Presentation on theme: "Properties of Real Numbers"— Presentation transcript:
1 Properties of Real Numbers Unit 2, Lesson 1Online Algebra 1Cami Craig
2 PropertiesIn this lesson we are going to look at properties, rules of mathematics that can be proven.We will be looking at properties of equality, and real number properties.
3 Properties of Real Numbers Properties of Real Numbers include:Commutative PropertyAssociative PropertyIdentity PropertyDistributive PropertyInverse PropertyClosure Property
4 Commutative Properties Commutative Properties deal with order. Order in multiplication and addition do not matter!Additiona + b = b + aOr5 + 9 = 9 + 5…..Is this true? Try it!Of course it is 14 = 14Multiplicationab = baOr5(9) = 9(5)…..Is this true? Try it!Of course it is 45 = 45
5 Commutative Properties and Subtraction Does the Commutative Property hold for subtraction and division?Let’s try:If the commutative property holds for subtraction then the following should be true:6 – 3 = 3 – 6But we know it isn’t true6 – 3 = 3 and 3 – 6 = -3So the commutative property does NOT work for subtraction.
6 Commutative Properties and Division Does the Commutative Property hold for division?Let’s try:If the commutative property holds for division then the following should be true:10 ÷ 5 = 5 ÷ 10But we know it isn’t true10 ÷ 5 = 2 and 5 ÷ 10 = 0.5So the commutative property does NOT work for division.
7 Associative Properties Associative Properties deal with Grouping. Grouping in multiplication and addition do not matter! But just like the commutative properties the associative property does not apply to subtraction.Multiplicationa(bc) = (ab)cOr4 x (6 x 2) = (4 x 6) x 2…..Is this true? Try it!4 x 12 = 24 x 248 = 48Additiona + (b + c) = (a + b) + cOr11 +(5 + 9) = (11 + 5) + 9…..Is this true? Try it!11 + ( 5 + 9) = (11 + 5) + 9=25 = 25
8 Associative and Commutative Properties How can you tell these properties apart?Commutative Properties=4 x 5 = 5 x 4Associative Properties3 + ( 6 + 4) = (3 + 6) + 47 x (3 x 5) = (7 x 3) x 5Notice that in the commutative property the order of the numbers change, while in the associative properties the order stays the same, but the grouping changes.
9 Distributive Property The distributive property is:a(b + c) = ab + acOr2( 4 + 5) = 2 x x 5I like to call the Distributive Property the fair share property, because the number on the outside of the parentheses is multiplied to both numbers in the parentheses.
10 Identity PropertiesThe Identity Properties deal with getting back the same thing.AdditionWhen we add 0 to a number, we get that original number back:For example:A + 0 = A= -4We actually call 0 the Additive Identity Element.MultiplicationWhen we multiply 1 to a number, we get that original number back:For example:1a = a-15(1) = -15We actually call 1 the Multiplicative Identity Element.
11 Inverse and Closure Properties The inverse property for addition states that a+ -a = 0.The inverse property for multiplication states that a x 1/a = 1.The closure property for addition states that a + b is a real number.The closure property for multiplication states that a x b is a real number.
12 Let’s see what you have learned so far Let’s see what you have learned so far! What property does each example represent?-2(3 + 4) = -2 x x 45 + (3 + 6) = (3 + 6) + 55(1) = 517 x (8 x 2) = (17 x 8) x 29 + 0 = 94 x ¼ = 1Distributive PropertyCommutative PropertyNotice that although there are parentheses, it is the order that changes not the grouping.Identity Property of MultiplicationAssociative PropertyIdentity Property of AdditionInverse Property of Mult.
13 Properties of Equality Properties of equality include the following:The Reflexive PropertyThe Symmetric PropertyThe Transitive Property
14 Properties of Equality Reflexive Property of Equalitya = a-9 = -9Symmetric Property of EqualityIf a = b, then b = a.If 15x = 45, then 45 = 15xTransitive Property of EqualityIf a = b and b = c, then a = cIf d = 3y and 3y = 6, then d = 6.
15 Review What property is each of the following an example of? 9 = 9a + 8 = 8 + aIf x + 8 = 9, and 9 = 4 + 5, then x + 8 = 4 + 53(x – 7) = 3x – 215 x 1 = 5If 16 = 4x, then 4x = 16= 0ReflexiveCommutativeTransitiveDistributiveIdentitySymmetricInverse