Problem 2 - 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. $$ -14 $$

Step-by-Step Solution

Answer

-14 belongs to integers, \(Z\).

Step 1: Identify the Number

The number given is \(-14\). It is a real number, and we need to determine which subsets of real numbers it belongs to.

Step 2: Check if Natural Number

Natural numbers are positive whole numbers starting from 1 (1, 2, 3,...). Since \(-14\) is negative, it does not belong to natural numbers, represented by \(N\).

Step 3: Check if Whole Number

Whole numbers are all natural numbers including zero (0, 1, 2..). Since \(-14\) is negative, it is not a whole number, thus it does not belong to \(W\).

Step 4: Check if Integer

Integers include all whole numbers and their negative counterparts (..., -3, -2, -1, 0, 1, 2, 3...). Since \(-14\) is a negative whole number, it belongs to the set of integers, represented by \(Z\).

Key Concepts

Natural Numbers

Natural numbers are the counting numbers that you start using naturally when you begin to count. These numbers begin at 1, continue to 2, 3, 4, and go on infinitely in a positive direction.
Natural numbers are simple and straightforward, great for counting items. However, it's essential to remember that they do not include zero or any negative numbers, so a number like \(-14\) would not be considered a natural number. Natural numbers are denoted by the letter 'N'.

• Positive and non-fractional numbers
• Starts from 1 and goes on indefinitely
• No inclusion of zero or negative numbers

Understanding this concept is step one to grasping the broader category of real numbers.

Whole Numbers

Whole numbers extend the sequence of natural numbers by including zero. So, these numbers start from 0, 1, 2, 3, and so on. The inclusion of zero is the main element that separates whole numbers from natural numbers.
Whole numbers still exclude negative numbers, which means they remain non-negative. With this definition, \(-14\) is not a whole number because it is negative.
Use the letter 'W' to represent whole numbers:

• Includes natural numbers plus zero
• Non-negative integers only
• No fractions or decimals

Whole numbers are crucial when dealing with many daily mathematical situations, like counting people or objects.

Integers

Integers introduce the concept of negative numbers to the mix of whole numbers. This means integers include all whole numbers—both positive and negative—along with zero. So, the integer set is \{..., -3, -2, -1, 0, 1, 2, 3, ... \}.
This set of numbers allows for a more extensive range, accommodating calculations that require a balance between positive and negative values.
The letter 'Z' represents integers, which includes numbers like \(-14\):

• Contains negative and positive whole numbers
• Includes zero
• No fractions or decimals again

Understanding integers fully is pivotal, as they form the basis for numerous mathematical concepts and give real-world problems more depth and flexibility.