Properties of Real Numbers

Transcription

1 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). Objectives Commutative Properties of Addition and Multiplication. Associative Properties of Addition and Multiplication. Identity Properties of Addition and Multiplication. Inverse Property. Distributive Property.

2 Commutative property for addition and multiplication The commutative properties state that the order in which you add or multiply numbers does not change their sum or product. a b b a For any numbers a and b, and a b b a and Associative property for addition and multiplication The associative properties state that the way you group three or more numbers when adding or multiplying does not change their sum or product. For any number a, b, and c ( a b) c a ( b c) and ( ab) c a( bc) ( 5 6) 4 (5 4) 6 ( 3 5)2 2(5 3)

3 Addition properties You can rearrange and group the factors to make mental calculations easier Evaluate ( ) ( ) Multiplication properties commutative associative add add You can rearrange and group the factors to make mental calculations easier Evaluate ( 2 5) (8 3) commutative associative multiply multiply

4 Identity Properties of Addition and Multiplication The sum of any number and 0 is equal to the number. This is called the The additive identity a + 0 = 0 + a = a The product of any number and 1 is equal to the number. a 1 a or 1 a a is called the multiplicative identity Inverse Property a ( a) 0 and -a a 0 addition (inverse) a 1 a 1 and 1 a a 1 ( a 0) Multiplication (Reciprocal)

5 PROPERTIES Name the properties illustrated by the following 3(x + y) = 3x + 3y Distributive Prop.

6 PROPERTIES Name the properties illustrated by the following 5 + x = x + 5 Commutative

7 PROPERTIES Name the properties illustrated by the following 3x(1) = 3x Mult. Identity

8 PROPERTIES Name the properties illustrated by the following (3x + y)(0) = 0 Mult. Prop. of Zero

9 PROPERTIES Name the properties illustrated by the following = 48 Add. Identity

10 PROPERTIES Name the properties illustrated by the following If a = b + c and a = 7 then (b+c) = 7 Substitution

11 SIMPLIFY 3(5x 2y) 15x 6y

12 SIMPLIFY -5(x + y) + 3(x + 2y) -2x + y

13 SIMPLIFY 6x 2 + 3x 5x 6x 2 2x

14 SIMPLIFY -3(4m + n) + 2m -10m - 3n

15 Distributive Property 3(X + 2) 3x 6 3(X 2) 3x 6

16 The Distributive Property For any numbers, a, b, and c, a ( b + c ) = ab +ac and ( b + c ) a = ab +ac a ( b c ) = ab - ac and ( b c ) a = ab -ac

17 The Distributive Property Rewrite 8(10 + 4) using the Distributive Property. Then evaluate Rewrite (12 4)6 using the Distributive Property Then evaluate. using the Distributive Property. Then evaluate. Distributive Property.

18 Examples: a. Use the Distributive Property to find. b. Rewrite 4(r 6) using the Distributive Property. Then simplify. 4 r 6 r r 24 c. Rewrite using the Distributive Property. Then simplify.

19 Simplifying Expressions. Objectives Simplify expressions. Identify terms and numerical coefficients. Identify like terms. Combine like terms.

20 All algebraic expressions are composed of terms separated by signs of addition or subtraction 2x 2 6x 5 three terms A term is a, number a, variable or a product or quotient of numbers and variables. Example: y, 4a, p 3, and 8g 2 h are all terms.

21 variable Like terms are terms that contain the same, with corresponding variables having the same power y 6y 5y like terms Example: unlike terms 5n + 7n = 12n 5n and 7n are. like terms An expression is in simplest form when it is replaced by an equivalent expression having no like terms or. parentheses

22 The COEFFICIENT of a term is its numerical factor. EXAMPLE In the term -5x 2 y, the coefficient is -5. What is the coefficient of xy? 1 What is the coefficient of? What is the coefficient of -n? -1

23 ADDING TERMS: Add the coefficients of like terms. Simplify: 6a + 3a Simplify: 7w 3w Simplify: 2a + 6b 9a 4w Cannot be simplified; no like terms. An expression is in SIMPLEST FORM when it is replaced by an equivalent expression having no like terms or parentheses. SIMPLIFY: 5x + 3(x + 2y) 5x + 3x + 6y Distributive Prop. 8x + 6y Add like terms This is in simplest form!

24 Associative (+)

25 Simplify expressions involving like terms. TRY THIS ONE! SIMPLIFY: 2(3a + b) + 3(a + 5b) 6a + 2b + 3a + 15b 9a + 17b Distributive Prop. Add like terms Simplest form!

26 THE DISTRIBUTIVE PROPERTY TRY THIS ONE! Rewrite Property. Then simplify. using the Distributive Answer:

27 THE DISTRIBUTIVE PROPERTY TRY ANOTHER ONE! 3w(4x 3y) 12wx 9wy

28 THE DISTRIBUTIVE PROPERTY

29 Multiplicative property of -1 For any number a, -1( x ) = - x x(-1) = -x Example:

30 THE DISTRIBUTIVE PROPERTY ONE MORE! - 5(3a 2 4a + 3) -15a a 15

31 Distributive property Simplify Distributive Property Multiply. Commutative Property (+) Answer:

32 Distributive property Simplify. Answer: