clemath.com ﻭ ﻭ ﻭ ﺍﻤﻮﻋﺎﺕ IR Q ID Z ﳎﻤﻮﻋﺔ ﺍﻷﻋﺪﺍﺩ ﺍﻟﺼﺤﻴﺤﺔ ﺍﻟﻨﺴ 1
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ﺍﻤﻮﻋﺎﺕ Zﻭ IDﻭ Qﻭ IR
(1ﳎﻤﻮﻋﺔ ﺍﻷﻋﺪﺍﺩ ﺍﻟﺼﺤﻴﺤﺔ ﺍﻟﻨﺴﺒﻴﺔ Z/
}Z/ = {..... − 3;−2;−1;0;1;2;3.....
ﺃﻣﺜﻠﺔ:
−1275 ∈ Z/
؛
I ⊂ Z/
−13,5 ∉ Z/
؛
3
∉ Z/
5
2 ∉ Z/
؛
(2ﳎﻤﻮﻋﺔ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻌﺸﺮﻳﺔ ﺍﻟﻨﺴﺒﻴﺔ . ID
p
n ∈ I و ID = n / p ∈ Z/
10
13205 13205
= 13,205
=
ﺃﻣﺜﻠﺔ∈ ID :
1000
10 3
I ⊂ Z/ ⊂ ID
ﻭ
1
= 0,33333...... ∉ ID
3
Q
(3ﳎﻤﻮﻋﺔ ﺍﻷﻋﺪﺍﺩ ﺍﳉﺬﺭﻳﺔ /
p
Q/ =
q ∈ Z/ * و / p ∈ Z/
q
3
2
ﺃﻣﺜﻠﺔ:
ﻭ − ∈ Q/ﻭ
∈ Q/
7
5
I ⊂ Z/ ⊂ ID ⊂ Q/
2501
∈ Q/
1000
= 2,501
ﻭ
31
∈ Q/
1
= 31
(4ﳎﻤﻮﻋﺔ ﺍﻷﻋﺪﺍﺩ ﺍﳊﻘﻴﻘﻴﺔ IR
ﺗﻮﺟﺪ ﻗﻴﺎﺳﺎﺕ ﻻ ﳝﻜﻦ ﺍﻟﺘﻌﺒﲑ ﻋﻨﻬﺎ ﺑﺄﻋﺪﺍﺩ ﺟﺬﺭﻳﺔ ،ﻭﻗﻴﻤﻬﺎ ﺍﻟﻌﺪﺩﻳﺔ ﺗﺴﻤﻰ ﺃﻋﺪﺍﺩﺍ ﻻﺟﺬﺭﻳﺔ .ﻣﺜﻼ ﺍﻟﻌﺪﺩ
ﺍﻟﻼﺟﺬﺭﻱ . 2
ﺗﻌﺮﻳﻒ ﳎﻤﻮﻋﺔ ﺍﻷﻋﺪﺍﺩ ﺍﳊﻘﻴﻘﻴﺔ IRﻫﻲ ﺍﻤﻮﻋﺔ ﺍﳌﻜﻮﻧﺔ ﻣﻦ ﺍﻷﻋﺪﺍﺩ ﺍﳉﺬﺭﻳﺔ ﻭﺍﻷﻋﺪﺍﺩ ﺍﻟﻼﺟﺬﺭﻳﺔ.
I ⊂ Z/ ⊂ ID ⊂ Q/ ⊂ IR
(5ﺍﻟﻌﻤﻠﻴﺎﺕ ﰲ ﺍﻤﻮﻋﺔ IR
(1-5ﺍﳉﻤﻊ ﻭﺍﻟﻀﺮﺏ ﰲ IR
ﻟﻜﻞ aﻭ bﻭ cﻭ dﻣﻦ IRﻟﺪﻳﻨﺎ:
• a+b =b+a
• a + (b + c) = (a + b) + c
• a+0= a
) ﺗﺒﺎﺩﻟﻴﺔ ﺍﳉﻤﻊ (
) ﲡﻤﻴﻌﻴﺔ ﺍﳉﻤﻊ (
• (−a) + a = a + (−a ) = 0
) 0ﻳﺴﻤﻰ ﺍﻟﻌﻨﺼﺮ ﺍﶈﺎﻳﺪ ﻟﻠﺠﻤﻊ (
) −aﻳﺴﻤﻰ ﻣﻘﺎﺑﻞ ( a
• ab = ba
) ﺗﺒﺎﺩﻟﻴﺔ ﺍﻟﻀﺮﺏ (
1
L. L.
clemath.com
a (bc) = (ab)c •
() ﲡﻤﻴﻌﻴﺔ ﺍﻟﻀﺮﺏ
( ﻳﺴﻤﻰ ﺍﻟﻌﻨﺼﺮ ﺍﶈﺎﻳﺪ ﻟﻠﻀﺮﺏ1 )
( a ﻳﺴﻤﻰ ﻣﻘﻠﻮﺏ
1× a = a •
1
) (a ≠ 0)
a
( ) ﺗﻮﺯﻳﻌﻴﺔ ﺍﻟﻀﺮﺏ ﻋﻠﻰ ﺍﳉﻤﻊ
1
1
×a = a× =1 •
a
a
a (b + c) = ab + ac •
(a + b)c = ac + bc
a −a
a
(b ≠ 0 )
− =
=
b
b
−b
a c
( d ≠ 0 ﻭb ≠ 0 ) ad = bc ﻳﻜﺎﻓﺊ
=
b d
a ca
(c ≠ 0 ﻭb ≠ 0 )
=
b cb
a c a+c
(m ≠ 0)
+ =
m m
m
a c ad + bc
( d ≠0 ﻭb≠0)
+ =
b d
bd
a c ac
( d ≠0 ﻭb≠0)
× =
b d bd
1 b
=
( b ≠ 0 ﻭa ≠ 0)
a a
b
a
b = a × d = ad
(d ≠ 0 ﻭc ≠ 0 ﻭb ≠ 0)
c b c bc
d
•
•
•
•
•
•
•
•
•
IR ( ﻗﻮﺍﻋﺪ ﺍﳊﺴﺎﺏ ﰲ2-5
: ﻟﺪﻳﻨﺎIR ﻣﻦc ﻭb ﻭa ﻟﻜﻞ
(b ≠ 0 )
(c ≠ 0)
L. L.
2
a = c−b
ﻳﻜﺎﻓﺊ
a+b =c •
a=
c
b
a+c =b+c
ﻳﻜﺎﻓﺊ
ab = c •
a=b •
ac = bc
ﻳﻜﺎﻓﺊ
ﻳﻜﺎﻓﺊ
b = 0 ﺃﻭa = 0
ﻳﻜﺎﻓﺊ
ab = 0 •
b≠0 ﻭa≠0
ﻳﻜﺎﻓﺊ
ab ≠ 0 •
a=b •
clemath.com
ﺍﻟﺘﻤﺮﻳﻦ .1 1ﺑﲔ ﺃﻥ
1
1
1
+
+
=0
)(a − b)( a − c) (b − a)(b − c) (c − a)(c − b
2
4 1
1
1+
−
5 ×3 6
×2
1
1 3 1
8−
−
3+
9
2 4 10
5−
.2ﺃﺣﺴﺐ:
(3-5ﺍﳉﺬﻭﺭ ﺍﳌﺮﺑﻌﺔ
ﺗﻌﺮﻳﻒ
aﻋﺪﺩ ﺣﻘﻴﻘﻲ ﻣﻮﺟﺐ .ﺍﳉﺬﺭ ﺍﳌﺮﺑﻊ ﻟﻠﻌﺪﺩ aﻫﻮ ﺍﻟﻌﺪﺩ ﺍﳊﻘﻴﻘﻲ ﺍﳌﻮﺟﺐ bﺍﻟﺬﻱ ﻣﺮﺑﻌﻪ
ﻳﺴﺎﻭﻱ ، aﻭﻳﺮﻣﺰ ﻟﻪ ﺑﺎﻟﺮﻣﺰ a
ﺃﻣﺜﻠﺔ 9 = 3 :
ﻷﻥ 32 = 9
ﻭﻧﻜﺘﺐa = b :
ﻳﻌﲏ b 2 = a
. 2 = 1,4142135...
؛
ﺧﺎﺻﻴﺎﺕ • ﻟﻜﻞ xﻭ yﻣﻦ : IR +
-
x. y
y -
= x. y
=x
x
؛
x
=
y
y
) ( y≠0
؛
=x
)( x
2
x=y
ﻳﻜﺎﻓﺊ
• ﻟﻜﻞ xﻣﻦ : IR +
x2 = x
• ﻟﻜﻞ xﻣﻦ : IR −
x2 = −x
ﺍﻟﺘﻤﺮﻳﻦ .1 2ﺃﺣﺴﺐ ﺍﳉﺬﺭ ﺍﳌﺮﺑﻊ ﻟﻜﻞ ﻣﻦ ﺍﻷﻋﺪﺍﺩ ﺍﻟﺘﺎﻟﻴﺔ:
81
64
625؛ 1024؛ 108؛
3 121 + 3
.2ﺑﺴﻂ ﺍﻷﻋﺪﺍﺩ ﺍﻟﺘﺎﻟﻴﺔ:
؛ . 0,16
؛
75 × 27
. 6 2 + 82
؛
(4-5ﻗﻮﻯ ﻋﺪﺩ ﺣﻘﻴﻘﻲ.
ﺗﻌﺮﻳﻒ
a ∈ IR
ﻭ * . n ∈ I
a n = a × a × ..... × a
a0 =1
ﻭ
ﻭ
1
n
a
= ( a ≠ 0 ) a −n
ﺍﻟﻌﺪﺩ a nﻳﺴﻤﻰ ﻗﻮﺓ ﺍﻟﻌﺪﺩ aﺫﺍﺕ ﺍﻷﺱ . n
ﻗﻮﻯ ﺍﻟﻌﺪﺩ 10
4 ) 10 4 = 10000ﺃﺻﻔﺎﺭ ( ﻭ
−5
5 ) 10 = 0.00001ﺃﺻﻔﺎﺭ (
3
L. L.
clemath.com
ﻧﺘﺎﺋﺞ
a ∈ IR؛ b ∈ IR؛
a n × a m = a n+ m
n ∈ Z/
n
a n × b n = (ab) n
؛
(a n ) m = a n×m
ﻭ
n
= a n−m
؛
a
am
an
)( a ≠ 0
a
=
bn b
) (b ≠ 0
= an
)( a ≥ 0
)( a
n
؛
m ∈ Z/
ﺍﻟﻜﺘﺎﺑﺔ ﺍﻟﻌﻠﻤﻴﺔ ﻟﻌﺪﺩ ﻋﺸﺮﻱ
ﺃﻣﺜﻠﺔ :
34210000 = 3,421 × 10 7
−5
؛
0,0000351 = 3,51×10
ﺍﻟﺘﻤﺮﻳﻦ 3ﺑﺴﻂ ﺍﻟﺘﻌﺎﺑﲑ ﺍﻟﺘﺎﻟﻴﺔ:
−4
2
7 × 55 × 2
(49 × 121) − 2
7 3 × 21−5 × 352 × (5−1 ) 4؛ 32 × (9 2 ) −4 × 27 × 81؛
(5-5ﺍﳌﺘﻄﺎﺑﻘﺎﺕ ﺍﳍﺎﻣﺔ
ﺧﺎﺻﻴﺎﺕ ﻟﻴﻜﻦ a ∈ IRﻭ : b ∈ IR
3
2
3
2
3
(a + b) = a + 3a b + 3ab + b
(a + b) 2 = a 2 + 2ab + b 2
(a − b) 3 = a 3 − 3a 2 b + 3ab 2 − b 3
(a − b) 2 = a 2 − 2ab + b 2
) a 3 − b 3 = (a − b)(a 2 + ab + b 2
2
3
2
3
) a + b = (a + b)(a − ab + b
)a 2 − b 2 = (a − b)(a + b
(6-5ﺍﻟﻨﺸﺮ ﻭﺍﻟﺘﻌﻤﻴﻞ
ﻧﺸﺮ ﺟﺪﺍﺀ ﻫﻮ ﲢﻮﻳﻠﻪ ﺇﱃ ﲨﻊ ﻭﺗﻌﻤﻴﻞ ﳎﻤﻮﻉ ﻫﻮ ﲢﻮﻳﻠﻪ ﺇﱃ ﺟﺪﺍﺀ.
1
1
ﺍﻟﺘﻤﺮﻳﻦ .1 4ﺃﻧﺸﺮ ﺍﻟﺘﻌﺒﲑﻳﻦ ﺍﻟﺘﺎﻟﻴﲔ ( x + 2)( x 2 − 2 x + 3) :؛ ) ( x − )(5 x −
3
4
3
2
.2ﻋﻤﻞ ﺍﻟﺘﻌﺒﲑﻳﻦ ﺍﻟﺘﺎﻟﻴﲔ (7 x − 1) + (49 x 2 − 1) :؛ x + 2 x − x − 2
ﺍﻟﺘﻤﺮﻳﻦ 5ﻋﻤﻞ ﺍﻟﺘﻌﺎﺑﲑ ﺍﻟﺘﺎﻟﻴﺔ:
3
a − b3 + a 2 − b 2
ﺍﻟﺘﻤﺮﻳﻦ 6ﺍﻧﺸﺮ ﺍﻟﺘﻌﺎﺑﲑ ﺍﻟﺘﺎﻟﻴﺔ:
()
؛
2
3
2
3
a +b + a −b
؛
)(a + b) − (a − b
3
3
)
)(2 a − 3 b )(4a + 6 ab + 9b
)(3 a + 5 b )(9a − 15 ab + 25b
a −1 a + a +1
(
4
L. L.
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