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Qoldiqli bo‘lish

a  p  q  r (0  r  p) ,
bu yerda a –bo‘linuvchi, p –bo‘luvchi, q –bo‘linma, r –qoldiq.

O‘lchov birliklari

1. 1 km=1000m. 7. 1m³=1000dm³=1000000sm³.
2. 1 m=10 dm=100sm. 8. 1dm³=1000sm³.
3. 1 sm=10 mm. 9. 1l=1dm³=1000sm³.
4. 1 km²=1000000m². 10. 1t=10s=1000kg.
5. 1 m²=100dm²=10000sm². 11. 1kg=1000g.
6. 1 ga=100ar=10000m². 12. 1g=1000mg.

Oddiy kasrlar ustida amallar

a _ c _ a  c _; a _ c

_ ad  bc _; a _ c

_ a  c _;

a _: c

_ a _ d

_ a  d _.

b b b
b d bd
b d b  d
b d b c b  c

Agar
a : b  c : d

Proporsiya

bo‘lsa, quyidagilar o‘rinli:

1. 
   ad  bc ; 2) ^{ma  nb}  ^{mc  nd} ;
   

pa  qb pc  qd

₃₎ a  b _ b _{; 4)} a  c

 ^a ;

c  d d b  d b

Davriy o‘nli kasrlarni oddiy kasrga aylantirish

a , a a a a


(b b b b



_{) } a₀a₁a₂a₃....a_kb₁b₂b₃....b_n  a₀a₁a₂a₃ a_k

0 1 2 3
k 1 2 3 n
9_9_–9_...9_–990_0_–0_...0_–00
n k

Sonli tengsizliklar

1. 
   a  b 
   

a  b  0 , a  b
 b  a ;

2. 
   a  b,
   
3. 
   a  b,
   

4. 
   a  b,
   

b  c с  0
с  0
 a  c , a  b 
 ac  bc, ^a  ^b ;
c c
 ac  bc, ^a  ^b ;
c c
a  с  b  c ;

5) a  b  0
 aⁿ  bⁿ ,

• 
  (n ) ;
  

6) a  b  0,
c  0 
a _ b _, c _ c _.

c c a b

Agar


x₁, x₂ ,..., x_n

O‘rta qiymatlar

musbat sonlar bo‘lsa, ularning:

1. o‘rta arifmetigi:

_{A } x₁  x₂  ...  x_{n ;}
n

2. 
   o‘rta geometrigi: G  ;
   

3. o‘rta garmonigi: H 

1
x₁
n _;

x

x


¹  ...  ¹
2 n

x ²  x ²  ...  x ²

4. 
   o‘rta kvadratigi: K  ^{1 2 n} ;
   

n
Koshi teoremasi: H  G  A  K .

Protsentlar

1) a sonining P protsentini topish:
_{x } a  P _;
100

2. 
   P protsenti a ga teng sonni topish:
   

_{x } a 100 _;
P

2. 
   a soni b sonining
   

^a 100%
b
ini tashkil etadi;

2. 
   a soni a + b yig‘indining
   

a a  b
100%
ini tashkil etadi.

Daraja va uning xossalari

1) aⁿ  a__–a_–_a _–  _– a _{; 2)}
n
a⁰  1,
a¹  a _;

3) a^{ p}  ¹
_a p
; 4)
a ^p  a^q  a ^pq ;

2. 
   a ^p : a^q  a^pq ; 6) _a ^p _q  a ^pq ;
   

⁷⁾ ^{a  b}^p
 ^a  ^p
 a^p  b^p ; 8) _{ }
 ^b 
_a p


_bp .

Bu yerda
n  N ,
a  0,
b  0 ;
q, p R .

Haqiqiy sonning moduli


a  ^
a, agar
a  0 bo'lsa;

1) ab 
a  b ;
^a, agar

2)

a  0 bo'lsa.


 ;

3) a  b 
a  b ; 4)
a  b 
a  b ; 5)
a ²  a² .

Arifmetik ildiz va uning xossalari

^ a  0,



_ⁿ a _  ^


 0,
_  a ,
n  2k, k  N ,




n
^_^{n a} 


 a.
^a,
n  2k 1,
k  N.

1)   ; 2)  ;

n


3) a^m ; 4) ^mp a^np
5)  aⁿ a ^p ; 6)
7)  ^mnp a^npb^pc ; 8)

 ^m aⁿ ;

 ^mn a ;
_ n1 _{a ;}

9)  
^{A  m} ; m  .
2

Qisqa ko‘paytirish formulalari

¹⁾ ^{a  b}^{2  a2  2ab  b2 ; 3)} ^{a  b}^{3  a3  3a2b  3ab2  b3 ;}

^{2) a2  b2 } ^{a  b} ^{a  b} ^{; 4)
5) a4  b4 } ^{a  b} ^{a  b} _^{a2  b2} _^{; 6)
a3  b3 } ^{a  b} _^{a2 ∓ ab  b2} _^;
(a  b  c)²  a²  b²  c²  2(ab  ac  bc) .

Kombinatorika elementlari

1. 
   m ta elementdan n tadan barcha o‘rinlashtirishlar soni:
   

m

A
ⁿ  m(m 1)(m  2)...(m  n  1) 
m! _.
(m  n)!

2. 
   n ta elementdan barcha o‘rin almashtirishlari soni:
   

P_n  n!  1 2  3  4 ...  n .

3. 
   m ta elementdan n tadan barcha gruppalashlar soni:
   

A



m

n

C
_{n m } m(m 1)(m  2)...(m  n  1) _
P_n 1 2  3   n
m! _.
(m  n)! n!

Chiziqli tenglama

ax  b  0 .

1. a  0,
2. a  0,
b  R b  0
bo‘lsa, yagona yechimga ega: bo‘lsa, yechimi yo‘q;
x   ^b ;
a

3. a  0,
b  0
bo‘lsa, yechimi cheksiz ko‘p: x  R .

Kvadrat tenglama

ax²  bx  c  0 ( a  0 ).

1. D  b²  4ac  0
bo‘lsa, 2 ta har xil haqiqiy yechimi bor:
b 

x_1,2  _2a .

2. D  b²  4ac  0
bo‘lsa, 1 ta ikki karrali haqiqiy yechimi bor:

x  ^b .

3. D  b²  4ac  0
1,2 _2a
bo‘lsa, haqiqiy yechimi yo‘q.

Yechimlarining xossalari:

Agar
x₁ va
x₂ sonlar
ax²  bx  c  0
tenglamaning ildizlari bo‘lsa, u holda:

1. 
   x  x   ^b , x  x  ^c