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

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.

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.

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

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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