1. Properties of Real Numbers
Section 1-4

2. 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.

3. Vocabulary
Equivalent Expression
Deductive reasoning
Counterexample

4. 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.

5. 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.

6. 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 {}.

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

8. 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

10. 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

11. 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 (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

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

13. 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:
( ) + ( ) + ( )
(40) + (40) + (40) = 120

14. 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

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

16. 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

17. 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

18. 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

19. 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.

20. 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.

21. 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

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

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

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

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

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

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

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

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

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

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

32. Identify which property that justifies each of the following.

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

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

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

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

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

38. 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).

39. 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.

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

41. 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.

42. 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.

43. 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.

44. 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.

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