Classify, graph, and compare real numbers. Find and estimate square roots Identify and apply properties of real numbers.

Real Numbers and The Number Line Properties of Real Numbers Classify, graph, and compare real numbers. Find and estimate square roots Identify and apply properties of real numbers. Square root, radicand, radical, perfect square, set, element of a set, subset, rational numbers, natural numbers, whole numbers, integers, irrational numbers, real numbers, inequality

Square Root A square root is one of two equal factors of a number. For example one square root of 64 is 8 since 88 or 8 is 64 2 A number like 64, whose square root is a rational number is called a perfect square 5 5or 5 66 or 6 2 2 is 25 is 36

Square Root The symbol called a radical sign, is used to indicate the principal square root of the expression under the radical sign For example: indicates 64 8 the principalsquare root of 64 8 indicates the negative square root of 64 64 8 indicates both square roots of 64 Note that is not the same as. 64 64 64 is not a real number since no real number multiplied by itself is negative 64

Find Square Roots a. 16 9 4 3 note: a b a b b. 0.81 0.9 c. 0.0144 0.12 d. 121 121 is undefined for real numbers

REAL NUMBERS NATURAL NUMBERS {1, 2, 3, } WHOLE NUMBERS {0, 1, 2, 3, } INTEGERS { -2, -1, 0, 1, 2, } RATIONAL NUMBERS: Any number that can be written in fraction form is a rational number. This includes integers, terminating decimals, and repeating decimals. 2 1 3. 25 4 0 0.33 Irrational numbers Numbers that can not be expressed as rational numbers. 3 1.73205080... 24 4.89897948... The set of Real Numbers consists of the set of rational numbers and the set of irrational numbers.

Real Numbers Natural Numbers 2 3 Rational Numbers 2 3 1 4 0.25 Integers 1 2 0.33 3 0 4... Whole Numbers 1 4... Irrational Numbers 3 1.73205... 2 5 3.14159...

Completeness Property Each point on the number line corresponds to exactly one real number Example: Graph each set of numbers: 5 ( -4, -2, 0,,4, 48, ) 2 48 6.9282032... 6.93

RATIONAL NUMBERS GRAPHING THE NUMBER SETS ON THE NUMBER LINE -6-4 -2 0 2 4 6 NATURAL NUMBERS (or counting numbers)

RATIONAL NUMBERS GRAPHING THE NUMBER SETS ON THE NUMBER LINE -6-4 -2 0 2 4 6 WHOLE NUMBERS

RATIONAL NUMBERS GRAPHING THE NUMBER SETS ON THE NUMBER LINE -6-4 -2 0 2 4 6 INTEGERS

RATIONAL NUMBERS EXAMPLES OF RATIONAL NUMBERS 5 5 1 2 5 already fraction a - 7 7 1 0.3 3 10 0 0 1 0.3 1 3

RATIONAL NUMBERS GRAPHING THE NUMBERS ON THE NUMBER LINE -6-4 -2 0 2 4 6 0.5

RATIONAL NUMBERS GRAPHING THE NUMBERS ON THE NUMBER LINE -6-4 -2 0 2 4 6 8 / 3

RATIONAL NUMBERS GRAPHING THE NUMBERS ON THE NUMBER LINE -6-4 -2 0 2 4 6 3 2 / 5

RATIONAL NUMBERS GRAPHING THE NUMBERS ON THE NUMBER LINE -6-4 -2 0 2 4 6 {-3, -2, 0, 5}

RATIONAL NUMBERS GRAPHING THE NUMBERS ON THE NUMBER LINE -6-4 -2 0 2 4 6 {0, 2, 4, 6, }

REVIEW TRUE OR FALSE 0 is the smallest natural number. NO, 1 is the smallest

REVIEW TRUE OR FALSE -1 is the largest negative integer. YES

REVIEW TRUE OR FALSE Name a rational number that is not an integer. Any pure fraction ½, 0.6, 2.3, etc

REVIEW Name a whole number that is not a natural number. 0

REVIEW Name a whole number that is not an integer. NOT POSSIBLE

a. b. c. d. Your turn Example: classify real numbers 5 12 121 56 36 4 = 0.22727272.. ( Repeating decimal)-rational number =11 (natural, whole integer number) Rational number =7.48331477 (which is not repeating or terminating decimal) Irrational number = - 9 (integer) Rational number

An inequality is a mathematical sentence that compares the values of two expressions using an inequality symbol. less than less than or equal to greater than greater than or equal to

Compare Real numbers Replace the with <, >, or = to make the sentence true. Answer:

Order Real numbers from least to greatest 2 Write 4, 0.4,, 2,and -1.5in order from least togreatest 3 Write each number as a decimal 4 2 0.4-0.67 2 1.41-1.5 From the least to the greatest 1.5 0.67 0.4 1.41 2

Properties of Real Numbers Essential Understanding. Relationships that are always true for real numbers are called properties, which are rules used to rewrite and compare expressions. Two algebraic expressions are equivalent expressions if they have the same value for all values of the variable (s).

Properties of Real Numbers. Commutative Properties of Addition and Multiplication Let be a and b any real numbers a b b a Associative Properties of Addition and Multiplication Let a,b and c any real numbers a b c a b c and a bc ab c Identity Properties of Addition and Multiplication Zero property of multiplication. Multiplication property of -1. and a b a 0 a and a 1 a a0 0 1a a ba