### Learning Outcome

- Classify a real number as a natural, whole, integer, rational, or irrational number

The numbers we use for counting, or enumerating items, are the **natural numbers**: [latex]1, 2, 3, 4, 5,[/latex] and so on. We describe them in set notation as [latex]\{1, 2, 3, …\}[/latex] where the ellipsis (…) indicates that the numbers continue to infinity. The natural numbers are, of course, also called the *counting numbers*. Any time we enumerate the members of a team, count the coins in a collection, or tally the trees in a grove, we are using the set of natural numbers. The set of **whole numbers** is the set of natural numbers and zero: [latex]\{0, 1, 2, 3,…\}[/latex].

The set of **integers** adds the opposites of the natural numbers to the set of whole numbers: [latex]\{…-3, -2, -1, 0, 1, 2, 3,…\}[/latex]. It is useful to note that the set of integers is made up of three distinct subsets: negative integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the integers.

The set of **rational numbers** is written as [latex]\left\{\dfrac{m}{n}\normalsize |m\text{ and }{n}\text{ are integers and }{n}\ne{ 0 }\right\}[/latex]. Notice from the definition that rational numbers are fractions (or quotients) containing integers in both the numerator and the denominator, and the denominator is never [latex]0[/latex]. We can also see that every natural number, whole number, and integer is a rational number with a denominator of [latex]1[/latex].

Because they are fractions, any rational number can also be expressed in decimal form. Any rational number can be represented as either:

- a terminating decimal: [latex]\dfrac{15}{8}\normalsize =1.875[/latex], or
- a repeating decimal: [latex]\dfrac{4}{11}\normalsize =0.36363636\dots =0.\overline{36}[/latex]

We use a line drawn over the repeating block of numbers instead of writing the group multiple times.

### Example

Write each of the following as a rational number.

- [latex]7[/latex]
- [latex]0[/latex]
- [latex]–8[/latex]

### Example

Write each of the following rational numbers as either a terminating or repeating decimal.

- [latex]-\dfrac{5}{7}[/latex]
- [latex]\dfrac{15}{5}[/latex]
- [latex]\dfrac{13}{25}[/latex]

## Irrational Numbers

At some point in the ancient past, someone discovered that not all numbers are rational numbers. A builder, for instance, may have found that the diagonal of a square with unit sides was not [latex]2[/latex] or even [latex]\dfrac{3}{2}[/latex], but was something else. Or a garment maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit more than [latex]3[/latex], but still not a rational number. Such numbers are said to be *irrational* because they cannot be written as fractions. These numbers make up the set of **irrational numbers**. Irrational numbers cannot be expressed as a fraction of two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if it is not rational. So we write this as shown.

### Example

Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.

- [latex]\sqrt{25}[/latex]
- [latex]\dfrac{33}{9}[/latex]
- [latex]\sqrt{11}[/latex]
- [latex]\dfrac{17}{34}[/latex]
- [latex]0.3033033303333\dots[/latex]

## Real Numbers

Given any number *n*, we know that *n* is either rational or irrational. It cannot be both. The sets of rational and irrational numbers together make up the set of **real numbers**. As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and irrational numbers according to their algebraic sign (+ or –). Zero is considered neither positive nor negative.

The real numbers can be visualized on a horizontal number line with an arbitrary point chosen as [latex]0[/latex], with negative numbers to the left of [latex]0[/latex] and positive numbers to the right of [latex]0[/latex]. A fixed unit distance is then used to mark off each integer (or other basic value) on either side of [latex]0[/latex]. Any real number corresponds to a unique position on the number line.The converse is also true: Each location on the number line corresponds to exactly one real number. This is known as a one-to-one correspondence. We refer to this as the **real number line** as shown below.

### Example

Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line?

- [latex]-\dfrac{10}{3}[/latex]
- [latex]\sqrt{5}[/latex]
- [latex]-\sqrt{289}[/latex]
- [latex]-6\pi[/latex]
- [latex]0.615384615384\dots[/latex]

## Sets of Numbers as Subsets

Beginning with the natural numbers, we have expanded each set to form a larger set, meaning that there is a subset relationship between the sets of numbers we have encountered so far. These relationships become more obvious when seen as a diagram.

### A General Note: Sets of Numbers

The set of **natural numbers** includes the numbers used for counting: [latex]\{1,2,3,\dots\}[/latex].

The set of **whole numbers** is the set of natural numbers and zero: [latex]\{0,1,2,3,\dots\}[/latex].

The set of **integers** adds the negative natural numbers to the set of whole numbers: [latex]\{\dots,-3,-2,-1,0,1,2,3,\dots\}[/latex].

The set of **rational numbers** includes fractions written as [latex]\{\dfrac{m}{n}\normalsize |m\text{ and }n\text{ are integers and }n\ne 0\}[/latex].

The set of **irrational numbers** is the set of numbers that are not rational. They are nonrepeating and nonterminating decimals: [latex]\{h|h\text{ is not a rational number}\}[/latex].

### Example

Classify each number as being a natural number (*N*), whole number (*W*), integer (*I*), rational number (*Q*), and/or irrational number (*Q’*).

- [latex]\sqrt{36}[/latex]
- [latex]\dfrac{8}{3}[/latex]
- [latex]\sqrt{73}[/latex]
- [latex]-6[/latex]
- [latex]3.2121121112\dots [/latex]