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Integers form a fundamental part of mathematics, especially when dealing with whole numbers. Simply put, an integer is a number without any fractional or decimal component. In the number line, integers span both directions from zero, which is center-point, considered an integer itself.

Integers include two main categories: positive numbers, which are greater than zero (e.g., 1, 2, 3, ...), and negative numbers, which are less than zero (e.g., -1, -2, -3, ...). It’s important to remember that the whole numbers span infinitely in both positive and negative directions, thus providing endless possibilities when working with them.

When we are asked to find integers between two specific numbers, like in the exercise between 15 and 22, it includes all the whole numbers from 16 to 21 excluding the endpoints. Understanding this concept is crucial as it forms the basis for set creation using the roster method.

Every integer can be classified as either positive, negative, or zero—the neutral element. Positive numbers are those that are above zero on the number line. They represent quantities that can be counted like apples in a basket or steps taken forward.

On the other hand, negative numbers are less than zero, often symbolizing a deficiency or a reverse direction, like a debt owed or steps taken backward. The number zero is a special integer that represents neutrality—it’s neither positive nor negative and serves as the dividing line between positive and negative numbers on the number line.

Understanding the nature of positive and negative numbers is vital when learning about integers because they are foundational for mathematical operations like addition and subtraction, where they obey specific rules that differ from the rules of their positive counterparts.

The roster method is one of the simplest ways to portray a set. This method involves listing each element of the set within curly brackets, separated by commas, and without repetition of elements. For instance, to express the set of all vowels in the English alphabet, you would write \( \(\text{A, E, I, O, U}\) \).

When writing sets of numbers, such as the integers between 15 and 22 as asked in the exercise, the set in roster form would be \( \(\{16, 17, 18, 19, 20, 21\}\) \) because it includes every integer between the two given numbers without including the endpoints themselves. This method is efficient, neat, and widely used for sets with finite or clearly defined members.

Understanding how to construct sets with the roster method is a crucial skill in mathematics, as it is often the starting step in set theory and helps in visualizing and organizing data or mathematical concepts.