Problem 13 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Indicate whether the statement is true or false. Every natural number is an integer.

Step-by-Step Solution

Answer

The statement "Every natural number is an integer" is true, as natural numbers are a subset of integers, with integers including negative numbers and zero as well.

Step 1: Definition of Natural Numbers

Natural numbers are the set of positive whole numbers, often used for counting or ordering. It typically starts from 1 and includes all positive whole numbers excluding zero, i.e., {1, 2, 3, 4, 5, 6,...}.

Step 2: Definition of Integers

Integers are the set of whole numbers, which include positive, negative numbers, and zero. In mathematical notation, integers can be represented as {..., -3, -2, -1, 0, 1, 2, 3,...}.

Step 3: Comparing Natural Numbers and Integers

When comparing the definitions of natural numbers and integers, we can see that natural numbers are a subset of integers. Every natural number is included in the set of integers, although integers also include negative numbers and zero, which are not part of the natural numbers.

Step 4: Conclusion

Based on the definitions and comparison between natural numbers and integers, we can conclude that the statement "Every natural number is an integer" is true.

Key Concepts

Set of Positive Whole numbers

Understanding the concept of positive whole numbers is fundamental in mathematics. These are the numbers you first learn to count with as a child and are formally known as natural numbers. They are used not only for counting items but also for assigning an order to them, such as first, second, third, and so on.

A set of positive whole numbers includes all the numbers starting from 1 and moving upwards without any end. In a more technical sense, this set is expressed as \( {1, 2, 3, 4, 5, ...} \), where the ellipsis (...) indicates that the numbers go on indefinitely. One important point to note is that this set does not include zero, as zero represents neither a positive nor a negative quantity.

Furthermore, when writing or speaking about these numbers, we often use terms like 'greater than' and 'less than' to establish a hierarchy or order among them, which forms the basis for arranging objects or values in ascending or descending sequence.

Counting and Ordering

The process of counting involves identifying the quantity of items in a set, while ordering is the arrangement of these items in a logical sequence. These are basic operations we perform with natural numbers, and they are crucial for understanding more advanced mathematical concepts.

When counting, we start from 1 and proceed to higher numbers. For example, if there are five apples on the table, we count them as 1, 2, 3, 4, 5. This operation is intuitive and forms the backbone of arithmetic.

On the other hand, ordering involves organizing these counts, such as arranging the same apples by size or ripeness. We might arrange them from smallest to largest or vice versa. Ordering helps us systematize data and make it ready for analysis or presentation. It serves as a prelude to the concept of ordering numbers on a number line or solving inequalities.

Subset of Integers

Natural numbers are a specific subset of the larger set called integers. When we examine the realm of integers, we find all the whole numbers both positive and negative, as well as zero. Integers can be written in the form \( {..., -3, -2, -1, 0, 1, 2, 3, ...} \) and illustrate the full range of whole numbers without any fractional or decimal part.

In this context, a subset is a portion or part of a larger set that contains some or all of the elements of the larger set. Accordingly, since every natural number is a non-negative whole number, they are completely included within the integers. Therefore, you can think of natural numbers as being the sunny side of the integer family, where every value is positive. On the other side, integers include the natural numbers but extend in both directions along the number line, including zero and the 'less sunny' negative numbers.