**Rational and Irrational Numbers**

**Both rational and irrational numbers are real numbers.**

This Venn Diagram shows the relationships between sets of numbers. Notice that rational and irrational numbers are contained in the large blue rectangle representing the set of Real Numbers.

### Rational Numbers

A** rational number** is a number that can be expressed as a fraction or ratio.

The numerator and the denominator of the fraction are both integers.

**Examples of rational numbers are:**

A rational number can be expressed as a ratio (fraction) with integers in both the top and the bottom of the fraction.

When the fraction is divided out, it becomes a terminating or repeating decimal. (The repeating decimal portion may be one number or a billion numbers.)

**Rational numbers on number line:**

A number line is a straight line diagram on which every point corresponds to a real number.

Since rational numbers are real numbers, they have a specific location on a number line.

**To convert a repeating decimal to a fraction:**

**To show that the rational numbers are “dense”:**

(The term “dense” means that between any two rational numbers there is another rational number.)

### Irrational Numbers

An **irrational number** cannot be expressed as a fraction.

- Irrational numbers cannot be represented as terminating or repeating decimals.
- Irrational numbers are non-terminating, non-repeating decimals.
**Examples of irrational numbers are:**

**Note:** Many students think that π is the terminating decimal, 3.14, but it is not. Yes, certain math problems ask you to use π as 3.14, but that problem is rounding the value of to make your calculations easier. π is actually a non-ending decimal and is an irrational number.

There are certain radical values which fall into the irrational number category.

For example, √2 cannot be written as a “simple fraction”

which has integers in the numerator and the denominator.

As a decimal, √2 = 1.414213562373095048801688624 …

which is a non-ending and non-repeating decimal, making √2 irrational.

**Irrational Numbers on a Number Line:**

By definition, a number line is a straight line diagram on which every point corresponds to a real number.

Since irrational numbers are a subset of the real numbers, and real numbers can be represented on a number line, one might assume that each irrational number has a “specific” location on the number line.

“Estimates” of the locations of irrational numbers on number line:

Maths