1300 Math Formulas

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1300 Math Formulas - Alex Svirin

Transcript of 1300 Math Formulas

  • 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 =