1 real number
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THE SET OF REALTHE SET OF REALNUMBERS (NUMBERS (RR))
THE SET OF REALTHE SET OF REALNUMBERS (NUMBERS (RR))
Review of the Subsets of theReview of the Subsets of theea um er ys emea um er ys em
HASFAZILAH AHMATJAN 2010
ObjectivesObjectivesIdenti the Subsets o the Set oIdenti the Subsets o the Set oReal NumbersReal Numbers
Make a Venn DiagramMake a Venn Diagramrepresentation of the Set of Realrepresentation of the Set of RealNumbersNumbers
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REAL NUMBERS (REAL NUMBERS (RR))e n on: e n on:
REAL NUMBERS (REAL NUMBERS (R))-- Set of all rational andSet of all rational and
irrational numbers.irrational numbers.
e n on: e n on:
REAL NUMBERS (REAL NUMBERS (R))-- Set of all rational andSet of all rational and
irrational numbers.irrational numbers.
HASFAZILAH AHMATJAN 2010
SUBSETS ofSUBSETS of RR
RATIONAL NUMBERS (RATIONAL NUMBERS (Q))-- numbers that can be expressednumbers that can be expressedas a quotientas a quotient a/ba/b, where, where aa andand bbare integers.are integers.
RATIONAL NUMBERS (RATIONAL NUMBERS (Q))-- numbers that can be expressednumbers that can be expressedas a quotientas a quotient a/ba/b, where, where aa andand bbare integers.are integers.
-- erm na ng or repea ng ec ma serm na ng or repea ng ec ma s-- Ex: {1/2, 55/230,Ex: {1/2, 55/230, --205/39}205/39}-- erm na ng or repea ng ec ma serm na ng or repea ng ec ma s-- Ex: {1/2, 55/230,Ex: {1/2, 55/230, --205/39}205/39}
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SUBSETS ofSUBSETS of RRDefinition:Definition:Definition:Definition:
INTEGERS (INTEGERS (Z))-- numbers that consist of positivenumbers that consist of positiveintegers, negative integers, andintegers, negative integers, andzero,zero,
INTEGERS (INTEGERS (Z))-- numbers that consist of positivenumbers that consist of positiveintegers, negative integers, andintegers, negative integers, andzero,zero,
-- {,{, --2,2, --1, 0, 1, 2 ,}1, 0, 1, 2 ,}-- {,{, --2,2, --1, 0, 1, 2 ,}1, 0, 1, 2 ,}
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SUBSETS ofSUBSETS of RRDefinition:Definition:Definition:Definition:
NATURAL NUMBERS (NATURAL NUMBERS (N))-- counting numberscounting numbers
-- positive integerspositive integers
-- 1 21 2
NATURAL NUMBERS (NATURAL NUMBERS (N))-- counting numberscounting numbers
-- positive integerspositive integers
-- 1 21 2, , , ,, , , ,, , , ,, , , ,
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SUBSETS ofSUBSETS of RRDefinition:Definition:Definition:Definition:
WHOLE NUMBERS (WHOLE NUMBERS (W))-- nonnegative integersnonnegative integers
-- { 0 }{ 0 } {1, 2, 3, 4, .}{1, 2, 3, 4, .}--
WHOLE NUMBERS (WHOLE NUMBERS (W))-- nonnegative integersnonnegative integers
-- { 0 }{ 0 } {1, 2, 3, 4, .}{1, 2, 3, 4, .}-- , , , , , , , , , , , , , , , , , , , ,
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SUBSETS ofSUBSETS of RRDefinition:Definition:Definition:Definition:
IRRATIONAL NUMBERS (IRRATIONAL NUMBERS (Q))-- nonnon--terminating and nonterminating and non--repeating decimalsrepeating decimals-- transcendental numberstranscendental numbers
IRRATIONAL NUMBERS (IRRATIONAL NUMBERS (Q))-- nonnon--terminating and nonterminating and non--repeating decimalsrepeating decimals-- transcendental numberstranscendental numbers
-- x: p , sqr ,x: p , sqr , -- . ... ..-- x: p , sqr ,x: p , sqr , -- . ... ..
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The Set of Real NumbersThe Set of Real NumbersThe Set of Real NumbersThe Set of Real Numbers
Q''Q
Q
QZW
N
HASFAZILAH AHMATJAN 2010
Addi t ional Exam ples 1: Classi fy ing Real Numbers
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Try This: Exam ple 1
Write all names that apply to each number.
9
whole, integer, rational, real
35.9 is a terminating decimal.35.9
A.
B.
9 = 3
,
813
whole, integer, rational, real
= = 393
813
C.
HASFAZILAH AHMATJAN 2010
is not a perfect square.
irrational
03
03
= 0
.
B.
ra ona
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State if the number is rational, irrational,
not a real number
9C.
or not a rea num er.
4
9
rational
2
3
=2
3
4
9D.
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Try This: Examples 2
State if the number is rational, irrational,
23 is a whole number thatis not a perfect square.
23
irrational
9
A.
.
0
not a number, so not a real number
.
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State if the number is rational, irrational,
not a real number
7
C.
or not a real number.
81
rational
9
=
9
81D.
HASFAZILAH AHMATJAN 2010
Write all names that apply to each number.
1. 2. 2 16
State if the number is rational, irrational, or nota real number.
3. 4. 4 9
2
250
real, irrational real, integer, rational
Find a real number between 2 and 2 .38
345.
not a real number rational
Possible answer 2 .58 HASFAZILAH AHMATJAN 2010