Making Sense of Rational and Irrational Numbers

Objectives: Identify number sets.Objectives: Identify number sets.Write decimals as fractions.Write decimals as fractions.Write fractions as decimals.Write fractions as decimals.

The set of real numbers is all numbers that can be written on a number line. It consists of the set of rational numbers and the set of irrational numbers.

Irrational numbersRational numbers

Real Numbers

Integers

Wholenumbers

Recall that rational numbers can be written as the quotient of two integers (a fraction) or as either terminating or repeating decimals.

3 = 3.84 5

= 0.623

1.44 = 1.2

Whole numbers and their opposites.

Natural Numbers - Natural counting numbers.

1, 2, 3, 4 …

Whole Numbers - Natural counting numbers and zero.

0, 1, 2, 3 …

Integers -… -3, -2, -1, 0, 1, 2, 3 …

Integers, fractions, and decimals.Rational Numbers -

Ex: -0.76, -6/13, 0.08, 2/3

Rational Numbers

AnimalReptile

Biologists classify animals based on shared characteristics. The horned lizard is an animal, a reptile, a lizard, and a gecko. Rational Numbers are classified this way as well!

You already know that some numbers can be classified as whole numbers, integers, or rational numbers. The number 2 is a whole number, an integer, and a rational number. It is also a real number.Lizard

Gecko

Venn Diagram: Naturals, Wholes, Integers, Rationals

Naturals1, 2, 3...

Wholes0

Integers11 5

Rationals

6.7

59

0.8

327

Real Numbers

Name all the sets of numbers to which the givennumber belongs. Circle the most specific set.

1) 5

22) 3

3) 16

4) 0

5) 0.7

Integers, Rationals

Rationals

Rationals

, Integers, RationalsNaturals , Wholes

, Integers, RationalsWholes

A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits.

Caution!

Irrational numbers can be written only as decimals that do not terminate or repeat. They cannot be written as the quotient of two integers. If a whole number is not a perfect square, then its square root is an irrational number. For example, 2 is not a perfect square, so 2 is irrational.

ReminderReminder

• Real numbers are all the positive, negative, fraction, and decimal numbers you have heard of.

• They are also called Rational Numbers.

• IRRATIONAL NUMBERS are usually decimals that do not terminate or repeat. They go on forever.

• Examples: π

3

2

Identify each root as rational or irrational.

1) 10

2) 25

3) 15

4) 49

6) 62

7) 81

8) 16

9) 99 irrational

irrational

irrational

rational

rationalrational

irrational

rational

5) 50 10) 121 rationalirrational

Decimal to Fraction: A skill Decimal to Fraction: A skill you will need for this unit!you will need for this unit!

• To change a decimal to a fraction, take the To change a decimal to a fraction, take the place value and reduce!place value and reduce!

• 0.5 means 5 tenths, so 5/10.

• Now reduce 5/10 = ½

• 0.5 = 1/2

Converting Fractions and DecimalsFraction Decimal

38

means 3 8

8 3.0000 3

2460

7

5640

5

400

0.375

To change a fraction to a decimal, take the top divided by the bottom, or numerator divided by the denominator.

Complete the table.Fraction Decimal

45 0.8

3100

0.03

720 0.35

7610

6.7

198

9.125

Repeating Decimals

Fraction Decimal13

means 1 3

3 1.0000 3

910

3

910

3

91

0.3...

0.33

Every rational number (fraction) either terminatesor repeats when written as a decimal.

Repeating Decimals

Fraction Decimal5

11

means 5 11

11 5.000000 4

4460

5

5550

4

44

0.454...

0.454

60555044

54

6

0.45

Repeating Decimals

Fraction Decimal56

means 5 6

6 5.0000 8

4820

3

1820

3

182

0.83...

0.833

0.83