### Properties of Rational versus Irrational Numbers

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,

#### Identifying Rational and Irrational Numbers

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

a) 23.7

23.7 can be written as

b) 2.8956

2.8956 can be written as

c)

d)

**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,

#### Expressing Decimals as Fractions

Express the following decimals as fractions.

a.) 0.439

0.439 can be expressed as

b.)

**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**.

*Classify the following real numbers.*

a) 0

Integer

b) -1

Integer

c)

Irrational (Although it's written as a fraction,

d)

Irrational

e)

Rational (It simplifies to

### Example

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

#### Example 1

Since

Since

Since the

Since

This means that the ordering is:

### Review

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

0.25−−−−√ 1.35−−−−√ 20−−√ 25−−√ 100−−−√ π2−−√ 2⋅18−−−−√ - Write 0.6278 as a fraction.
- Place the following numbers in numerical order, from lowest to highest.
6√261501.5−−−√1613 - Use the marked points on the number line and identify each proper fraction.

### Review (Answers)

To view the Review answers, open this PDF file and look for section 2.10.