1300 Math Formulas
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Transcript of 1300 Math Formulas
1300 Math Formulas ==============================
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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===
==
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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=
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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=
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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=
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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=
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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= = = =
==
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=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~=
+= ===
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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 =