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1 THE NUMBER LINE AND NUMBER SYSTEMS Standards 6, 25 PROPERTIES OF REAL NUMBERS EXAMPLES OF PROPERTIES OF REAL NUMBERS SIMPLIFYING EXPRESSIONS APPLYING PROPERTIES END SHOW PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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2 Standard 6: Students add, subtract, multiply, and divide complex numbers. Estándar 6: Los estudiantes suman, restan, multiplican, y dividen números complejos. Standard 25: Students use properties from number systems to justify steps in combining and simplifying functions. Estándar 25: Los estudiantes usan propiedades de los sistemas numéricos para justificar pasos al combinar y simplificar funciones. ALGEBRA II STANDARDS THIS LESSON AIMS: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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3 Standards 6, 25 012 3 4 5 6 7 8 9 10 11 12 -2 -3 -4 -5 -6-7 -10 -11-12 -8-9 WHOLE NUMBERS NATURAL NUMBERS POSITIVE INTEGERS INTEGERS NEGATIVE INTEGERS THE NUMBER LINE NATURAL NUMBERS:1, 2, 3, 4, … WHOLE NUMBERS: 0, 1, 2, 3, 4, … POSITIVE INTEGERS: 1, 2, 3, 4, … NEGATIVE INTEGERS: -1, -2, -3, -4, … INTEGERS: …, -3, -2, -1, 0, 1, 2, 3, … PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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4 Standards 6, 25 RATIONAL NUMBERS: m n where: m and n are integers. 0.25 0.75 7 10 25 400 1 4 = 3 4 = 7 1 = 10 1 = 5 = 20 = IRRATIONAL NUMBERS -8 1 = 0 0 1 = 2 7 3.1416 1.4142 2.6457 None can be expressed as a fraction! The following numbers can be expressed as fractions and therefore they are Rational numbers: 5 1 = 20 1 = PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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5 REAL NUMBERS R= reals I I= irrationals Q Q= rationals Z Z= integers W W= Wholes N N= naturals Standards 6, 25 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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6 PROPERTIES OF REAL NUMBERS COMMUTATIVE PROPERTY: Addition:a + b = b + a 5 + 7 = 7 + 5 1 + 6 = 6 + 1 3.6 + 1.1 = 1.1 + 3.6 Multiplication: 9 6 4 20 = 6 9 6.4 5.2 = 20 4 = 5.2 6.4 For any real numbers a, b, and c: a b = b a Standards 6, 25 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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7 PROPERTIES OF REAL NUMBERS ASSOCIATIVE PROPERTY: Addition:(a + b) + c = a + (b + c) (3 + 4) +1 = 3 + (4 + 1) (2 + 5) + 7 = 2 + (5 + 7) (6.2 + 4.1) +3.3 = 6.2 + (4.1 + 3.3) Multiplication: For any real numbers a, b, and c: 15 4 7 2 3 5 4 7 2 3 5 = 34 45 6 = 34 45 6 5.7 7.2 2.3 = a b c= a b c Standards 6, 25 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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8 PROPERTIES OF REAL NUMBERS IDENTITY PROPERTY: Addition:a + 0 = 0 + a=a 5 + 0 = 0 + 5 1 + 0 = 0 + 1 3.6 + 0 = 0 + 3.6 Multiplication: 9 1 4 1 = 1 9 6.4 1 = 1 4 = 1 6.4 For any real numbers a, b, and c: a 1 = 1 a = a = 9 = 4 = 6.4 = 5 = 1 = 3.6 Standards 6, 25 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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9 PROPERTIES OF REAL NUMBERS INVERSE PROPERTY: Addition:a + (-a) = (-a) + a=0 5 + (-5) = (-5) + 5 3 + (-3) = (-3) + 3 3.6 + (-3.6) = (-3.6)+ 3.6 Multiplication: For any real numbers a, b, and c: = 1 = 0 a = a = 1 1 a 1 a If a=0 then 3 5 5 3 1 5 5 1 2 2 = 2 1 2 = 5 3 3 5 1 5 = 5 Standards 6, 25 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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10 PROPERTIES OF REAL NUMBERS DISTRIBUTIVE PROPERTY: Distributive: For any real numbers a, b, and c: a(b+c) = ab + ac (b+c)a = ba + ca and 3(5+1) = 3(5) + 3(1) (5+1)3 = 5(3) + 1(3) and 4(2+6) = 4(2) + 4(6) (2+6)4 = 2(4) + 6(4) and Standards 6, 25 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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11 Name the property shown at each equation: 1 45 = 45 a) 56 + 34 = 34 + 56 b) (-3) + 3 = 0 c) 5(9 +2) = 45 + 10 d) (2 + 1) +b= 2 + (1 + b) e) -34(23) = 23(-34) f) Identity property (X) Commutative property (+) Inverse property (+) Distributive property Associative property (+) Commutative property (X) Standards 6, 25 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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12 Simplify 3(4c -7d) + 5(2c + 9c) 3(4c -7d) + 5(2c + 9d) = 3(4c) – 3(7d) +5(2c) +5(9d) =12c – 21d + 10c +45d = 12c + 10c – 21d + 45d = 22c +24d Use distributive property Multiply Use commutative property to group like terms Add like terms Simplify 1 4 (12-4x) 3 5 (15x-10) + 1 4 (12-4x) 3 5 (15x-10) + =( )(12) – ( )(4x) + ( )(15x) – ( )(10) 1 4 1 4 3 5 3 5 = 3 – x + 9x -6 = 3 -6 - x + 9x = 8x-3 Use distributive property Multiply Use commutative property to group like terms Add like terms and commutative property Standards 6, 25 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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