1300 Math Formulas
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qĩỏở=ộ~ệẫ=ỏở=ỏồớẫồớỏỗồ~ọọú=ọẫẹớ=ọ~ồõK=

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Preface
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qĩỏở= ĩ~ồầỗỗõ= ỏở= ~= ỗóộọẫớẫ= ầẫởõớỗộ= ờẫẹẫờẫồẫ= ẹỗờ= ởớỡầẫồớở= ~ồầ= ẫồệỏồẫẫờởK= fớ= ĩ~ở= ẫợẫờúớĩỏồệ= ẹờỗó= ĩỏệĩ= ởĩỗỗọ=
ó~ớĩ=ớỗ=ó~ớĩ=ẹỗờ=~ầợ~ồẫầ=ỡồầẫờệờ~ầỡ~ớẫở=ỏồ=ẫồệỏồẫẫờỏồệI=
ẫỗồỗóỏởI=ộĩúởỏ~ọ=ởỏẫồẫởI=~ồầ=ó~ớĩẫó~ớỏởK=qĩẫ=ẫỗỗõ=
ỗồớ~ỏồở= ĩỡồầờẫầở= ỗẹ= ẹỗờóỡọ~ởI= ớ~ọẫởI= ~ồầ= ẹỏệỡờẫở= ẹờỗó=
kỡóẫờ= pẫớởI= ^ọệẫờ~I= dẫỗóẫớờúI= qờỏệỗồỗóẫớờúI= j~ớờỏẫở=
~ồầ= aẫớẫờóỏồ~ồớởI= sẫớỗờởI= ^ồ~ọúớỏ= dẫỗóẫớờúI= `~ọỡọỡởI=
aỏẹẹẫờẫồớỏ~ọ=bốỡ~ớỏỗồởI=pẫờỏẫởI=~ồầ=mờỗ~ỏọỏớú=qĩẫỗờúK==
qĩẫ= ởớờỡớỡờẫầ= ớ~ọẫ= ỗẹ= ỗồớẫồớởI= ọỏồõởI= ~ồầ= ọ~úỗỡớ= ó~õẫ=
ẹỏồầỏồệ= ớĩẫ= ờẫọẫợ~ồớ= ỏồẹỗờó~ớỏỗồ= ốỡỏõ= ~ồầ= ộ~ỏồọẫởởI= ởỗ= ỏớ=
~ồ=ẫ=ỡởẫầ=~ở=~ồ=ẫợẫờúầ~ú=ỗồọỏồẫ=ờẫẹẫờẫồẫ=ệỡỏầẫK===

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Contents
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1 krj_bo=pbqp=
NKN= pẫớ=fầẫồớỏớỏẫở==1=
NKO= pẫớở=ỗẹ=kỡóẫờở==5=
NKP= _~ởỏ=fầẫồớỏớỏẫở==7=
NKQ= `ỗóộọẫủ=kỡóẫờở==8=
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2 ^idb_o^=
OKN= c~ớỗờỏồệ=cỗờóỡọ~ở==12=
OKO= mờỗầỡớ=cỗờóỡọ~ở==13=
OKP= mỗùẫờở==14=
OKQ= oỗỗớở==15=
OKR= iỗệ~ờỏớĩóở==16=
OKS= bốỡ~ớỏỗồở==18=
OKT= fồẫốỡ~ọỏớỏẫở==19=
OKU= `ỗóộỗỡồầ=fồớẫờẫởớ=cỗờóỡọ~ở==22=
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3 dbljbqov=
PKN= oỏệĩớ=qờỏ~ồệọẫ==24=

PKO= fởỗởẫọẫở=qờỏ~ồệọẫ==27=
PKP= bốỡỏọ~ớẫờ~ọ=qờỏ~ồệọẫ==28=
PKQ= p~ọẫồẫ=qờỏ~ồệọẫ==29=
PKR= pốỡ~ờẫ==33=
PKS= oẫớ~ồệọẫ==34=
PKT= m~ờ~ọọẫọỗệờ~ó==35=
PKU= oĩỗóỡở==36=
PKV= qờ~ộẫũỗỏầ==37=
PKNM= fởỗởẫọẫở=qờ~ộẫũỗỏầ==38=
PKNN= fởỗởẫọẫở=qờ~ộẫũỗỏầ=ùỏớĩ=fồởờỏẫầ=`ỏờọẫ==40=
PKNO= qờ~ộẫũỗỏầ=ùỏớĩ=fồởờỏẫầ=`ỏờọẫ==41=

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PKNP= hỏớẫ==42=
PKNQ= `úọỏ=nỡ~ầờỏọ~ớẫờ~ọ==43=
PKNR= q~ồệẫồớỏ~ọ=nỡ~ầờỏọ~ớẫờ~ọ==45=
PKNS= dẫồẫờ~ọ=nỡ~ầờỏọ~ớẫờ~ọ==46=
PKNT= oẫệỡọ~ờ=eẫủ~ệỗồ==47=
PKNU= oẫệỡọ~ờ=mỗọúệỗồ==48=
PKNV= `ỏờọẫ==50=
PKOM= pẫớỗờ=ỗẹ=~=`ỏờọẫ==53=
PKON= pẫệóẫồớ=ỗẹ=~=`ỏờọẫ==54=
PKOO= `ỡẫ==55=
PKOP= oẫớ~ồệỡọ~ờ=m~ờ~ọọẫọẫộỏộẫầ==56=
PKOQ= mờỏởó==57=
PKOR= oẫệỡọ~ờ=qẫớờ~ĩẫầờỗồ==58=
PKOS= oẫệỡọ~ờ=múờ~óỏầ==59=

PKOT= cờỡởớỡó=ỗẹ=~=oẫệỡọ~ờ=múờ~óỏầ==61=
PKOU= oẫớ~ồệỡọ~ờ=oỏệĩớ=tẫầệẫ==62=
PKOV= mọ~ớỗồỏ=pỗọỏầở==63=
PKPM= oỏệĩớ=`ỏờỡọ~ờ=`úọỏồầẫờ==66=
PKPN= oỏệĩớ=`ỏờỡọ~ờ=`úọỏồầẫờ=ùỏớĩ=~ồ=lọỏốỡẫ=mọ~ồẫ=c~ẫ==68=
PKPO= oỏệĩớ=`ỏờỡọ~ờ=`ỗồẫ==69=
PKPP= cờỡởớỡó=ỗẹ=~=oỏệĩớ=`ỏờỡọ~ờ=`ỗồẫ==70=
PKPQ= pộĩẫờẫ==72=
PKPR= pộĩẫờỏ~ọ=`~ộ==72=
PKPS= pộĩẫờỏ~ọ=pẫớỗờ==73=
PKPT= pộĩẫờỏ~ọ=pẫệóẫồớ==74=
PKPU= pộĩẫờỏ~ọ=tẫầệẫ==75=
PKPV= bọọỏộởỗỏầ==76=
PKQM= `ỏờỡọ~ờ=qỗờỡở==78=
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4 qofdlkljbqov=
QKN= o~ầỏ~ồ=~ồầ=aẫệờẫẫ=jẫ~ởỡờẫở=ỗẹ=^ồệọẫở==80=
QKO= aẫẹỏồỏớỏỗồở=~ồầ=dờ~ộĩở=ỗẹ=qờỏệỗồỗóẫớờỏ=cỡồớỏỗồở==81=
QKP= pỏệồở=ỗẹ=qờỏệỗồỗóẫớờỏ=cỡồớỏỗồở==86=
QKQ= qờỏệỗồỗóẫớờỏ=cỡồớỏỗồở=ỗẹ=`ỗóóỗồ=^ồệọẫở==87=
QKR= jỗởớ=fóộỗờớ~ồớ=cỗờóỡọ~ở==88=

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QKS= oẫầỡớỏỗồ=cỗờóỡọ~ở==89=
QKT= mẫờỏỗầỏỏớú=ỗẹ=qờỏệỗồỗóẫớờỏ=cỡồớỏỗồở==90=
QKU= oẫọ~ớỏỗồở=ẫớùẫẫồ=qờỏệỗồỗóẫớờỏ=cỡồớỏỗồở==90=

QKV= ^ầầỏớỏỗồ=~ồầ=pỡớờ~ớỏỗồ=cỗờóỡọ~ở==91=
QKNM= aỗỡọẫ=^ồệọẫ=cỗờóỡọ~ở==92=
QKNN= jỡọớỏộọẫ=^ồệọẫ=cỗờóỡọ~ở==93=
QKNO= e~ọẹ=^ồệọẫ=cỗờóỡọ~ở==94=
QKNP= e~ọẹ=^ồệọẫ=q~ồệẫồớ=fầẫồớỏớỏẫở==94=
QKNQ= qờ~ồởẹỗờóỏồệ=ỗẹ=qờỏệỗồỗóẫớờỏ=bủộờẫởởỏỗồở=ớỗ=mờỗầỡớ==95=
QKNR= qờ~ồởẹỗờóỏồệ=ỗẹ=qờỏệỗồỗóẫớờỏ=bủộờẫởởỏỗồở=ớỗ=pỡó==97===
QKNS= mỗùẫờở=ỗẹ=qờỏệỗồỗóẫớờỏ=cỡồớỏỗồở==98=
QKNT= dờ~ộĩở=ỗẹ=fồợẫờởẫ=qờỏệỗồỗóẫớờỏ=cỡồớỏỗồở==99=
QKNU= mờỏồỏộ~ọ=s~ọỡẫở=ỗẹ=fồợẫờởẫ=qờỏệỗồỗóẫớờỏ=cỡồớỏỗồở==102=
QKNV= oẫọ~ớỏỗồở=ẫớùẫẫồ=fồợẫờởẫ=qờỏệỗồỗóẫớờỏ=cỡồớỏỗồở==103=
QKOM= qờỏệỗồỗóẫớờỏ=bốỡ~ớỏỗồở==106=
QKON= oẫọ~ớỏỗồở=ớỗ=eúộẫờỗọỏ=cỡồớỏỗồở==106=
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5 j^qof`bp=^ka=abqbojfk^kqp=
RKN= aẫớẫờóỏồ~ồớở==107=
RKO= mờỗộẫờớỏẫở=ỗẹ=aẫớẫờóỏồ~ồớở==109=
RKP= j~ớờỏẫở==110=
RKQ= lộẫờ~ớỏỗồở=ùỏớĩ=j~ớờỏẫở==111=
RKR= púởớẫóở=ỗẹ=iỏồẫ~ờ=bốỡ~ớỏỗồở==114=
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6 sb`qlop=
SKN= sẫớỗờ=`ỗỗờầỏồ~ớẫở==118=
SKO= sẫớỗờ=^ầầỏớỏỗồ==120=
SKP= sẫớỗờ=pỡớờ~ớỏỗồ==122=
SKQ= p~ọỏồệ=sẫớỗờở==122=
SKR= p~ọ~ờ=mờỗầỡớ==123=
SKS= sẫớỗờ=mờỗầỡớ==125=

SKT= qờỏộọẫ=mờỗầỡớ=127=
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7 ^k^ivqf`=dbljbqov=
TKN= lồẫ=-aỏóẫồởỏỗồ~ọ=`ỗỗờầỏồ~ớẫ=púởớẫó==130=

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TKO= qùỗ=-aỏóẫồởỏỗồ~ọ=`ỗỗờầỏồ~ớẫ=púởớẫó==131=
TKP= pớờ~ỏệĩớ=iỏồẫ=ỏồ=mọ~ồẫ==139=
TKQ= `ỏờọẫ==149=
TKR= bọọỏộởẫ==152=
TKS= eúộẫờỗọ~==154=
TKT= m~ờ~ỗọ~==158=
TKU= qĩờẫẫ=-aỏóẫồởỏỗồ~ọ=`ỗỗờầỏồ~ớẫ=púởớẫó==161=
TKV= mọ~ồẫ==165=
TKNM= pớờ~ỏệĩớ=iỏồẫ=ỏồ=pộ~ẫ==175=
TKNN= nỡ~ầờỏ=pỡờẹ~ẫở==180=
TKNO= pộĩẫờẫ==189=
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8 afccbobkqf^i=`^i`rirp=
UKN= cỡồớỏỗồở=~ồầ=qĩẫỏờ=dờ~ộĩở==191=
UKO= iỏóỏớở=ỗẹ=cỡồớỏỗồở==208=
UKP= aẫẹỏồỏớỏỗồ=~ồầ=mờỗộẫờớỏẫở=ỗẹ=ớĩẫ=aẫờỏợ~ớỏợẫ==209=
UKQ= q~ọẫ=ỗẹ=aẫờỏợ~ớỏợẫở==211=
UKR= eỏệĩẫờ=lờầẫờ=aẫờỏợ~ớỏợẫở==215=
UKS= ^ộộọỏ~ớỏỗồở=ỗẹ=aẫờỏợ~ớỏợẫ==217=

UKT= aỏẹẹẫờẫồớỏ~ọ==221=
UKU= jỡọớỏợ~ờỏ~ọẫ=cỡồớỏỗồở==222=
UKV= aỏẹẹẫờẫồớỏ~ọ=lộẫờ~ớỗờở==225=
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9 fkqbdo^i=`^i`rirp=
VKN= fồầẫẹỏồỏớẫ=fồớẫệờ~ọ==227=
VKO= fồớẫệờ~ọở=ỗẹ=o~ớỏỗồ~ọ=cỡồớỏỗồở==228=
VKP= fồớẫệờ~ọở=ỗẹ=fờờ~ớỏỗồ~ọ=cỡồớỏỗồở==231=
VKQ= fồớẫệờ~ọở=ỗẹ=qờỏệỗồỗóẫớờỏ=cỡồớỏỗồở==237=
VKR= fồớẫệờ~ọở=ỗẹ=eúộẫờỗọỏ=cỡồớỏỗồở==241=
VKS= fồớẫệờ~ọở=ỗẹ=bủộỗồẫồớỏ~ọ=~ồầ=iỗệ~ờỏớĩóỏ=cỡồớỏỗồở==242=
VKT= oẫầỡớỏỗồ=cỗờóỡọ~ở==243=
VKU= aẫẹỏồỏớẫ=fồớẫệờ~ọ==247=
VKV= fóộờỗộẫờ=fồớẫệờ~ọ==253=
VKNM= aỗỡọẫ=fồớẫệờ~ọ==257=
VKNN= qờỏộọẫ=fồớẫệờ~ọ==269=

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VKNO= iỏồẫ=fồớẫệờ~ọ==275=
VKNP= pỡờẹ~ẫ=fồớẫệờ~ọ==285=
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10 afccbobkqf^i=bnr^qflkp=
NMKN= cỏờởớ=lờầẫờ=lờầỏồ~ờú=aỏẹẹẫờẫồớỏ~ọ=bốỡ~ớỏỗồở==295=
NMKO= pẫỗồầ=lờầẫờ=lờầỏồ~ờú=aỏẹẹẫờẫồớỏ~ọ=bốỡ~ớỏỗồở==298=
NMKP= pỗóẫ=m~ờớỏ~ọ=aỏẹẹẫờẫồớỏ~ọ=bốỡ~ớỏỗồở==302=

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11 pbofbp=
NNKN= ^ờỏớĩóẫớỏ=pẫờỏẫở==304=
NNKO= dẫỗóẫớờỏ=pẫờỏẫở==305=
NNKP= pỗóẫ=cỏồỏớẫ=pẫờỏẫở==305=
NNKQ= fồẹỏồỏớẫ=pẫờỏẫở==307=
NNKR= mờỗộẫờớỏẫở=ỗẹ=`ỗồợẫờệẫồớ=pẫờỏẫở==307=
NNKS= `ỗồợẫờệẫồẫ=qẫởớở==308=
NNKT= ^ọớẫờồ~ớỏồệ=pẫờỏẫở==310=
NNKU= mỗùẫờ=pẫờỏẫở==311=
NNKV= aỏẹẹẫờẫồớỏ~ớỏỗồ=~ồầ=fồớẫệờ~ớỏỗồ=ỗẹ=mỗùẫờ=pẫờỏẫở==312=
NNKNM= q~úọỗờ=~ồầ=j~ọ~ỡờỏồ=pẫờỏẫở==313=
NNKNN= mỗùẫờ=pẫờỏẫở=bủộ~ồởỏỗồở=ẹỗờ=pỗóẫ=cỡồớỏỗồở==314=
NNKNO= _ỏồỗóỏ~ọ=pẫờỏẫở==316=
NNKNP= cỗỡờỏẫờ=pẫờỏẫở==316=
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12 mol_^_fifqv=
NOKN= mẫờóỡớ~ớỏỗồở=~ồầ=`ỗóỏồ~ớỏỗồở==318=
NOKO= mờỗ~ỏọỏớú=cỗờóỡọ~ở==319=
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qĩỏở=ộ~ệẫ=ỏở=ỏồớẫồớỏỗồ~ọọú=ọẫẹớ=ọ~ồõK=
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Chapter 1

Number Sets
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1.1 Set Identities
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pẫớởW=^I=_I=`=
rồỏợẫờở~ọ=ởẫớW=f=
`ỗóộọẫóẫồớ=W= ^ =
mờỗộẫờ=ởỡởẫớW= ^ _ ==
bóộớú=ởẫớW= =
rồỏỗồ=ỗẹ=ởẫớởW= ^ _ =
fồớẫờởẫớỏỗồ=ỗẹ=ởẫớởW= ^ _ =
aỏẹẹẫờẫồẫ=ỗẹ=ởẫớởW= ^ y _ =
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1.

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4.
5.

^ f=
^ ^=

^ = _ =ỏẹ= ^ _ =~ồầ= _ ^ .=

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bóộớú=pẫớ=
^=
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rồỏỗồ=ỗẹ=pẫớở==
` = ^ ∪ _ = {đ ư đ ∈ ^ çê ñ ∈ _}=
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CHAPTER 1. NUMBER SETS

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Figure 1.

6.

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8.

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`ỗóóỡớ~ớỏợỏớú=
^_ = _^=
^ởởỗỏ~ớỏợỏớú=
^ (_ ` ) = (^ _ ) ` =
fồớẫờởẫớỏỗồ=ỗẹ=pẫớở=
` = ^ ∪ _ = {đ ư đ ∈ ^ ~ồầ ủ _} =
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Figure 2.

9.


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10.

=

=
`ỗóóỡớ~ớỏợỏớú=
^_ = _^=
^ởởỗỏ~ớỏợỏớú=
^ (_ ∩ ` ) = (^ ∩ _ ) ∩ ` =

=

2
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CHAPTER 1. NUMBER SETS

11.

=
12.
=
13.
=
14.

ấëíêáÄìíáỵáíó=

^ ∪ (_ ∩ ` ) = (^ ∪ _ ) ∩ (^ ∪ ` ) I=
^ ∩ (_ ∪ ` ) = (^ ∩ _ ) (^ ` ) K=
fầẫóộỗớẫồú=
^ ^ = ^ I==
^^ = ^=
aỗóỏồ~ớỏỗồ=
^ = I=
^f= f=
fầẫồớỏớú=
^ = ^ I==
^f= ^
=

15.

16.

17.

18.

`ỗóộọẫóẫồớ=
^ = {ủ f ử ủ ^}
=
`ỗóộọẫóẫồớ=ỗẹ=fồớẫờởẫớỏỗồ=~ồầ=rồỏỗồ
^ ^ = f I==
^ ^ = =
=
aẫ=jỗờệ~ồở=i~ùở
(^ _ ) = ^ _ I==

(^ _ ) = ^ _ =
=
aỏẹẹẫờẫồẫ=ỗẹ=pẫớở
` = _ y ^ = {đ ư đ ∈ _ ~åÇ ñ ∉ ^} =
=

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CHAPTER 1. NUMBER SETS

=

=====

=
Figure 3.

=
19.

_ y ^ = _ y (^ ∩ _ )
=

20.

_ y ^ = _ ∩ ^′

21.


^y^=∅

22.

^ y _ = ^ =áÑ= ^ ∩ _ = ∅ .

=
=

=

=====

=
Figure 4.

=
23.

(^ y _) ∩ ` = (^ ∩ `) y (_ `)

24.

^ = f y ^

25.

`~ờớẫởỏ~ồ=mờỗầỡớ
` = ^ × _ = {(đ I ó ) ư đ ∈ ^ ~åÇ ó ∈ _}

=
=

4
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=


CHAPTER 1. NUMBER SETS

1.2 Sets of Numbers
=

26.

27.

=
28.

=
29.

=
30.

k~ớỡờ~ọ=ồỡóẫờởW=k=
tĩỗọẫ=ồỡóẫờởW= kM =
fồớẫệẫờởW=w=

mỗởỏớỏợẫ=ỏồớẫệẫờởW= w + =
kẫệ~ớỏợẫ=ỏồớẫệẫờởW= w =
o~ớỏỗồ~ọ=ồỡóẫờởW=n=
oẫ~ọ=ồỡóẫờởW=o==
`ỗóộọẫủ=ồỡóẫờởW=`==
=
=
k~ớỡờ~ọ=kỡóẫờở
`ỗỡồớỏồệ=ồỡóẫờởW k = {NI OI PI K} K=
tĩỗọẫ=kỡóẫờở
`ỗỡồớỏồệ=ồỡóẫờở=~ồầ=ũẫờỗW= k M = {MI NI OI PI K} K=
fồớẫệẫờở
tĩỗọẫ=ồỡóẫờở=~ồầ=ớĩẫỏờ=ỗộộỗởỏớẫở=~ồầ=ũẫờỗW=
w + = k = {NI OI PI K}I=
w = {KI − PI − OI − N} I=
w = w − ∪ {M} ∪ w + = {KI − PI − OI − NI MI NI OI PI K} K=
o~ớỏỗồ~ọ=kỡóẫờở
oẫộẫ~ớỏồệ=ỗờ=ớẫờóỏồ~ớỏồệ=ầẫỏó~ọởW==
~


n = ủ ử ủ = ~ồầ ~ w ~ồầ w ~ồầ M K=




fờờ~ớỏỗồ~ọ=kỡóẫờở
kỗồờẫộẫ~ớỏồệ=~ồầ=ồỗồớẫờóỏồ~ớỏồệ=ầẫỏó~ọởK

=


5
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CHAPTER 1. NUMBER SETS

31.

oẫ~ọ=kỡóẫờở==
rồỏỗồ=ỗẹ=ờ~ớỏỗồ~ọ=~ồầ=ỏờờ~ớỏỗồ~ọ=ồỡóẫờởW=oK=

=
32.

`ỗóộọẫủ=kỡóẫờở
` = {ủ + ỏú ử ủ ∈ o ~åÇ ó ∈ o}I==
ïÜÉêÉ=á=áë=íÜÉ=áã~Ưáå~êó=ìåáíK

=
33.

k⊂ w⊂n⊂ o ⊂ `=
=

===

=

=

Figure 5.

=
=
=
=
=
=

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CHAPTER 1. NUMBER SETS

1.3 Basic Identities
=
oẫ~ọ=ồỡóẫờởW=~I=I==
=
=
34.

^ầầỏớỏợẫ=fầẫồớỏớú=
~+M=~ =
=

35.

^ầầỏớỏợẫ=fồợẫờởẫ=
~ + ( ~ ) = M =

=

36.

`ỗóóỡớ~ớỏợẫ=ỗẹ=^ầầỏớỏỗồ=
~ += +~ =

37.

^ởởỗỏ~ớỏợẫ=ỗẹ=^ầầỏớỏỗồ=
(~ + ) + = ~ + ( + ) =

=

=
38.

aẫẹỏồỏớỏỗồ=ỗẹ=pỡớờ~ớỏỗồ=
~ = ~ + ( ) =
=

39.

=
40.

41.
42.

jỡọớỏộọỏ~ớỏợẫ=fầẫồớỏớú=

~ N = ~ =
jỡọớỏộọỏ~ớỏợẫ=fồợẫờởẫ=
N
~ = N I= ~ M
~
=
jỡọớỏộọỏ~ớỏỗồ=qỏóẫở=M
~ M = M
=
`ỗóóỡớ~ớỏợẫ=ỗẹ=jỡọớỏộọỏ~ớỏỗồ=
~ = ~
=
=

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CHAPTER 1. NUMBER SETS

43.

^ởởỗỏ~ớỏợẫ=ỗẹ=jỡọớỏộọỏ~ớỏỗồ=
(~ ) = ~ (Ä ⋅ Å )
=
ấëíêáÄìíáỵÉ=i~ï=
~ (Ä + Å ) = ~Ä + ~ =

44.


=
45.

aẫẹỏồỏớỏỗồ=ỗẹ=aỏợỏởỏỗồ=
~
N
= ~ =


=
=
=

1.4 Complex Numbers
=
k~ớỡờ~ọ=ồỡóẫờW=ồ=
fó~ệỏồ~ờú=ỡồỏớW=ỏ=
`ỗóộọẫủ=ồỡóẫờW=ũ=
oẫ~ọ=ộ~ờớW=~I==
fó~ệỏồ~ờú=ộ~ờớW=ỏI=ầỏ=
jỗầỡọỡở=ỗẹ=~=ỗóộọẫủ=ồỡóẫờW=ờI= ờN I= ờO =
^ờệỡóẫồớ=ỗẹ=~=ỗóộọẫủ=ồỡóẫờW= I= N I= O =
=
=
46.

=
47.
=
48.


ỏN = ỏ =
ỏ O = −N =
á P = −á =
áQ = N=

áR = á =
á S = −N =
á T = −á =
áU = N =

á Q å +N = á =
á Q å+ O = −N =
á Q å + P = −á =
á Qå = N =

ò = ~ + ỏ =

`ỗóộọẫủ=mọ~ồẫ=
=

8
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CHAPTER 1. NUMBER SETS

=

=====


=
Figure 6.

=
49.
=
50.
=
51.
=

(~ + Äá ) − (Å + Çá ) = (~ − Å ) + (Ä − Ç)á =
(~ + Äá )(Å + Çá ) = (~Å − ÄÇ ) + (~Ç + ÄÅ )á =
~ + Äá ~Å + ÄÇ ÄÅ − ~Ç
=
+
⋅á =
Å + Çá Å O + Ç O Å O + Ç O

52.

=
53.

(~ + Äá ) + (Å + Çá ) = (~ + ) + ( + ầ )ỏ =

`ỗồỡệ~ớẫ=`ỗóộọẫủ=kỡóẫờở=
|||||||


~ + ỏ = ~ ỏ =
=
54.

~ = ờ ỗở ϕ I= Ä = ê ëáå ϕ ==
=

9
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CHAPTER 1. NUMBER SETS

=

=
Figure 7.

55.

=
56.

=
mỗọ~ờ=mờẫởẫồớ~ớỏỗồ=ỗẹ=`ỗóộọẫủ=kỡóẫờở=
~ + ỏ = ờ(ỗở + ỏ ởỏồ ) =
jỗầỡọỡở=~ồầ=^ờệỡóẫồớ=ỗẹ=~=`ỗóộọẫủ=kỡóẫờ=
fẹ= ~ + ỏ =ỏở=~=ỗóộọẫủ=ồỡóẫờI=ớĩẫồ=
ờ = ~ O + O =EóỗầỡọỡởFI==


= ~ờớ~ồ =E~ờệỡóẫồớFK=
~

=
57.

=
58.

mờỗầỡớ=ỏồ=mỗọ~ờ=oẫộờẫởẫồớ~ớỏỗồ=
ũ N ũ O = ờN (ỗở N + ỏ ởỏồ N ) ờO (ỗở O + ỏ ởỏồ O ) =
= ờNờO [ỗở(N + O ) + ỏ ởỏồ(N + O )] =
`ỗồỡệ~ớẫ=kỡóẫờở=ỏồ=mỗọ~ờ=oẫộờẫởẫồớ~ớỏỗồ=
|||||||||||||||||||||

ờ(ỗở + ỏ ởỏồ ) = ờ[ỗở( ) + ỏ ởỏồ( )] =

=
59.

fồợẫờởẫ=ỗẹ=~=`ỗóộọẫủ=kỡóẫờ=ỏồ=mỗọ~ờ=oẫộờẫởẫồớ~ớỏỗồ=
N
N
= [ỗở( ) + ỏ ởỏồ( )] =
ờ(ỗở + ỏ ởỏồ ) ờ

10
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CHAPTER 1. NUMBER SETS

60.

=
61.
=
62.
=
63.

=
64.

nỡỗớỏẫồớ=ỏồ=mỗọ~ờ=oẫộờẫởẫồớ~ớỏỗồ=
ũ N ờN (ỗở N + ỏ ởỏồ N ) ờN
= [ỗở(N O ) + ỏ ởỏồ(N O )] =
=
ũ O ờO (ỗở O + ỏ ởỏồ O ) ờO
mỗùẫờ=ỗẹ=~=`ỗóộọẫủ=kỡóẫờ=
ồ
ũ ồ = [ờ(ỗở + ỏ ởỏồ )] = ờ ồ [ỗở(ồ) + ỏ ởỏồ(ồ)] =
cỗờóỡọ~=aẫ=jỗỏợờẫ=
(ỗở + ỏ ởỏồ )ồ = ỗở(ồ) + ỏ ởỏồ(ồ) =
kớĩ=oỗỗớ=ỗẹ=~=`ỗóộọẫủ=kỡóẫờ=
+ Oõ
+ Oõ

ồ
ũ = ồ ờ(ỗở + ỏ ởỏồ ) = ồ ờ ỗở

+ á ëáå
 I==
å
å 

ïÜÉêÉ==
â = MI NI OI KI å N K==
bỡọẫờở=cỗờóỡọ~=
ẫ ỏủ = ỗở ủ + ỏ ởỏồ ñ =
=
=

11
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Chapter 2

Algebra
=
=
=
=

2.1 Factoring Formulas
=
~ä=åìãÄÉêëW=~I=ÄI=Å==
k~íìê~ä=åìãÄÉêW=å=
=
=

65.
=
66.
=
67.
=
68.
=
69.
=
70.
=
71.

=
72.

~ O − ÄO = (~ + Ä)(~ − Ä) =
~ P − ÄP = (~ − Ä)(~ O + ~Ä + ÄO ) =
~ P + ÄP = (~ + Ä)(~ O − ~Ä + ÄO ) =
~ Q − ÄQ = (~ O − ÄO )(~ O + ÄO ) = (~ − Ä)(~ + Ä)(~ O + ÄO ) =
~ R − ÄR = (~ − Ä)(~ Q + ~ P Ä + ~ O ÄO + ~ÄP + ÄQ ) =
~ R + ÄR = (~ + Ä)(~ Q − ~ P Ä + ~ O O ~P + Q ) =
fẹ=ồ=ỏở=ỗầầI=ớĩẫồ=
~ å + Äå = (~ + Ä)(~ å−N − ~ å −O Ä + ~ å −P ÄO − K − ~Äå −O + Äå −N ) K==
fĐ=å=áë=ÉỵÉåI=íÜÉå==
~ å − Äå = (~ − Ä)(~ å −N + ~ å −O Ä + ~ å −P ÄO + K + ~Äå−O + Äå −N ) I==

12
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CHAPTER 2. ALGEBRA

~ å + Äå = (~ + Ä)(~ å−N − ~ å −O Ä + ~ å −P ÄO − K + ~Äå−O − Äå −N ) K=
=
=
=

2.2 Product Formulas

73.
=
74.
=
75.
=
76.
=
77.
=
78.
=
79.

=
80.
=
81.


oẫ~ọ=ồỡóẫờởW=~I=I===
tĩỗọẫ=ồỡóẫờởW=ồI=õ=
=
=
(~ )O = ~ O O~ + ÄO =

(~ + Ä)O = ~ O + O~Ä + ÄO =
(~ − Ä)P = ~ P − P~ O Ä + P~ÄO − ÄP =
(~ + Ä)P = ~ P + P~ OÄ + P~ÄO + ÄP =
(~ − Ä)Q = ~ Q − Q~ P Ä + S~ O ÄO − Q~ÄP + ÄQ =
(~ + Ä)Q = ~ Q + Q~ P Ä + S~ OÄO + Q~P + Q =
_ỏồỗóỏ~ọ=cỗờóỡọ~=
(~ + )ồ = ồ` M~ ồ + å`N~ å−NÄ + å` O~ å−OÄO + K + å` å−N~Äå−N + å` å Äå I
å>
ïÜÉêÉ= å ` â =
=~ờẫ=ớĩẫ=ỏồỗóỏ~ọ=ỗẫẹẹỏỏẫồớởK=
õ> (ồ õ )>

(~ + + )O = ~ O + ÄO + Å O + O~Ä + O~Å + OÄÅ =
(~ + Ä + Å + K + ì + ỵ )O = ~ O + ÄO + Å O + K + ì O + î O + =
+ O(~Ä + ~Å + K + ~ì + ~ỵ + ÄÅ + K + Äì + Äỵ + K + ìỵ ) =

13
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CHAPTER 2. ALGEBRA

2.3 Powers
=

_~ởẫở=Eộỗởỏớỏợẫ=ờẫ~ọ=ồỡóẫờởFW=~I===
mỗùẫờở=Eờ~ớỏỗồ~ọ=ồỡóẫờởFW=ồI=ó=
=
=
~ ó ~ ồ = ~ ó+ồ =

82.
=
83.

~ã
= ~ ã −å =
å
~

=
84.
=

(~Ä)ã = ~ ã Äã =

85.

~ã
~
  = ã =
Ä
 Ä

ã


=
86.
=
87.
=
88.
=

(~ )

ã å

= ~ ãå =

~ M = N I= ~ ≠ M =
~N = N =
~ −ã =

89.

N
=
~ã

=
ã
å

~ = å ~ã =


90.

=
=
=
=
=

14
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CHAPTER 2. ALGEBRA

2.4 Roots
=

91.
=

_~ởẫởW=~I===
mỗùẫờở=Eờ~ớỏỗồ~ọ=ồỡóẫờởFW=ồI=ó=
~ I M =ẹỗờ=ẫợẫồ=ờỗỗớở=E ồ = Oâ I= â ∈ k F=
=
=
å
~Ä = å ~ å Ä =

92.

=

å

~ ã Ä = åã ~ ã Äå =

93.

å

~ å~
=
I= Ä ≠ M =
Ä åÄ

=
94.

=
95.
=
96.
=

~ åã ~ ã åã ~ ã
I= Ä ≠ M K=
=
=
ã
Äå

Ä åã Äå
å

(~ )
å

ã

( ~)
å

å

é

= å ~ ãé =

=~=
åé

97.
=

å

~ã =

98.
=


å

~ =~ =

99.
=

ã å

100.
=

ã
å

ã

~ = ãå ~ =

( ~)
å

~ ãé =

ã

= å ~ã =

15
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