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**(as opposed to fake numbers?)**

REAL NUMBERS (as opposed to fake numbers?)

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**Objective TSW identify the parts of the Real Number System**

TSW define rational and irrational numbers TSW classify numbers as rational or irrational

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**Real Numbers Real Numbers are every number.**

Therefore, any number that you can find on the number line. Real Numbers have two categories.

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**What does it Mean? The number line goes on forever.**

Every point on the line is a REAL number. There are no gaps on the number line. Between the whole numbers and the fractions there are numbers that are decimals but they don’t terminate and are not recurring decimals. They go on forever.

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**Real Numbers REAL NUMBERS 154,769,852,354 1.333**

-5, … -8 61 π 49%

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**Two Kinds of Real Numbers**

Rational Numbers Irrational Numbers

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Rational Numbers A rational number is a real number that can be written as a fraction. A rational number written in decimal form is terminating or repeating.

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**Examples of Rational Numbers**

16 1/2 3.56 -8 1.3333… - 3/4

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**One of the subsets of rational numbers**

Integers One of the subsets of rational numbers

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**What are integers? Integers are the whole numbers and their opposites.**

Examples of integers are 6 -12 186 -934

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**Integers are rational numbers because they can be written as fraction with 1 as the denominator.**

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Types of Integers Natural Numbers(N): Natural Numbers are counting numbers from 1,2,3,4,5, N = {1,2,3,4,5, } Whole Numbers (W): Whole numbers are natural numbers including zero. They are 0,1,2,3,4,5, W = {0,1,2,3,4,5, } W = 0 + N

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REAL NUMBERS NATURAL Numbers WHOLE Numbers IRRATIONAL Numbers INTEGERS RATIONAL Numbers

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Irrational Numbers An irrational number is a number that cannot be written as a fraction of two integers. Irrational numbers written as decimals are non-terminating and non-repeating.

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

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**Examples of Irrational Numbers**

Pi

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**Try this! a) Irrational b) Irrational c) Rational d) Rational**

e) Irrational

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**Additional Example 1: Classifying Real Numbers**

Write all classifications that apply to each number. A. 5 5 is a whole number that is not a perfect square. irrational, real B. –12.75 –12.75 is a terminating decimal. rational, real 16 2 = = 2 4 2 16 2 C. whole, integer, rational, real

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A fraction with a denominator of 0 is undefined because you cannot divide by zero. So it is not a number at all.

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**Additional Example 2: Determining the Classification of All Numbers**

State if each number is rational, irrational, or not a real number. A. 21 irrational 0 3 0 3 = 0 B. rational

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**Additional Example 2: Determining the Classification of All Numbers**

State if each number is rational, irrational, or not a real number. 4 0 C. not a real number

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**Objective TSW compare rational and irrational numbers**

TSW order rational and irrational numbers on a number line

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**Comparing Rational and Irrational Numbers**

When comparing different forms of rational and irrational numbers, convert the numbers to the same form. Compare and (convert -3 to … … > 3 7 3 7

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Practice

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**Ordering Rational and Irrational Numbers**

To order rational and irrational numbers, convert all of the numbers to the same form. You can also find the approximate locations of rational and irrational numbers on a number line.

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**Example Order these numbers from least to greatest.**

¹/₄, 75%, .04, 10%, ⁹/₇ ¹/₄ becomes 0.25 75% becomes 0.75 0.04 stays 0.04 10% becomes 0.10 ⁹/₇ becomes … Answer: , 10%, ¹/₄, 75%, ⁹/₇

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Practice Order these from least to greatest:

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