2 Real Numbers

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REAL NUMBERS

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REAL NUMBERS

The set N of natural numbers

N = {1, 2, 3, 4, . . .}These are numbers for counting things.

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The set Z of integers

Z = {. . ., -3, -2, -1, 0, 1, 2, 3, . . .}

N Z Z ∪∪= −

}0{


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REAL NUMBERS

The set Q of rational numbers

This set includes fractions as well as

},,|{ Z bab

a x xQ ∈==

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integers.All rational numbers are characterized

by having repeating decimal form.


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REAL NUMBERS

The set Q’ of irrational numbers

Q’ = {x|x ∉∉∉∉ Q}

...414213562.12 =

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...1415926535.3=

The set R of real numbers

R = Q ∪ Q’


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PROPERTIES OF REALNUMBERS

Determine whether each of these statements is true or false.

Note: I = set of irrational numbers.

False

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False

True

False


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Determine whether each of these statements is true or false.

True

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True

False

True


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Closure Properties

Closure Property of Addition

If a b ∈∈∈∈ R then a + b ∈∈∈∈ R.

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Closure Property of Multiplication

If a, b ∈∈∈∈ R, then a • b ∈∈∈∈ R.


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Associative Properties

Associative Property of Addition

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(a + b) + c = a + (b + c).

Associative Property of Multiplication

For every a, b, c ∈∈∈∈ R,

(ab)c = a(bc).


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Distributive Property

For every a, b , c ∈ R,

a ( b + c) = ab + ac

(b + c)a = ba + ca

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Identity Properties

Identity Property of Addition

For every a ∈∈∈∈ R, a + 0 = 0 + a = a.

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Identity Property of Multiplication

For every a ∈∈∈∈ R, a • 1 = 1 • a = a.


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Inverse Properties

Inverse Property of Addition

For every a ∈∈∈∈ R, there exists –a called

the additive inverse of a, such that

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a + –a = –a + a = 0.

Inverse Property of Multiplication

For every a ∈∈∈∈ R and a ≠ 0, there exists

1/a called the multiplicative inverse of a,such that a • 1/a = 1/a • a = 1.


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Multiplication Property of Zero For every a ∈∈∈∈ R, a • 0 = 0 • a = 0.

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INTERVALS OF REALNUMBERS

An interval of real numbers is the set of real

numbers that lie between two real numbers, which

are called endpoints of the interval.

Interval notation is used to represent intervals.

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The interval notation (2, 3) is used to represent the

real numbers that lie between 2 and 3 on the number

line.

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Parentheses are used to indicate that the endpoints

do not belong to the interval, whereas brackets are

used to indicate that the endpoints do belong to the

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interval.


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The interval notation [2, 3] consists of the real numbers

between 2 and 3 including the endpoints.

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Some intervals do not have endpoints. The infinity

symbol, ∞∞∞∞ , is used to indicate that an interval does not

end.

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For example, the interval (3, ∞∞∞∞ ), consists of the realnumbers greater than 3 and extending infinitely far to

the right of the number line


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The interval (–∞∞∞∞ , 3), consists of the real numbers less

than 3.

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The set of real numbers is written in the interval

notation (–∞∞∞∞ , ∞∞∞∞ ).


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ACTIVITY 1

Write each interval of real numbers in interval

notation and graph it.

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OPERATIONS ON REALNUMBERS

Perform these computations.

1362 +− 49−

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)39(88 −+−

)25(32 −−−

)25(32 −−

127−

7−

57−


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15− 15

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75−−

−−

4

1

2

1

75−

4

3


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1957 ÷− 3−

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−÷−

3820

−+− 2

121

215−

1−


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Evaluate each expression.

325 •+ 113

29 • 72

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( )2246 − 100

352840 •÷•÷ 6


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ORDER OF OPERATIONS

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ORDER OF OPERATIONS


5220 •÷ 50

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74 +

)87(6 −−−

)95()86( −−−

53

5−

2


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ORDER OF OPERATIONS


)526(53 •−− 23

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394 +−−

34635 •−•+

4295

−

−

1−

23

2


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2 Real Numbers