counting apples, 1, 2, 3... would be natural numbers.
In mathematical notation, this set is often represented by the letter \(N\).
When thinking of natural numbers, picture a simple number line starting at 1 and going off to infinity.
Since -805 is negative, it cannot be part of the natural numbers, as they're strictly positive.

Whole Numbers

Whole numbers expand upon the natural numbers by including zero.
They start from 0 and extend to infinity just like natural numbers.

• Whole numbers are non-negative.
• This set includes 0, 1, 2, 3, and so forth.

Zero is part of this set because it serves as a foundational number in many mathematical scenarios.
These numbers, represented by the letter \(W\), help in counting things that might originate from zero,
such as starting a stopwatch at zero seconds.
So, while whole numbers do include zero, they do not include any negative numbers.
Because -805 is negative, it cannot belong to the set of whole numbers.

Integers

Integers form a broader category that includes all whole numbers and their negatives.
This set stretches in both positive and negative directions on the number line.

• Integers encompass ..., -3, -2, -1, 0, 1, 2, 3, and so on.
• They include every natural and whole number, as well as their negative counterparts.

The set of integers, denoted by \(Z\), provides a complete picture of whole numbers with positive, negative, and zero included.
These numbers come into play when measuring changes or balances, such as debts or temperature drops.
For the number -805, it falls comfortably within the integers category because it's a negative whole number.