1300 Math Formulas ==============================

fp_k= =VVQVNMTTQN==

`==OMMQ=^KpK=^=o=oK=

i

=q=~==~==~K=

ii

Preface ====q= ~= = ~= = = = = -= ~= K= f= ~= = = = =~==~==~~=~~==I=I=~=I=~=~~K=q==~= = = ~I= ~I= ~= = =k= pI= ^~I= dI= qI= j~=~= a~I= sI= ^~= dI= `~I=a~=b~I=pI=~=m~=qK==q= = ~= = I= I= ~= ~= ~== = ~= ~= = ~= ~I= = =~===~=~=~===K===

==

iii

Contents ====

1 krj_bo=pbqp= NKN= p=f==1= NKO= p==k==5= NKP= _~=f==7= NKQ= `=k==8= =2 ^idb_o^= OKN= c~=c~==12= OKO= m=c~==13= OKP= m==14= OKQ= o==15= OKR= i~==16= OKS= b~==18= OKT= f~==19= OKU= `=f=c~==22= =3 dbljbqov= PKN= o=q~==24= PKO= f=q~==27= PKP= b~~=q~==28= PKQ= p~=q~==29= PKR= p~==33= PKS= o~==34= PKT= m~~~==35= PKU= o==36= PKV= q~==37= PKNM= f=q~==38= PKNN= f=q~==f=`==40= PKNO= q~==f=`==41=

iv

PKNP= h==42= PKNQ= `=n~~~==43= PKNR= q~~=n~~~==45= PKNS= d~=n~~~==46= PKNT= o~=e~==47= PKNU= o~=m==48= PKNV= `==50= PKOM= p==~=`==53= PKON= p==~=`==54= PKOO= `==55= PKOP= o~~=m~~==56= PKOQ= m==57= PKOR= o~=q~==58= PKOS= o~=m~==59= PKOT= c==~=o~=m~==61= PKOU= o~~=o=t==62= PKOV= m~=p==63= PKPM= o=`~=`==66= PKPN= o=`~=`==~=l=m~=c~==68= PKPO= o=`~=`==69= PKPP= c==~=o=`~=`==70= PKPQ= p==72= PKPR= p~=`~==72= PKPS= p~=p==73= PKPT= p~=p==74= PKPU= p~=t==75= PKPV= b==76= PKQM= `~=q==78= = =4 qofdlkljbqov= QKN= o~~=~=a=j~==^==80= QKO= a=~=d~==q=c==81= QKP= p==q=c==86= QKQ= q=c==`=^==87= QKR= j=f~=c~==88=

v

QKS= o=c~==89= QKT= m==q=c==90= QKU= o~==q=c==90= QKV= ^=~=p~=c~==91= QKNM= a=^=c~==92= QKNN= j=^=c~==93= QKNO= e~=^=c~==94= QKNP= e~=^=q~=f==94= QKNQ= q~==q=b==m==95= QKNR= q~==q=b==p==97=== QKNS= m==q=c==98= QKNT= d~==f=q=c==99= QKNU= m~=s~==f=q=c==102= QKNV= o~==f=q=c==103= QKOM= q=b~==106= QKON= o~==e=c==106= = =5 j^qof`bp=^ka=abqbojfk^kqp= RKN= a~==107= RKO= m==a~==109= RKP= j~==110= RKQ= l~==j~==111= RKR= p==i~=b~==114= = =6 sb`qlop= SKN= s=`~==118= SKO= s=^==120= SKP= s=p~==122= SKQ= p~=s==122= SKR= p~~=m==123= SKS= s=m==125= SKT= q=m=127= = =7 ^k^ivqf`=dbljbqov= TKN= l=-a~=`~=p==130=

vi

TKO= q=-a~=`~=p==131= TKP= p~=i==m~==139= TKQ= `==149= TKR= b==152= TKS= e~==154= TKT= m~~~==158= TKU= q=-a~=`~=p==161= TKV= m~==165= TKNM= p~=i==p~==175= TKNN= n~=p~==180= TKNO= p==189= = =8 afccbobkqf^i=`^i`rirp= UKN= c=~=q=d~==191= UKO= i==c==208= UKP= a=~=m===a~==209= UKQ= q~==a~==211= UKR= e=l=a~==215= UKS= ^~==a~==217= UKT= a~==221= UKU= j~~=c==222= UKV= a~=l~==225= = =9 fkqbdo^i=`^i`rirp= VKN= f=f~==227= VKO= f~==o~~=c==228= VKP= f~==f~~=c==231= VKQ= f~==q=c==237= VKR= f~==e=c==241= VKS= f~==b~=~=i~=c==242= VKT= o=c~==243= VKU= a=f~==247= VKV= f=f~==253= VKNM= a=f~==257= VKNN= q=f~==269=

vii

VKNO= i=f~==275= VKNP= p~=f~==285= = =10 afccbobkqf^i=bnr^qflkp= NMKN= c=l=l~=a~=b~==295= NMKO= p=l=l~=a~=b~==298= NMKP= p=m~~=a~=b~==302= = =11 pbofbp= NNKN= ^=p==304= NNKO= d=p==305= NNKP= p=c=p==305= NNKQ= f=p==307= NNKR= m==`=p==307= NNKS= `=q==308= NNKT= ^~=p==310= NNKU= m=p==311= NNKV= a~=~=f~==m=p==312= NNKNM= q~=~=j~~=p==313= NNKNN= m=p=b~==p=c==314= NNKNO= _~=p==316= NNKNP= c=p==316= = =12 mol_^_fifqv= NOKN= m~=~=`~==318= NOKO= m~=c~==319= = = =

==

viii

=q=~==~==~K=

=

1

Chapter 1

Number Sets ====

1.1 Set Identities =pW=^I=_I=`=r~=W=f=`=W= ^ =m=W= _^ ==b=W= =r==W= _^ =f==W= _^ =a==W= _y^ ==

=1. f^ =

=2. ^^ =

=3. _^ = == _^ =~= ^_ .=

=4. b=p=

^ ==

5. r==p=={ }_^_^` == =

=

CHAPTER 1. NUMBER SETS

2

===== ==

Figure 1. =

6. `~=^__^ = =

=7. ^~=

( ) ( ) `_^`_^ = ==8. f==p=

{ }_~^_^` == = ==

===== ==

Figure 2. =

9. `~=^__^ = =

=10. ^~=

( ) ( ) `_^`_^ = ==

CHAPTER 1. NUMBER SETS

3

11. a=( ) ( ) ( )`^_^`_^ = I=( ) ( ) ( )`^_^`_^ = K=

=12. f=

^^^ = I==^^^ = =

=13. a~=

=^ I=ff^ = =

=14. f=

^^ = I==^f^ =

=15. `=

{ }^f^ = =

16. `==f=~=r f^^ = I===^^ =

=17. a=j~=i~

( ) _^_^ = I==( ) _^_^ = ==

18. a==p { }^~_^y_` == =

=

CHAPTER 1. NUMBER SETS

4

===== ==

Figure 3. =

19. ( )_^y_^y_ = =

20. ^_^y_ = =

21. =^y^ =

22. ^_y^ = == =_^ . =

===== ==

Figure 4. =

23. ( ) ( ) ( )`_y`^`_y^ = 24. ^yf^ = 25. `~~=m

( ){ }_~^I_^` == ==

CHAPTER 1. NUMBER SETS

5

1.2 Sets of Numbers =k~~=W=k=t=W= Mk =

fW=w=m=W= +w =k~=W= w =o~~=W=n=o~=W=o==`=W=`====

26. k~~=k `=W { }KIPIOINk = K=

27. t=k

`==~=W= { }KIPIOINIMkM = K==28. f

t==~===~=W={ }KIPIOINkw ==+ I=

{ }NIOIPIw = K I={ } { }KK IPIOINIMINIOIPIwMww == + K=

=29. o~~=k

o~==~=~W==

== M~w~w~~

~n K=

=30. f~~=k

k~=~=~=~K =

CHAPTER 1. NUMBER SETS

6

31. o~=k==r==~~=~=~~=W=oK=

=32. `=k

{ }o~o` += I======~~=K

=33. `onwk =

=

=== ==

Figure 5. ======

CHAPTER 1. NUMBER SETS

7

1.3 Basic Identities =o~=W=~I=I==

==

34. ^=f=~M~ =+ =

=35. ^=f=

( ) M~~ =+ ==

36. `~==^=~~ +=+ =

=37. ^~==^=

( ) ( )~~ ++=++ ==38. a==p~=

( )~~ += ==

39. j~=f=~N~ = =

=40. j~=f=

N~

N~ = I= M~

=41. j~=q=M

MM~ = =

42. `~==j~=~~ =

==

CHAPTER 1. NUMBER SETS

8

43. ^~==j~=( ) ( )~~ =

=44. a=i~=

( ) ~~~ +=+ ==45. a==a=

N~

~= =

===

1.4 Complex Numbers =k~~=W==f~~=W==`=W==o~=~W=~I==f~~=~W=I==j==~==W=I= N I= O =^==~==W= I= N I= O =

==

N = = R = = NQ =+ =NO = = NS = = N OQ =+ =P = = T = = PQ =+ =

46.

NQ = = NU = = N Q = ==47. ~ += ==48. `=m~=

=

CHAPTER 1. NUMBER SETS

9

===== ==

Figure 6. =

49. ( ) ( ) ( ) ( )~~ +++=+++ ==50. ( ) ( ) ( ) ( )~~ +=++ ==51. ( )( ) ( ) ( )~~~ ++=++ ==

52.

~

~

~OOOO

+

+++

=++

=

=53. `~=`=k=

~~|||||||

=+ ==54. = ~ I= = ==

=

CHAPTER 1. NUMBER SETS

10

==

Figure 7. =

55. m~=m~==`=k=( )+=+ ~ =

=56. j=~=^==~=`=k=

f= ~ + ==~==I==OO ~ += =EFI==

~

~~= =E~FK=

=57. m==m~=o~=

( ) ( )OOONNNON ++= =( ) ( )[ ]ONONON +++= =

=58. `~=k==m~=o~=

( ) ( ) ( )[ ]+=+ |||||||||||||||||||||

==59. f==~=`=k==m~=o~=

( ) ( ) ( )[ ]+=+ N

N=

CHAPTER 1. NUMBER SETS

11

60. n==m~=o~=( )( ) ( ) ( )[ ]ONONO

N

OOO

NNN

O

N

+=

++

= =

=61. m==~=`=k=

( )[ ] ( ) ( )[ ]+=+= ==62. c~=a=j=

( ) ( ) ( )+=+ ==63. k=o==~=`=k=

( )

++

+=+=

O

O I==

==NIIOINIM = K K==

=64. b=c~=

+= ===

12

Chapter 2

Algebra ====

2.1 Factoring Formulas =o~=W=~I=I===k~~=W==

==

65. ( )( )~~~ OO += ==

66. ( )( )OOPP ~~~~ ++= ==67. ( )( )OOPP ~~~~ ++=+ ==68. ( )( ) ( )( )( )OOOOOOQQ ~~~~~~ ++=+= ==

69. ( )( )QPOOPQRR ~~~~~~ ++++= ==70. ( )( )QPOOPQRR ~~~~~~ +++=+ ==71. f===I==

( )( )NOOPON ~~~~~~ +++=+ K K===72. f===I===

( )( )NOOPON ~~~~~~ +++++= K I==

CHAPTER 2. ALGEBRA

13

( )( )NOOPON ~~~~~~ +++=+ K K====

2.2 Product Formulas o~=W=~I=I===t=W=I====

73. ( ) OOO ~O~~ += ==

74. ( ) OOO ~O~~ ++=+ ==

75. ( ) POOPP ~P~P~~ += ==

76. ( ) POOPP ~P~P~~ +++=+ ==

77. ( ) QPOOPQQ ~Q~S~Q~~ ++= ==

78. ( ) QPOOPQQ ~Q~S~Q~~ ++++=+ ==79. _~=c~=

( ) I`~`~`~`~`~ NNOOONNM +++++=+ K

= ( )>>>

`

= =~==~=K=

=

80. ( ) O~O~O~~ OOOO +++++=++ ==

81. ( ) ++++++=+++++ OOOOOO ~~ KK =( )~~~~O +++++++++++ KKK =

CHAPTER 2. ALGEBRA

14

2.3 Powers =_~=E=~=FW=~I===m=E~~=FW=I==

==

82. ~~~ += ==

83.

~~

~ = =

=

84. ( ) ~~ = ==

85.

~

~=

=

=

86. ( ) ~~ = ==87. N~M = I= M~ ==88. N~N = ==

89.

~

N~ = =

=

90.

~~ = ======

CHAPTER 2. ALGEBRA

15

2.4 Roots =_~W=~I===m=E~~=FW=I==

MI~ ====E O = I= k F===

91. ~~ = ==

92. ~~ = ==

93.

~

~= I= M =

=

94.

~

~

~== I= M K=

=

95. ( ) ~~ = ==

96. ( ) ~~ = ==

97. ~~ = ==

98.

~~ = ==

99. ~~ = ==

100. ( ) ~~ = ==

CHAPTER 2. ALGEBRA

16

101. ~

~

~

N N

= I= M~ K=

=

102. O

~~

O

~~~

OO

+= =

=

103. ~

~

~

N

=

m

=

===

2.5 Logarithms =m=~=W=I=I=~I=I==k~~=W=====

104. a==i~= ~= ==~===

~ = I= M~ > I= N~ K==

105. MN~ = ==

106. N~~ = ==

107.

=N~

N~M~ =

=108. ( ) ~~~ += =

=

109.

~~~ = =

CHAPTER 2. ALGEBRA

17

110. ( ) ~~ = ==

111.

N ~

~ = =

=

112. ~

~

~ == I= M > I= N K=

=

113. ~

N

~ = =

=114. ~~ = =

=115. i~==_~=NM=

NM = ==

116. k~~=i~= = I==

= KTNUOUNUOUKO

NN

=

+=

=

=

117. QPQOVQKMNM

N == =

=

118. PMORURKO

N == =

=====

CHAPTER 2. ALGEBRA

18

2.6 Equations =o~=W=~I=I=I=I=I=I==pW= N I= O I= N I= O I= P =

==

119. i~=b~==l=s~~=

M~ =+ I=~

= K==

=120. n~~=b~=

M~O =++ I=~O

~Q

O

OIN

= K=

=121. a~=

~Qa O =