4-7 The Real Numbers

REAL NUMBERS BY: Miss Osiris Rincon

4-7 The Real Numbers

Recall that rational numbers can be written as fractions. Rational numbers can also be written as decimals that either terminate or repeat. 4 2 3 = 3.8 = 0.6 1.44 = 1.2 5 3

4-7 The Real Numbers Irrational numbers can only be written as decimals that do not terminate or repeat. If a whole number is not a perfect square, then its square root is an irrational number. 2 â&#x2030;&#x2C6;1.4142135623730950488016â&#x20AC;Ś Caution! A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits.

4-7 The setReal of realNumbers numbers consists of the set of The

rational numbers and the set of irrational numbers.

4-7 The Real Numbers Example 1: Classifying Real Numbers

Identify as rational or irratioanl. A.

5 irrational

B. â&#x20AC;&#x201C;12.75 rational C.

16 2

â&#x20AC;&#x201C;12.75 is a terminating decimal.

16 4 = =2 2 2 rational

4-7 The Real Numbers Example 2

Write all names that apply to each number. A.

9

9

=3

rational B.

C.

â&#x20AC;&#x201C;35.9 â&#x20AC;&#x201C;35.9 is a terminating decimal. rational 81 3 rational

81 3

=

9 =3 3

4-7 The Real Numbers Example 3: Determining the Classification of All Numbers

State if each number is rational, irrational, or not a real number. A.

21 irrational

B.

0 3 rational

0 =0 3

4-7 The Real Numbers Example 4: Determining the Classification of All Numbers

State if each number is rational, irrational, or not a real number. C.

â&#x20AC;&#x201C;4 not a real number

D.

4 9 rational

4-7 The Real Numbers Example 5

State if each number is rational, irrational, or not a real number. A.

23 irrational

9 B. 0 undefined, so not a real number

4-7 The Real Numbers Example 6

State if each number is rational, irrational, or not a real number. C.

â&#x20AC;&#x201C;7 not a real number

D.

64 81 rational

4-7 The Real Numbers The Density Property of real numbers states that between any two real numbers is another real number. This property is also true for rational numbers, but not for whole numbers or integers. For instance, there is no integer between â&#x20AC;&#x201C;2 and â&#x20AC;&#x201C;3.

4-7 The Real Numbers Example 7: Applying the Density Property of Real Numbers 2 3 Find a real number between 3 and 3 . 5 5 There are many solutions. One solution is halfway between the two numbers. To find it, add the numbers and divide by 2. 2 3 5 3 +3 รท2 =6 รท2 5 5 5 3

1

2

3

4

=7รท2=3

1 2

3 5 3 5 1 3 5 35 4 32 3 2 1 A real number between 3 and 3 is 3 . 5 5 2

4-7 The Real Numbers Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems

4-7 The Real Numbers Lesson Quiz Write all names that apply to each number.

1.

2. – 16

2

2 rational

irrational

State if each number is rational, irrational, or not a real number. 4. 4 • 9 3. 25 0 rational not a real number

5. Find a real number between –2 3 and –2 3 . Possible answer –25 . 8

4

8

4-7 The Real Numbers Lesson Quiz for Student Response Systems 1. Identify rational, irrational, not real. A. irrational B. rational C. Not real D. All of the above

4-7 The Real Numbers Lesson Quiz for Student Response Systems 2. Identify the name that applies to A. irrational B. rational C. not a real number D. none

.

4-7 The Real Numbers Lesson Quiz for Student Response Systems 3. Identify a real number between A. â&#x20AC;&#x201C;4 B. C. D.

.

Real numbers