Types of Numbers

Whole numbers are almost identical to natural numbers with one important addition: zero. The set of whole numbers includes $0, 1, 2, 3, 4, 5$, and so on. By including zero, we gain the ability to represent the concept of having none of something.

The symbol for the set of whole numbers is $\mathbb{W}$, and it looks like this: $\{0, 1, 2, 3, 4, \ldots\}$. Notice that every natural number is also a whole number. The whole numbers simply extend the family by adding one new member at the bottom: zero.

Zero is a remarkable concept. For centuries, many cultures did not have a symbol for it. The idea that 'nothing' could be a number was a genuine intellectual breakthrough. Today, zero is essential. Without it, you cannot describe an empty bank account, a starting position, or the freezing temperature in Celsius. Because whole numbers include zero, they cover any situation where a count or quantity can reach a minimum of nothing.

Integers take the whole numbers and extend them in the opposite direction, below zero. The set of integers includes all the negative whole numbers, zero, and all the positive whole numbers. In other words, integers go: $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$.

The symbol for integers is $\mathbb{Z}$, from the German word 'Zahlen' meaning 'numbers.' Negative integers are numbers less than zero. They are written with a minus sign in front, like $-5$ or $-12$. Negative numbers might feel abstract at first, but they model real situations perfectly: a temperature of $-8$ degrees Celsius is 8 degrees below freezing, a bank account with $-20$ dollars means you owe 20 dollars, and an elevator at floor $-2$ is two floors underground.

Just like the positive side goes on forever, the negative side also has no end. There is no smallest integer because you can always subtract one more. This means integers extend infinitely in both directions.

The number line is the clearest way to visualize how these three sets relate to each other. Draw a horizontal line. Mark zero in the center. To the right, place the positive numbers: $1, 2, 3$, and so on. To the left, place the negative numbers: $-1, -2, -3$, and so on.

Natural numbers occupy only the right side of the number line, starting at 1. Whole numbers include that same right side plus the zero point. Integers cover the entire number line in both directions. Every point you marked is an integer.

This spatial picture reveals something important: the natural numbers are tucked inside the whole numbers, and the whole numbers are tucked inside the integers. Each set is a complete version of the previous one with more numbers added. Mathematicians call this a nested relationship.

One of the most elegant ideas in mathematics is that number sets are nested inside each other like Russian dolls. Every natural number is also a whole number. Every whole number is also an integer. But the reverse is not always true: zero is a whole number but not a natural number, and $-4$ is an integer but not a whole number or a natural number.

This nesting can be described precisely. We say that $\mathbb{N}$ is a subset of $\mathbb{W}$, which means every member of $\mathbb{N}$ belongs to $\mathbb{W}$. Similarly, $\mathbb{W}$ is a subset of $\mathbb{Z}$. In notation: $\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z}$.

When you are asked to classify a number, always check from the smallest set outward. Start by asking: is it a natural number? If yes, it is also a whole number and an integer. If no, ask: is it zero? If yes, it is a whole number and an integer, but not natural. If the number is negative, it can only be an integer.

Classifying a number means identifying which sets it belongs to. Since the sets are nested, a number can belong to one, two, or all three. The key is to apply the classification questions carefully and remember that belonging to a smaller set automatically means belonging to all larger sets around it.

Consider the number $7$. It is a positive whole value, so it is a natural number. Since every natural number is a whole number, it is also whole. Since every whole number is an integer, it is also an integer. So $7$ belongs to all three sets.

Now consider $0$. It is not positive, so it is not natural. But it is a defined member of the whole numbers. And since every whole number is an integer, zero is also an integer. So $0$ belongs to two sets: whole numbers and integers.

Finally, consider $-9$. It is negative, so it is not natural and not whole. But the integers include all negative whole values, so $-9$ is an integer. It belongs to only one of the three sets.