Transcript 1-6 Properties of Real Numbers

1-6 Properties of Real Numbers
Preview
Warm Up
California Standards
Lesson Presentation
1-6 Properties of Real Numbers
Warm Up
Add.
1. –6 + (–4)
–10
2. 17 + (–5)
12
3. (–9) + 7
–2
Subtract.
4. 12 – (–4)
16
5. –3 – (–5)
2
6. –7 – 15
–22
1-6 Properties of Real Numbers
California
Standards
1.0 Students identify and use the arithmetic
properties of subsets of integers and
rational, irrational, and real numbers,
including closure properties for the four
basic arithmetic operations where applicable.
24.3 Students use counterexamples to show
that an assertion is false and recognize that
a single counterexample is sufficient to
refute an assertion.
Also covered: 25.1
1-6 Properties of Real Numbers
Vocabulary
counterexample
closure
1-6 Properties of Real Numbers
The Commutative and Associative Properties of
Addition and Multiplication allow you to rearrange
an expression.
1-6 Properties of Real Numbers
1-6 Properties of Real Numbers
Additional Example 1: 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
1-6 Properties of Real Numbers
Check It Out! Example 1
Name the property that is illustrated in each
equation.
a. n + (–7) = –7 + n
Commutative Property of Addition
The order is
different.
b. 1.5 + (g + 2.3) = (1.5 + g) + 2.3 The grouping is
different.
Associative Property of Addition
c. (xy)z = (yx)z
The order is
different.
Commutative Property of Multiplication
1-6 Properties of Real Numbers
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.
1-6 Properties of Real Numbers
Caution!
One counterexample is enough to disprove a
statement, but one example is not enough to
prove a statement.
1-6 Properties of Real Numbers
Counterexamples
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.
1-6 Properties of Real Numbers
Additional Example 2: Finding Counterexamples to
Statements About Properties
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.
1-6 Properties of Real Numbers
Check It Out! Example 2
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.
1-6 Properties of Real Numbers
The Distributive Property also works with subtraction
because subtraction is the same as adding the
opposite.
1-6 Properties of Real Numbers
Additional Example 3: Using the Distributive
Property with Mental Math
Write each product using the Distributive
Property. Then simplify.
A. 5(71)
Rewrite 71 as 70 + 1.
5(71) = 5(70 + 1)
Use the Distributive Property.
= 5(70) + 5(1)
Multiply (mentally).
= 350 + 5
Add (mentally).
= 355
B. 4(38)
Rewrite 38 as 40 – 2.
4(38) = 4(40 – 2)
Use the Distributive Property.
= 4(40) – 4(2)
Multiply (mentally).
= 160 – 8
Subtract (mentally).
= 152
1-6 Properties of Real Numbers
Check It Out! Example 3
Write each product using the Distributive
Property. Then simplify.
a. 9(52)
9(52) = 9(50 + 2)
= 9(50) + 9(2)
= 450 + 18
= 468
b. 12(98)
12(98) = 12(100 – 2)
Rewrite 52 as 50 + 2.
Use the Distributive Property.
Multiply (mentally).
Add (mentally).
Rewrite 98 as 100 – 2.
= 12(100) – 12(2) Use the Distributive Property.
Multiply (mentally).
= 1200 – 24
Subtract (mentally).
= 1176
1-6 Properties of Real Numbers
Check It Out! Example 3
Write each product using the Distributive
Property. Then simplify.
c. 7(34)
7(34) = 7(30 + 4)
Rewrite 34 as 30 + 4.
= 7(30) + 7(4)
Use the Distributive Property.
= 210 + 28
Multiply (mentally).
= 238
Subtract (mentally).
1-6 Properties of Real Numbers
A set of numbers is said to be closed, or to have
closure, under an operation if the result of the
operation on any two numbers in the set is also in
the set.
1-6 Properties of Real Numbers
Closure Property of Real Numbers
1-6 Properties of Real Numbers
Additional Example 4: Finding Counterexamples to
Statements About Closure
Find a counterexample to show that each
statement is false.
A. The prime numbers are closed under addition.
Find two prime numbers, a and b, such that
their sum is not a prime number.
Try a = 3 and b = 5.
a+b=3+5=8
Since 8 is not a prime number, this is a
counterexample. The statement is false.
1-6 Properties of Real Numbers
Additional Example 4: Finding Counterexamples to
Statements About Closure
Find a counterexample to show that each
statement is false.
B. The set of odd numbers is closed under
subtraction.
Find two odd numbers, a and b, such that the
difference a – b is not an odd number.
Try a = 11 and b = 9.
a – b = 11 – 9 = 2
11 and 9 are odd numbers, but 11 – 9 = 2, which
is not an odd number. The statement is false.
1-6 Properties of Real Numbers
Check It Out! Example 4
Find a counterexample to show that each
statement is false.
a. The set of negative integers is closed
under multiplication.
Find two negative integers, a and b, such that
the product a  b is not a negative integer.
Try a = –2 and b = –1.
a  b = –2(–1) = 2
Since 2 is not a negative integer, this is a
counterexample. The statement is false.
1-6 Properties of Real Numbers
Check It Out! Example 4
Find a counterexample to show that each
statement is false.
b. The whole numbers are closed under the
operation of taking a square root.
Find a whole number, a, such that
whole number.
is not a
Try a = 15.
Since
is not a whole number, this is a
counterexample. The statement is false.
1-6 Properties of Real Numbers
Lesson Quiz: Part I
Name the property that is illustrated in each
equation.
1. 6(rs) = (6r)s
Associative Property of Multiplication
2. (3 + n) + p = (n + 3) + p
Commutative Property of Addition
3. (3 + n) + p = 3 + (n + p)
Associative Property of Addition
4. Find a counterexample to disprove the statement
“The Commutative Property is true for division.”
Possible answer: 3 ÷ 6 ≠ 6 ÷ 3
1-6 Properties of Real Numbers
Lesson Quiz: Part II
Write each product using the Distributive
Property. Then simplify.
5. 8(21)
8(20) + 8(1) = 168
6. 5(97)
5(100) – 5(3) = 485
Find a counterexample to show that each
statement is false.
7. The natural numbers are closed under subtraction.
Possible answer: 6 and 8 are natural, but 6 – 8 =
–2, which is not natural.
8. The set of even numbers is closed under division.
Possible answer: 12 and 4 are even, but 12 ÷ 4 =
3, which is not even.