Properties of Real Numbers Section 1-4

Goals Goal Rubric To identify and use properties of real numbers. Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary Equivalent Expression Deductive reasoning Counterexample

Definition Equivalent Expression – Two algebraic expressions are equivalent if they have the same value for all values of the variable(s). Expressions that look difference, but are equal. The Properties of Real Numbers can be used to show expressions that are equivalent for all real numbers.

Mathematical Properties Properties refer to rules that indicate a standard procedure or method to be followed. A proof is a demonstration of the truth of a statement in mathematics. Properties or rules in mathematics are the result from testing the truth or validity of something by experiment or trial to establish a proof. Therefore every mathematical problem from the easiest to the more complex can be solved by following step by step procedures that are identified as mathematical properties.

Commutative and Associative Properties Commutative Property – changing the order in which you add or multiply numbers does not change the sum or product. Associative Property – changing the grouping of numbers when adding or multiplying does not change their sum or product. Grouping symbols are typically parentheses (),but can include brackets [] or Braces {}.

Commutative Properties Commutative Property of Addition - (Order) For any numbers a and b , a + b = b + a 45 + 5 = 5 + 45 Commutative Property of Multiplication - (Order) 50 = 50 For any numbers a and b , a  b = b  a 6  8 = 8  6 48 = 48

Associative Properties Associative Property of Addition - (grouping symbols) For any numbers a, b, and c, (a + b) + c = a + (b + c) (2 + 4) + 5 = 2 + (4 + 5) (6) + 5 = 2 + (9) 11 = 11 Associative Property of Multiplication - (grouping symbols) For any numbers a, b, and c, (ab)c = a (bc) (2  3)  5 = 2  (3  5) (6)  5 = 2  (15) 30 = 30

Example: Identifying Properties Name the property that is illustrated in each equation. A. 7(mn) = (7m)n The grouping is different. Associative Property of Multiplication B. (a + 3) + b = a + (3 + b) The grouping is different. Associative Property of Addition C. x + (y + z) = x + (z + y) The order is different. Commutative Property of Addition

Your Turn: Name the property that is illustrated in each equation. The order is different. a. n + (–7) = –7 + n Commutative Property of Addition b. 1.5 + (g + 2.3) = (1.5 + g) + 2.3 The grouping is different. Associative Property of Addition The order is different. c. (xy)z = (yx)z Commutative Property of Multiplication

Note! The Commutative and Associative Properties of Addition and Multiplication allow you to rearrange an expression.

Commutative and Associative Properties Commutative and associative properties are very helpful to solve problems using mental math strategies. Rewrite the problem by grouping numbers that can be formed easily. (Associative property) This process may change the order in which the original problem was introduced. (Commutative property) Solve: 18 + 13 + 16 + 27 + 22 + 24 (18 + 22) + (16 + 24) + (13 + 27) (40) + (40) + (40) = 120

Commutative and Associative Properties Commutative and associative properties are very helpful to solve problems using mental math strategies. Solve: 4  7  25 Rewrite the problem by changing the order in which the original problem was introduced. (Commutative property) 4  25  7 Group numbers that can be formed easily. (Associative property) (4  25)  7 (100)  7 = 700

Identity and Inverse Properties Additive Identity Property Multiplicative Identity Property Multiplicative Property of Zero Multiplicative Inverse Property

Additive Identity Property For any number a, a + 0 = a. The sum of any number and zero is equal to that number. The number zero is called the additive identity. If a = 5 then 5 + 0 = 5

Multiplicative Identity Property For any number a, a  1 = a. The product of any number and one is equal to that number. The number one is called the multiplicative identity. If a = 6 then 6  1 = 6

Multiplicative Property of Zero For any number a, a  0 = 0. The product of any number and zero is equal to zero. If a = 6, then 6  0 = 0

Multiplicative Inverse Property Two numbers whose product is 1 are called multiplicative inverses or reciprocals. Zero has no reciprocal because any number times 0 is 0.

Identity and Inverse Properties Property Words Algebra Numbers Additive Identity Property The sum of a number and 0, the additive identity, is the original number. n + 0 = n 3 + 0 = 0 Multiplicative Identity Property The product of a number and 1, the multiplicative identity, is the original number. n  1 = n Additive Inverse Property The sum of a number and its opposite, or additive inverse, is 0. n + (–n) = 0 5 + (–5) = 0 Multiplicative Inverse Property The product of a nonzero number and its reciprocal, or multiplicative inverse, is 1.

Example: Writing Equivalent Expressions A. 4(6y) Use the Associative Property of Multiplication 4(6y) = (4•6)y =24y Simplify B. 6 + (4z + 3) Use the Commutative Property of Addition 6 + (4z + 3) = 6 + (3 + 4z) Use the Associative Property of Addition = (6 + 3) + 4z = 9 + 4z Simplify

Example: Writing Equivalent Expressions C. Rewrite the numerator using the Identity Property of Multiplication Use the rule for multiplying fractions Simplify the fractions Simplify

Your Turn: Simplify each expression. 4(8n) (3 + 5x) + 7 32n 10 + 5b 4y

Identify which property that justifies each of the following. 4  (8  2) = (4  8)  2

Identify which property that justifies each of the following. 4  (8  2) = (4  8)  2 Associative Property of Multiplication

Identify which property that justifies each of the following. 6 + 8 = 8 + 6

Identify which property that justifies each of the following. 6 + 8 = 8 + 6 Commutative Property of Addition

Identify which property that justifies each of the following. 12 + 0 = 12

Identify which property that justifies each of the following. 12 + 0 = 12 Additive Identity Property

Identify which property that justifies each of the following. 5 + (2 + 8) = (5 + 2) + 8

Identify which property that justifies each of the following. 5 + (2 + 8) = (5 + 2) + 8 Associative Property of Addition

Identify which property that justifies each of the following.

Identify which property that justifies each of the following. Multiplicative Inverse Property

Identify which property that justifies each of the following. 5  24 = 24  5

Identify which property that justifies each of the following. 5  24 = 24  5 Commutative Property of Multiplication

Identify which property that justifies each of the following. -34 1 = -34

Identify which property that justifies each of the following. -34 1 = -34 Multiplicative Identity Property

Deductive Reasoning Deductive Reasoning – a form of argument in which facts, rules, definitions, or properties are used to reach a logical conclusion (i.e. think Sherlock Holmes).

Counterexample The Commutative and Associative Properties are true for addition and multiplication. They may not be true for other operations. A counterexample is an example that disproves a statement, or shows that it is false. One counterexample is enough to disprove a statement.

Caution! One counterexample is enough to disprove a statement, but one example is not enough to prove a statement.

Example: Counterexample Statement Counterexample February has fewer than 30 days, so the statement is false. No month has fewer than 30 days. Every integer that is divisible by 2 is also divisible by 4. The integer 18 is divisible by 2 but is not by 4, so the statement is false.

Example: Counterexample Find a counterexample to disprove the statement “The Commutative Property is true for raising to a power.” Find four real numbers a, b, c, and d such that a³ = b and c² = d, so a³ ≠ c². Try a³ = 2³, and c² = 3². a³ = b c² = d 2³ = 8 3² = 9 Since 2³ ≠ 3², this is a counterexample. The statement is false.

Your Turn: Find a counterexample to disprove the statement “The Commutative Property is true for division.” Find two real numbers a and b, such that Try a = 4 and b = 8. Since , this is a counterexample. The statement is false.

Joke Time What do you call bears with no teeth? Gummi Bears. What do you call cheese that’s not yours? Nacho cheese. What do Winnie the Pooh and Jack the Ripper have in common? The same middle name.

Assignment 1.4 Exercises Pg. 29 – 31: #7 – 33 Read and take notes on Sec. 1.5, Pg. 37 - 40. Read and take notes on Sec. 1.6, Pg. 45 - 48.