2.2. properties of real numbers
DESCRIPTION
handoutsTRANSCRIPT
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Quiz 1
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Grade 4 paper, write last name first, recit section, date today
c
C
Determine the following:
1. 0,1,2 C 2. 2Z 3Z 3. Q Q
Using Venn Diagram, show: A B
A B C
A B
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2.2
Properties of Real
Numbers
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Learning Objectives At the end of the lesson, you should be able
to
define subtraction and division operations
enumerate the axioms
illustrate the closure property for real numbers
identify the identity and inverse elements for addition and subtraction
explain the density property
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Fundamental Operations
1. Addition
denoted by +
result is called sum
Example: 2 + 3 = 5
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Recall
2 + 7 = _____
(-3) + (-5) = _____
8 + (-4) = _____
4 + (-8) = _____
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Fundamental Operations
2. Multiplication
denoted by x or
result is called product
Example: 2 x 3 = 6
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Recall
2 x 8 = _____
(-3) x 6 = _____
3 x (-6) = _____
(-4) x (-5) = _____
Give the rules for multiplying signed numbers.
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Closure A set is closed (under an operation) if and only if the operation on two elements of the set produces another element of the set.
If an element outside the set is produced,
then the set is not closed under the operation.
Closure Property:
When you combine any two elements of the set, the result is also included in the set.
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Example
If you add two real numbers, you
will get another real number.
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Since this process is always true, it
is said that the the set of real numbers is
closed under the operation of
addition.
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Is the set of real numbers closed
under multiplication?
If you multiply two real numbers,
you will get another real number. Since this
process is always true, it is said that the the
set of real numbers is closed under the
operation of multiplication.
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Time to think
Consider the set P of prime numbers.
1) Is P closed under addition? No
2) Is P closed under multiplication? No
Consider the set C of composite positive numbers.
1) Is C closed under addition? No
2) Is C closed under multiplication? Yes
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Properties of Real Numbers
1. Commutative property a. Addition: For all real numbers a,
a + b = b + a we can add numbers in any order
a. Multiplication: For all real a, b a x b = b x a
we can multiply numbers in any order
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2. Associative property a) Addition: For all real numbers a, b, c,
a + (b + c) = (a + b) + c We can group numbers in a sum in any
way we want and still get the same answer.
a) Multiplication: For all real numbers a, b, c
(a x b) x c = a x (b x c) We can group numbers in a product in
any way we want and still get the same answer.
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3. Distributive property of multiplication over addition
For all real numbers a, b, c
a(b + c) = ab + ac (left-hand distributive law)
and
(a + b)c = ac + bc (right-hand distributive law)
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4. Existence of an Identity a) Addition: There exists a real number 0
such that for every real a, a + 0 = a Zero added to any number is the
number itself.
0 is called the additive identity b) Multiplication: There exists a real
number 1 such that for every real a,
a x 1 = a Any number multiplied by 1 gives the
number itself.
1 is called the multiplicative identity.
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5. Existence of an Inverse a) Additive Inverse (Opposite)
For every real number a there exists a real number, denoted (-a), such that
a + (-a) = 0
a) Multiplicative Inverse (Reciprocal)
For every real number a except 0 there exists a real number, denoted by 1/a, such that
a x (1/a) = 1 16
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Definition of Subtraction
a b = a + ( b)
subtracting b from a means adding the negative of b to a.
Thus, 5 3 = 5 + (-3) = 2
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Time to Think
Consider the set W of whole numbers.
Is W closed under subtraction? No
Is subtraction commutative on R? No
Is subtraction associative on R? No
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Definition of Division
If b is a nonzero real number,
a/b = a x (1/b)
Dividing a by b means multiplying a by the reciprocal of b.
Thus, 15/5 = 15 x (1/5) = 3
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Time to Think
Consider the set Z of integers. Is Z
closed under division? No
Is division commutative in R? No
Is division associative in R? No
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6. Density property
We can always find another real number
that lies between any two real numbers.
Example:
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SUMMARY
PROPERTY ADDITION MULTIPLICATION
Commutative a + b = b + a ab = ba
Associative (a+b)+c = a+(b+c) (ab)c = a(bc)
Identity a + 0 = 0 + a = a a x 1 = 1 x a = a
Inverse a + (-a) = (-a) + a = 0 a x (1/a ) = (1/a) x a = 1
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