What if you wanted to identify a number like ? Would you classify it as rational or irrational? After completing this Concept, you'll be able to decide which category numbers like this one fall into.

### Watch This

CK-12 Foundation: 0210S Irrational Numbers (H264)

### Guidance

Not all square roots are irrational, but any square root that can’t be reduced to a form with no radical signs in it is irrational. For example, is rational because it equals 7, but can’t be reduced farther than . That factor of is irrational, making the whole expression irrational.

#### Example A

*Identify which of the following are rational numbers and which are irrational numbers.*

a) 23.7

b) 2.8956

c)

d)

**Solution**

a) 23.7 can be written as , so it is rational.

b) 2.8956 can be written as , so it is rational.

c) We know from the definition of that the decimals do not terminate or repeat, so is irrational.

d) . We can’t reduce it to a form without radicals in it, so it is irrational.

**Repeating Decimals**

Any number whose decimal representation has a finite number of digits is rational, since each decimal place can be expressed as a fraction. For example, This decimal goes on forever, but it’s not random; it repeats in a predictable pattern. Repeating decimals are always rational; this one can actually be expressed as .

#### Example B

*Express the following decimals as fractions.*

a.) 0.439

b.)

**Solution:**

a.) 0.439 can be expressed as , or just . Also, any decimal that repeats is rational, and can be expressed as a fraction.

b.) can be expressed as , which is equivalent to .

**Classify Real Numbers**

We can now see how real numbers fall into one of several categories.

If a real number can be expressed as a rational number, it falls into one of two categories. If the denominator of its **simplest form** is one, then it is an **integer**. If not, it is a fraction (this term also includes decimals, since they can be written as fractions.)

If the number cannot be expressed as the ratio of two integers (i.e. as a fraction), it is **irrational**.

#### Example C

*Classify the following real numbers.*

a) 0

b) -1

c)

d)

e)

**Solution**

a) Integer

b) Integer

c) Irrational (Although it’s written as a fraction, is irrational, so any fraction with in it is also irrational.)

d) Irrational

e) Rational (It simplifies to , or .)

Watch this video for help with the Examples above.

CK-12 Foundation: Irrational Numbers

### Vocabulary

- The
**square root**of a number is a number which gives the original number when multiplied by itself. In algebraic terms, for two numbers and , if , then . - A square root can have two possible values: a positive value called the
**principal square root**, and a negative value (the opposite of the positive value). - A
**perfect square**is a number whose square root is an integer. - Some mathematical properties of square roots are:
- Square roots of numbers that are not perfect squares (or ratios of perfect squares) are
**irrational numbers**. They cannot be written as rational numbers (the ratio of two integers). In decimal form, they have an unending, seemingly random, string of numbers after the decimal point. - Computing a square root on a calculator will produce an
**approximate solution**since the calculator only shows a finite number of digits after the decimal point.

### Guided Practice

*Place the following numbers in numerical order, from lowest to highest.*

**Solution:**

Since is the only negative number, it is the smallest.

Since , .

Since the , then .

Since , then

This means that the ordering is:

### Practice

For questions 1-7, classify the following numbers as an integer, a rational number or an irrational number.

- Write 0.6278 as a fraction.
- Place the following numbers in numerical order, from lowest to highest.
- Use the marked points on the number line and identify each proper fraction.