Download Properties of Real Numbers - William H. Peacock, LCDR USN

Properties of Real Numbers
Section 1-4
Goals
Goal
• To identify and use
properties of real numbers.
Rubric
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
Commutative
Property of
Multiplication (Order)
45 + 5 = 5 + 45
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)
Associative Property
of Multiplication (grouping symbols)
11 = 11
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.
a. n + (–7) = –7 + n
Commutative Property of Addition
b. 1.5 + (g + 2.3) = (1.5 + g) + 2.3
Associative Property of Addition
c. (xy)z = (yx)z
Commutative Property of Multiplication
The order is
different.
The grouping is
different.
The order is
different.
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.
Solve: 18 + 13 + 16 + 27 + 22 + 24
Rewrite the problem by grouping numbers that
can be formed easily. (Associative property)
(18 + 22) + (16 + 24) + (13 + 27)
This process may change the order in which the
original problem was introduced. (Commutative
property)
(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
4  25  7
(4  25)  7
(100)  7 = 700
Rewrite the problem by changing the order in
which the original problem was introduced.
(Commutative property)
Group numbers that can be formed easily.
(Associative property)
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
a
, where a, b  0, there is exactly
b
b
a b
one number such that   1
a
b a
For every nonzero number
 Two numbers whose product is 1 are called multiplicative
inverses or reciprocals.
 Zero has no reciprocal because any number times 0 is 0.
Given the fraction
3
3 4 3  4 12
; then  
  1;
4
4 3 4  3 12
4
is the reciprocal.
3
Together the two fractions are multiplicative
inverses that are equal to the product 1.
the fraction
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
Multiplicative
Inverse
Property
The product of a
nonzero number and
its reciprocal, or
multiplicative inverse,
is 1.
5 + (–5) = 0
Example: Writing Equivalent
Expressions
A.
4(6y)
4(6y) = (4•6)y
=24y
B.
Use the Associative Property of
Multiplication
Simplify
6 + (4z + 3)
6 + (4z + 3) = 6 + (3 + 4z)
= (6 + 3) + 4z
= 9 + 4z
Use the Commutative
Property of Addition
Use the Associative
Property of Addition
Simplify
Example: Writing Equivalent
Expressions
C.
8m
12mn
8m
8  m 1

12mn 12  m  n
8 m 1
 
12 m n
2
1
 1
3
n
2

3n

Rewrite the numerator using the
Identity Property of Multiplication
Use the rule for multiplying fractions
a c ac
 
b d bd
Simplify the fractions
Simplify
Your Turn:
Simplify each expression.
A. 4(8n)
A. 32n
B. (3 + 5x) + 7
B. 10 + 5b
8 xy
C. 2 x
C. 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.
5 9
 1
9 5
Identify which property
that justifies each of the
following.
5 9
 1
9 5
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
No month has fewer than 30 days.
February has fewer than 30 days, so
the statement is false.
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.
c² = d
a³ = b
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: #8 – 44 even, 48
– 56.
• Read and take notes on Sec. 1.5, Pg. 37 40.
• Read and take notes on Sec. 1.6, Pg. 45 48.