Problem 13 - 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. -7

Step-by-Step Solution

Answer

-7 belongs to the set of integers (Z).

Step 1: Understand the Definition of Each Set

Before classifying the number, we need to understand what each set represents: \(N\) for natural numbers includes \(\{1, 2, 3, \ldots\}\), \(W\) for whole numbers includes \(\{0, 1, 2, 3, \ldots\}\), and \(Z\) for integers includes all positive and negative whole numbers and zero (\(\{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}\)).

Step 2: Analyze the Given Number

The given number is \(-7\). It is not positive, so it cannot be a natural number \(N\) or a whole number \(W\).

Step 3: Determine the Collection Membership

Since \(-7\) is a negative whole number, it belongs to the set of integers \(Z\).

Step 4: Conclusion

After analyzing the properties of \(-7\) and comparing them to the definitions of each set, we conclude that \(-7\) belongs to the set \(Z\).

Key Concepts

Natural Numbers

Natural numbers are the basic building blocks of our numerical system. They start from 1 and go on infinitely: 1, 2, 3, and so on. These numbers are used for counting objects, like apples or books, which is why they are sometimes called 'counting numbers'.
They don’t include zero or negative numbers. Think of natural numbers as the numbers you use when you're counting your fingers or how many steps you've taken.

• The set of natural numbers is typically denoted by \(N\).
• The main characteristic is that they're always positive.
• They do not include fractions or decimals, just whole numbers.

So whenever you're working with a positive number without fractions or decimals, you're likely dealing with a natural number.

Whole Numbers

Whole numbers take the idea of natural numbers a step further. Imagine if you start counting naturally, but someone starts from zero instead of one. That's the idea of whole numbers. They include all natural numbers such as 1, 2, 3, etc., but they also include 0.

• Whole numbers are represented by the set \(W\).
• They're similar to natural numbers, but with the important inclusion of zero.
• There are no negative numbers, fractions, or decimals in the set of whole numbers.

When you consider the possibility of having nothing (zero of something), this is when whole numbers come into play. They're the numbers you might use to count students sitting in a classroom, where zero students is a possibility.

Integers

Integers widen our number scope even further than whole numbers. They include all the whole numbers and their negative counterparts. In simple terms, integers encompass positive whole numbers, zero, and negative whole numbers like -1, -2, -3, and so forth.

• This set is denoted by \(Z\).
• Integers include zero, positive, and negative numbers, but like natural and whole numbers, there are still no fractions or decimals here.
• For example, -7 is an integer because it fits all the criteria: it's a whole number but on the negative side of zero.

Integers are used for expressing situations where you need to represent quantities less than zero, like temperatures dropping below freezing or debts. If you're working within the realm where negative values have meaning, you're likely dealing with integers.