In the world of mathematics we have categorized all the numbers that exist into certain sets. Let’s describe the certain sets that exist as well as their properties:

__The Set of Natural Numbers__

**Definition:** The set \(\mathbb{N}\) of natural numbers is defined by

\(\mathbb{N} = \left\{ {1,2,3, \ldots } \right\}\)

The set of natural numbers is often referred to as the set of *counting* numbers, because they are those numbers that we use to count. Notice the three dots in the set notation. That means that the set continues without end. Also notice that 0 is *not* included in the natural numbers.

__The Set of Integers__

**Definition:** The set \(\mathbb{Z}\) of integers is defined by

\(\mathbb{Z} = \left\{ { \ldots , - 3, - 2, - 1,0,1,2,3, \ldots } \right\}\)

The set of integers contains all whole numbers (including 0) and their opposites. Notice that the set of natural numbers is *contained* within the set of integers.

__The Set of Rational Numbers__

**Definition:** The set \(\mathbb{Q}\) of rational numbers is defined by

\(\mathbb{Q} = \large \left\{ {\frac{m}{n}:m,n{\text{ are integers}}} \right\}\)

Here is the set of rational numbers, all those numbers that can be expressed as a *ratio* of two integers. Furthermore, the set of rational numbers includes all those numbers whose decimal representation terminates or repeats.

__The Set of Irrational Numbers__

**Definition:** The set of Irrational Numbers is defined by those numbers whose decimal representations *never *terminates or repeats. Some famous irrational numbers include \(\pi \) and \(\sqrt 2 \). In fact, \(\sqrt n \) is irrational if \(n\) is not a *perfect square*.

Together all these sets combined make up the SET OF REAL NUMBERS.

**Example:** Name the set or sets to which 0 belongs.

**Solution:** By inspection we see that 0 is a member of the integers, the rational numbers (since \(\large \frac{0}{1} = 0\), and the real numbers.

**Example:** Name the set or sets to which \(\sqrt {82} \) belongs.

**Solution:** Since 82 is *not* a perfect square, \(\sqrt {82} \) is irrational. Therefore it cannot be rational, an integer, or a natural number. But it *is* a real number.

Below you can **download **some **free** math worksheets and practice.

Name the set or sets to which each number belongs.

This free worksheet contains 10 assignments each with 24 questions with answers.**Example of one question:**

**Watch below how to solve this example:**

Name the set or sets to which each number belongs.

This free worksheet contains 10 assignments each with 24 questions with answers.**Example of one question:**

**Watch below how to solve this example:**

Name the set or sets to which each number belongs.

This free worksheet contains 10 assignments each with 24 questions with answers.**Example of one question:**

**Watch below how to solve this example:**