Problem 4 - Fundamentals of Mathematics

Question

Next to each real number, note all subsets of the real numbers to which it belongs by writing \(N\) for natural numbers, \(W\) for whole numbers, or \(Z\) for integers. Some numbers may belong to more than one subset. $$ 1 $$

Step-by-Step Solution

Answer

The number 1 belongs to the subsets N, W, and Z.

Step 1: Identify the Number

The number given is "1". We need to determine to which subsets of the real numbers this belongs.

Step 2: Consider Natural Numbers

Natural numbers ( N ) are the positive integers starting from 1 and going upwards. Since 1 is traditionally included as the first natural number, "1" belongs to this subset.

Step 3: Whole Numbers Check

Whole numbers ( W ) consist of all the natural numbers along with zero (0). Consequently, "1" is a whole number since it includes all natural numbers.

Step 4: Integer Verification

Integers ( Z ) are all the whole numbers as well as their negative counterparts. Since "1" is a positive whole number, it is definitely categorized as an integer.

Key Concepts

Natural Numbers

Within the world of mathematics, natural numbers are often the first set of numbers we encounter. They are the counting numbers we use in everyday life. When you count objects like apples or days of the week, you are using natural numbers.

• Natural numbers start from 1 and go infinitely upwards: 1, 2, 3, 4, etc.
• No fractions, decimals, or negative numbers are included.
• This set does not include zero.

In formal mathematical notation, the set of natural numbers is often represented by the symbol \(N\). It's important to note that some definitions include 0 as a natural number, but most traditional definitions start from 1. Always consider the context of the problem to determine which definition applies.
By understanding these simple criteria, you will easily identify whether a number belongs to the set of natural numbers.

Whole Numbers

Whole numbers expand upon natural numbers by including zero. This slight modification creates a new set that is tremendously useful for various applications in mathematics. Think of whole numbers as the natural numbers plus zero.

• This set includes 0, 1, 2, 3, 4, and so on.
• There are no negative numbers allowed in this set.
• Fractions and decimals are not considered whole numbers.

Mathematicians use the symbol \(W\) to denote the set of whole numbers. This set is particularly relatable when considering things like scores or quantities that naturally start from zero. The inclusion of zero sets whole numbers apart from natural numbers and enables us to effectively measure "nothing" in quantitative terms.

Integers

Integers encompass a broader range than whole or natural numbers by including both the positive and negative counterparts of whole numbers. This makes the set of integers versatile and comprehensive.

• Integers include all whole numbers, such as 0, 1, 2, 3, as well as their negative counterparts, like -1, -2, -3.
• Fractions and decimals are excluded from this set.
• It creates a balanced spectrum from negative infinity to positive infinity.

The set of integers is represented by the symbol \(Z\). This notation comes from the German word "Zahlen," meaning "numbers." The ability to navigate both directions on the number line makes integers particularly useful in various mathematical operations and real-world problem-solving scenarios. Recognizing whether a number is an integer is crucial, as it clarifies the number's placement within the wider set of real numbers.