,,,UNIFI$ 5GLI% BRL,'
,,,GUIDEL9ES = TE*NICAL MAT]IAL,'
#BJJH V].N UPDAT$ ,AUGU/ #BJAD
,PRODUC$ #BJJH BY
,A3ESSIBLE ,=MAT ,PRODUC;N
,,RNZFB1 ,AUCKL&1 ,NEW ,Z1L&
,UPDAT$ #BJAD BY
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,9 #A ,VOLUME
,BRL PAGES P#A-P#AJ & #A-#ACC
,PR9T PAGES #A-#HG P#A
P#B
,TE*NICAL ,GUIDEL9ES
,AB ? ,DOCU;T
,? DOCU;T HAS BE5 PRODUC$ BY ! ,MA?S
,FOCUS ,GR\P1 A SUBGR\P ( ! ,,UEB ,RULES
,COMMITTEE )9 ! ,9T]NA;NAL ,C\NCIL ON
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,G5]AL ,ASSEMBLY 9 ,APRIL #BJJH X 0
AGRE$ T ! DOCU;T %D 2 REL1S$ = USE
9T]NA;NALLY1 & T FE$BACK %D 2 GA!R$ ) A
VIEW TO PRODUC+ A NEW $I;N PRIOR TO !
#BJAB ,G5]AL ,ASSEMBLY4
,! PURPOSE ( ? DOCU;T IS TO GIVE
TRANSCRIB]S 5 9=MA;N & EXAMPLES TO
PRODUCE ,MA?S1 ,SCI;E & ,COMPUT] NOTA;N
9 ,UNIFI$ ,5GLI% ,BRL4
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2 S\RC$ "? ! ,,ICEB REPRES5TATIVES ON YR
LOCAL ,BRL ,AU?ORITIES4
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,,ICEB1 AG "? ! ,BRL ,AU?OR;Y 9 YR [N
C.TRY4
P#C
,TE*NICAL ,GUIDEL9ES
,,,SPECIAL SYMBOLS US$ 9 ?
VOLUME,'
,? BOOK HAS BE5 PRODUC$ 9 ,UNIFI$
,5GLI% ,BRL4 ,MA!MATICAL SYMBOLS DEF9$
AT ! />T ( EA* *APT] H N BE5 9CLUD$ 9 !
LI/ 2L4
.= DOT LOCATOR
.=.2 ITALIC SYMBOL
.=^2 BOLDFACE SYMBOL
.=,- DA%
.=_4 BULLET
.="< OP5 R.D BRACKET "<P>5!SIS">
.="> CLOSE R.D BRACKET "<P>5!SIS">
.=@.< OP5 TRANSCRIB]'S NOTE SYMBOL
.=@.> CLOSE TRANSCRIB]'S NOTE SYMBOL
.=; GRADE #A SYMBOL
.=;; GRADE #A ^W
.=;;; GRADE #A PASSAGE
.=;' GRADE #A T]M9ATOR
P#D
,TE*NICAL ,GUIDEL9ES
,,3T5TS
@.<,PAGE NUMB]S REF] TO PR9T PAGE
NUMB]S4@.>
#A ,G5]AL ,PR9CIPLES """""""""""""""" #A
#A4A ,SPAC+ """"""""""""""""""""""" #A
#A4B ,"ULY+ RULES = NUMB]S & LRS "" #B
#A4C ,PR9T ,SYMBOLS """"""""""""""" #C
#A4D ,=MAT """""""""""""""""""""""" #C
#A4E ,TYPE=MS """"""""""""""""""""" #D
#A4F ,CAPITALISA;N """""""""""""""" #D
#A4G ,USE ( ,GRADE #A 9DICATORS """ #E
#B ,NUMB]S & ,A2REVIA;NS """""""""""" #H
#B4A ,:OLE NUMB]S """"""""""""""""" #H
#B4B ,DECIMALS """""""""""""""""""" #I
#B4C ,DATES """"""""""""""""""""""" #I
#B4D ,"T """"""""""""""""""""""""" #AJ
#B4E ,ORD9AL NUMB]S """""""""""""" #AJ
#B4F ,ROMAN ,NUM]ALS """"""""""""" #AA
#B4G ,EMPHASIS ( ,DIGITS """"""""" #AA
#B4H ,ANCI5T ,NUM]A;N SY/EMS """"" #AA
#B4I ,HEXADECIMAL NUMB]S """"""""" #AB
P#E
,TE*NICAL ,GUIDEL9ES
#B4AJ ,A2REVIA;NS """""""""""""""" #AB
#C ,SIGNS ( ,OP]A;N1 ,COMP>ISON &
,OMIS.N """""""""""""""""""""""" #AE
#C4A ,EXAMPLES """"""""""""""""""" #AF
#C4B ,ALGEBRAIC ,EXAMPLES """""""" #AG
#C4C ,USE ( ! BRL HYPH5 """""""""" #AG
#C4D ,POSITIVE & NEGATIVE NUMB]S #AH
#C4E ,CALCULATOR KEYS """""""""""" #AH
#C4F ,OMIS.N M>KS 9 MA!MATICAL
EXPRES.NS """""""""""""""""""" #AI
#D ,SPATIAL ,LAY\T & ,DIAGRAMS """"" #BJ
#D4A ,SPATIAL CALCULA;NS """"""""" #BJ
#D4B ,TALLY M>KS """"""""""""""""" #BE
#D4C ,TABLES """"""""""""""""""""" #BF
#D4D ,DIAGRAMS """"""""""""""""""" #BG
#E ,GR\P+ ,DEVICES "<,BRACKETS"> """ #CJ
#F ,FRAC;NS """""""""""""""""""""""" #CA
#F4A ,SIMPLE NUM]IC FRAC;NS """""" #CA
#F4B ,MIX$ NUMB]S """""""""""""""" #CA
#F4C ,FRAC;NS WRITT5 9 L9E> =M 9
PR9T """"""""""""""""""""""""" #CB
#F4D ,G5]AL FRAC;N 9DICATORS """"" #CB
#F4E ,EXTRA ,EXAMPLES """""""""""" #CC
P#F
,TE*NICAL ,GUIDEL9ES
#G ,SUP]SCRIPTS & SUBSCRIPTS """"""" #CD
#G4A ,DEF9I;N ( AN ITEM """""""""" #CD
#G4B ,SUP]SCRIPTS & SUBSCRIPTS )9
LIT]>Y TEXT """""""""""""""""" #CE
#G4C ,ALGEBRAIC EXPRES.NS 9VOLV+
SUP]SCRIPTS """""""""""""""""" #CE
#G4D ,MULTIPLE LEVELS """""""""""" #CG
#G4E ,NEGATIVE SUP]SCRIPTS """"""" #CG
#G4F ,EXAMPLES F ,*EMI/RY """""""" #CH
#G4G ,SIMULTANE\S SUP]SCRIPTS &
SUBSCRIPTS """"""""""""""""""" #CH
#G4H ,LEFT-4PLAC$ SUP]SCRIPTS OR
SUBSCRIPTS """"""""""""""""""" #CH
#G4I ,MODIFI]S DIRECTLY ABV OR 2L #CI
#H ,SQU>E ,ROOTS & O!R RADICALS """" #DJ
#H4A ,SQU>E ROOTS """""""""""""""" #DJ
#H4B ,CUBE ROOTS ETC """"""""""""" #DA
#H4C ,SQU>E ROOT SIGN ON XS [N """ #DA
#I ,FUNC;NS """""""""""""""""""""""" #DB
#I4A ,SPELL+ & CAPITALISA;N """""" #DB
#I4B ,ITALICS """""""""""""""""""" #DB
#I4C ,SPAC+ """""""""""""""""""""" #DC
#I4D ,TRIGONOMETRIC FUNC;NS """""" #DD
P#G
,TE*NICAL ,GUIDEL9ES
#I4E ,LOG>I?MIC FUNC;NS """""""""" #DE
#I4F ,! ,LIMIT FUNC;N """""""""""" #DF
#I4G ,/ATI/ICAL FUNC;NS """""""""" #DF
#I4H ,COMPLEX NUMB]S """"""""""""" #DG
#AJ ,SET ,!ORY1 ,GR\P ,!ORY &
,LOGIC """"""""""""""""""""""""" #DH
#AA ,MISCELLANE\S ,SYMBOLS """"""""" #EJ
#AA4A ,SPAC+ """"""""""""""""""""" #EA
#AA4B ,UNUSUAL ,PR9T SYMBOLS """"" #EA
#AA4C ,GRADE #A 9DICATORS """""""" #EA
#AA4D ,SYMBOLS : H M ?AN "O M1N+
9 PR9T """"""""""""""""""""""" #EA
#AA4E ,EXAMPLES """""""""""""""""" #EB
#AA4F ,EMBELLI%$ CAPITAL LRS """"" #EE
#AA4G ,GREEK LRS """"""""""""""""" #EF
#AB ,B>S & DOTS ETC4 OV] & "U """""" #EG
#AB4A ,! DEF9I;N ( AN ITEM """"""" #EG
#AB4B ,TWO 9DICATORS APPLI$ TO !
SAME ITEM """""""""""""""""""" #EI
#AC ,>R[S """""""""""""""""""""""""" #FJ
#AC4A ,SIMPLE >R[S """"""""""""""" #FJ
#AC4B ,>R[S ) UNUSUAL %AFTS & A
/&>D B>B$ TIP """""""""""""""" #FA
P#H
,TE*NICAL ,GUIDEL9ES
#AC4C ,>R[S ) UNUSUAL TIPS """"""" #FB
#AD ,%APE ,SYMBOLS & ,COMPOSITE
,SYMBOLS """"""""""""""""""""""" #FE
#AD4A ,USE ( ! %APE T]M9A;N
9DICATOR """"""""""""""""""""" #FF
#AD4B ,TRANSCRIB] DEF9$ %APES """" #FF
#AD4C ,COMB9$ %APES """""""""""""" #FG
#AE ,MATRICES & VECTORS """""""""""" #FI
#AE4A ,5L>G$ GR\P+ SYMBOLS """"""" #FI
#AE4B ,MATRICES """""""""""""""""" #FI
#AE4C ,DET]M9ANTS """""""""""""""" #GJ
#AE4D ,OMIS.N DOTS """"""""""""""" #GJ
#AE4E ,D1L+ ) WIDE MATRICES """""" #GA
#AE4F ,VECTORS """"""""""""""""""" #GB
#AE4G ,GR\P+ ( EQUA;NS """"""""""" #GC
#AF ,*EMI/RY """"""""""""""""""""""" #GD
#AF4A ,*EMICAL "NS """"""""""""""" #GE
#AF4B ,*EMICAL =MULAE """""""""""" #GE
#AF4C ,ATOMIC MASS NUMB]S """""""" #GF
#AF4D ,ELECTRONIC 3FIGURA;N """""" #GF
#AF4E ,*EMICAL ,EQUA;NS """""""""" #GG
#AF4F ,ELECTRONS """"""""""""""""" #GH
#AF4G ,/RUCTURAL ,=MULAE """"""""" #GH
P#I
,TE*NICAL ,GUIDEL9ES
#AG ,COMPUT] ,NOTA;N """"""""""""""" #HC
#AG4A ,DEF9I;N ( COMPUT] NOTA;N "" #HC
#AG4B ,L9E >RANGE;T & SPAC+ )9
COMPUT] NOTA;N """"""""""""""" #HC
#AG4C ,GRADE ( BRL 9 COMPUT]
NOTA;N """"""""""""""""""""""" #HF
P#AJ
,,,UNIFI$ 5GLI% BRL,' #A
,,,GUIDEL9ES = TE*NICAL MAT]IAL,'
#BJJH V].N UPDAT$ ,AUGU/ #BJAD
#A ,G5]AL ,PR9CIPLES
#A4A ,SPAC+
#A4A4A ,! LAY\T ( ! PR9T %D 2 PRES]V$
Z NE>LY Z POSSIBLE4 ,H["E C>E %D 2 TAK5
9 COPY+ PR9T SPAC+ AL;G A L9E Z ? IS (T5
SIMPLY A MATT] ( PR9T+ /YLE4 ,SPAC+ %D 2
US$ TO REFLECT ! /RUCTURE ( ! MA!MATICS4
,SPAC+ 9 PR9T "?\T A "W IS (T5 9CONSI/5T
& X IS N DESIRABLE 9 ! BRL TRANSCRIP;N T
? 9CONSI/5CY %D 2 PRES]V$4
#A4A4B ,= EA* "W1 A DECI.N M/ 2 MADE
ON ! SPAC+ ( OP]A;N SIGNS "<S* Z PLUS &
M9US"> & COMP>ISON SIGNS "<S* Z EQUALS &
LESS ?AN">4 ,:5 PRES5T+ BRL MA!MATICS TO
"Y] *N1 9CLUDE SPACES 2F & AF OP]A;N
SIGNS & 2F & AF COMP>ISON SIGNS4 ,= OLD]
/UD5TS :O >E TACKL+ L;G] ALGEBRAIC
EXPRES.NS "! NE$S TO 2 A BAL.E 2T CL>;Y
& COMPACT;S4 ,A GD APPROA* IS TO H !
OP]A;N SIGNS UNSPAC$ ON BO? SIDES B #A
#A ,G5]AL ,PR9CIPLES A#A
/ 9CLUDE A SPACE 2F & AF COMP>ISON
SIGNS4 ,? IS ! APPROA* US$ 9 MO/ ( !
EXAMPLES 9 ? DOCU;T4
#A4A4C ,"! >E AL SITUA;NS ": X IS
PREF]ABLE TO UNSPACE A COMP>ISON SIGN4
,"O IS :5 UNSPAC+ ! SIGN WD AVOID DIVID+
A COMPLEX EXPRES.N 2T L9ES 9 A
COMPLICAT$ MA!MATICAL >GU;T4 ,ANO!R IS
:5 ! COMP>ISON SIGN IS N ON ! BASE L9E
"<= EXAMPLE SIGMA NOTA;N ": I EQUALS #A
IS 9 A SMALL FONT DIRECTLY 2L">4
#A4A4D ,:5 ISOLAT$ CALCULA;NS APPE> 9
A LIT]>Y TEXT1 ! PR9T SPAC+ C 2 FOLL[$4
--------------------------------------#B
#A4B ,"ULY+ RULES = NUMB]S & LRS
,LI/$ 2L IS A SUMM>Y ( ! RULES =
,GRADE #A MODE & ,NUM]IC MODE Z !Y APPLY
TO ! BRAILL+ ( NUMB]S & LRS 9 MA!MATICS4
,REF] TO ! COMPLETE V].NS ( ^! RULES = M
DETAIL4
#A4B4A ,GRADE #A MODE
,A BRL SYMBOL MAY H BO? A GRADE #A
M1N+ & A 3TRAC;N "<I4E4 GRADE #B"> #B
#A ,G5]AL ,PR9CIPLES A#B
M1N+4 ,"S SYMBOLS MAY AL H A NUM]IC
M1N+4 ,A GRADE #A 9DICATOR IS US$ TO SET
GRADE #A MODE :5 ! GRADE #A M1N+ ( A
SYMBOL CD 2 MISR1D Z A 3TRAC;N M1N+ OR A
NUM]IC M1N+4
,NOTE T IF A S+LE LR "<EXCLUD+ A1 I &
O"> O3URS 9 AN ALGEBRAIC EXPRES.N1 X C 2
MISR1D Z A 3TRAC;N IF X IS 8/&+ AL"O0 S
MAY NE$ A GRADE #A 9DICATOR4 ,! SAME IS
TRUE ( A SEQU;E ( LRS 9 BRL T CD
REPRES5T A %ORT=M1 S* Z ;AB OR ;AC1 IF X
IS 8/&+ AL"O04
,A LR1 OR UNBROK5 SEQU;E ( LRS IS 8/&+
AL"O0 IF ! SYMBOLS 2F & AF ! LR OR
SEQU;E >E SPACES1 HYPH5S1 DA%ES1 OR ANY
COMB9A;N1 OR IF ON BO? SIDES ! ONLY
9T]V5+ SYMBOLS 2T ! LR OR SEQU;E & !
SPACE1 HYPH5 OR DA% >E COMMON LIT]>Y
PUNCTUA;N OR 9DICATOR SYMBOLS4 ,SEE !
,G5]AL ,RULE = A FULL DEF9I;N ( 8/&+
AL"O04
#A4B4B ,NUM]IC MODE
,NUM]IC MODE IS 9ITIAT$ BY ! #C
#A ,G5]AL ,PR9CIPLES B#B
8NUMB] SIGN0 "<DOTS #CDEF"> FOLL[$ BY "O
( ! T5 DIGITS1 ! COMMA OR ! DECIMAL
PO9T4
,! FOLL[+ SYMBOLS MAY O3UR 9 NUM]IC
MODE3 ! T5 DIGITS2 FULL /OP2 COMMA2 !
NUM]IC SPACE "<DOT #E :5 IMMLY FOLL[$ BY
A DIGIT">2 SIMPLE NUM]IC FRAC;N L9E2 & !
L9E 3T9UA;N 9DICATOR4 ,A SPACE OR ANY
SYMBOL N LI/$ "H T]M9ATES NUM]IC MODE1 =
EXAMPLE ! HYPH5 OR ! DA%4
,A NUM]IC MODE 9DICATOR AL SETS GRADE
#A MODE4 ,GRADE #A MODE1 :5 9ITIAT$ BY
NUM]IC MODE1 IS T]M9AT$ BY A SPACE1
HYPH5 OR DA%4 ,"!=E :ILE GRADE #A MODE
IS 9 E6ECT1 A GRADE #A 9DICATOR IS N
REQUIR$ EXCEPT = ANY "O ( ! L[]CASE LRS
A-;J IMMLY FOLL[+ A DIGIT1 A FULL /OP OR
A COMMA4 "<,NOTE T ,GRADE #A MODE1 :5
9ITIAT$ BY NUM]IC MODE1 IS N T]M9AT$ BY
! M9US SIGN1 "-4">
--------------------------------------#C
#A4C ,PR9T ,SYMBOLS
,"O ( ! "ULY+ DESIGN F1TURES ( ,,UEB
IS T EA* PR9T SYMBOL %D H "O & ONLY #D
#A ,G5]AL ,PR9CIPLES A#C
"O BRL EQUIVAL5T4 ,= EXAMPLE ! V]TICAL
B> IS US$ 9 PR9T TO REPRES5T ABSOLUTE
VALUE1 3DI;NAL PROBABIL;Y & ! ^WS 8S*
T01 TO GIVE J ?REE EXAMPLES4 ,! SAME BRL
SYMBOL %D 2 US$ 9 ALL ^! CASES1 & ANY
RULES = ! USE ( ! SYMBOL 9 BRL >E
9DEP5D5T ( ! SUBJECT >EA4 ,IF A PR9T
SYMBOL IS N DEF9$ 9 ,,UEB1 X C 2
REPRES5T$ EI US+ "O ( ! SEV5 TRANSCRIB]
DEF9$ PR9T SYMBOLS 9 ,SEC;N #AA1 OR BY
US+ ! TRANSCRIB] DEF9$ %APE SYMBOLS 9
,SEC;N #AD4
#A4D ,=MAT
.=" 3T9UA;N 9DICATOR
#A4D4A ,9 PR9T1 MA!MATICAL EXPRES.NS
>E "S"TS EMB$D$ 9 ! TEXT & "S"TS SET
A"P4 ,:5 AN EXPRES.N IS SET A"P1 ! BRL
=MAT %D 9DICATE ? BY SUITABLE 9D5TA;N1 =
EXAMPLE CELL #C ) OV]RUNS 9 #E OR CELL
#E ) OV]RUNS 9 #G4 ,AN EMB$D$ EXPRES.N :
DOES N FIT ON ! CURR5T BRL L9E %D ONLY 2
DIVID$ IF "! IS AN OBVI\S DIVID+ #E
#A ,G5]AL ,PR9CIPLES B#C
PO9T4 ,(T5 X IS BETT] TO MOVE ! :OLE
EXPRES.N TO ! NEXT BRL L9E4
#A4D4B ,:5 DIVID+ A MA!MATICAL
EXPRES.N1 *OICE ( A RUNOV] SITE %D FOLL[
MA!MATICAL /RUCTURE3
_4 2F COMP>ISON SIGNS
_4 2F OP]A;N SIGNS "<UN.S !Y >E )9 "O (
! MA!MATICAL UNITS 2L">
_4 2F A MA!MATICAL UNIT S* Z
_4 FRAC;NS "<& )9 ! FRAC;N 3SID] !
NUM]ATOR & D5OM9ATOR Z UNITS">
_4 FUNC;NS
_4 RADICALS
_4 ITEMS ) MODIFI]S S* Z SUP]SCRIPTS
OR B>S
_4 %APES OR >R[S
_4 ANY?+ 5CLOS$ 9 PR9T OR BRL GR\P+
SYMBOLS
_4 A NUMB] & XS A2REVIA;N OR
COORD9ATES
,USUALLY ! BE/ PLACE TO BR1K IS 2F A
COMP>ISON SIGN OR AN OP]A;N SIGN4 #F
#A ,G5]AL ,PR9CIPLES C#C
,BR1K+ 2T BRL PAGES %D 2 AVOID$4
--------------------------------------#D
#A4D4C ,:5 AN EXPRES.N W N FIT ON "O
BRL L9E & HAS TO 2 DIVID$1 ! USE (
9D5TA;N Z SU7E/$ 9 #A4D4A %D MAKE X CLE>
T ! OV]RUN IS "P ( ! SAME EXPRES.N4
,H["E 9 ! UNLIKELY CASE ": ! TWO POR;NS
CD 2 R1D Z TWO SEP>ATE EXPRES.NS !
3T9UA;N 9DICATOR "<DOT #E"> %D 2 PLAC$
IMMLY AF ! LA/ CELL ( ! 9ITIAL L9E4
,! SUM ( A1 ;B1 ;C1 ;D & ;E1 "TS ! SUM (
;F1 ;G1 ;H1 I & ;J1 EQUALS ! SUM ( #A
TO #E "TS ! SUM ( #F TO #AJ EQUALS
#FJJ4
"<A"6B"6C"6D"6E">"<F"6G"6H"6I"6J">
"7 "<#A"6#B"6#C"6#D"6#E">"
"<#F"6#G"6#H"6#I"6#AJ"> "7 #FJJ
#A4E ,TYPE=MS
,9 MA!MATICS1 ALGEBRAIC LRS >E
FREQU5TLY ITALICIS$ Z A 4T9C;N F ORD9>Y
TEXT4 ,X IS G5]ALLY N NEC TO #G
#A ,G5]AL ,PR9CIPLES A#D
9DICATE ? 9 BRL4 ,H["E1 :5 BOLD OR O!R
TYPEFACE IS US$ TO 4T+UI% DI6]5T TYPES (
MA!MATICAL LRS OR SIGNS F ORD9>Y
ALGEBRAIC LRS1 E4G4 = VECTORS OR
MATRICES1 ? 4T9C;N %D 2 RETA9$ 9 BRL BY
US+ ! APPROPRIATE TYPE=M 9DICATOR4 ,SEE
,SEC;N #B4G = ! EMPHASIS ( 9DIVIDUAL
DIGITS )9 NUMB]S4
#A4F ,CAPITALISA;N
,9 MA!MATICS & SCI;E1 /R+S ( CAPITAL
LRS (T5 O3UR1 = EXAMPLE 9 A GEOMETRICAL
"N1 9 A PHYSICS =MULA OR 9 G5ETICS4 ,S*
/R+S %D ALW 2 UNCONTRACT$4 ,CAPITAL ^W
9DICATORS "<D\BLE CAPS"> >E NORMALLY
US$4 ,SEE ,SEC;N #AF = ADVICE ON CAPITAL
LRS 9 *EMICAL =MULAE4 ,X IS PREF]ABLE TO
AL USE ? APPROA* 9 G5ETICS OR O!R TOPICS
": "! >E FREQU5T *ANGES ( CASE )9 A
SEQU;E ( LRS4
RECTANGLE ,,ABCD
;,V "7 ,,IR
,TRIANGLE ,,RST N ,,R/ #H
#A ,G5]AL ,PR9CIPLES B#D
,,AB;9#B
@.<,9 PR9T1 CAPITAL ,A1 CAPITAL
;,B1 SUP]SCRIPT #B4@.>
;,,AB1 ,,BC & ;,,AC
,,III,'RD
@.<,ROMAN CAPITAL ?IRD4@.>
,H,H,HH
@.<,PR9T %[S #C CAPITAL ;H'S & A
L[] CASE ;H4@.>
--------------------------------------#E
#A4G ,USE ( ,GRADE #A 9DICATORS
.=; GRADE #A SYMBOL 9DICATOR
.=;; GRADE #A ^W 9DICATOR
.=;;; GRADE #A PASSAGE 9DICATOR
.=;' GRADE #A PASSAGE T]M9ATOR
.=""=;;; GRADE #A PASSAGE 9DICATOR ON A
L9E ( XS [N
.=""=;' GRADE #A PASSAGE T]M9ATOR ON A
L9E ( XS [N
#A4G4A ,GRADE #A 9DICATORS W N 2 NE$$
= SIMPLE >I?METIC PROBLEMS 9VOLV+
NUMB]S1 OP]A;N SIGNS1 NUM]ICAL FRAC;NS &
MIX$ NUMB]S4 #I
#A ,G5]AL ,PR9CIPLES A#E
,EVALUATE ! FOLL[+3
#C M9US #B & A HALF EQUALS
,EVALUATE ! FOLL[+3
#C "- #B#A/B "7
#A4G4B ,SIMPLE ALGEBRAIC EQUA;NS :
9CLUDE LRS B NO FRAC;N OR SUP]SCRIPT
9DICATORS MAY NE$ GRADE #A SYMBOL
9DICATORS ": LRS /& AL"O OR FOLL[
NUMB]S4 "<,SEE ,SEC;N #A4B = ! "ULY+
RULES & ,SEC;N #C4B = M EXAMPLES4">
;Y EQUALS ;X PLUS #D;C
;Y "7 X"6#D;C
--------------------------------------#F
#A4G4C ,M COMPLEX ALGEBRAIC EQUA;NS >E
BE/ 5CLOS$ 9 GRADE #A PASSAGE 9DICATORS4
,? W 5SURE T ISOLAT$ LRS & 9DICATORS S*
Z SUP]SCRIPT1 SUBSCRIPT1 FRAC;NS1
RADICALS1 >R[S & %APES >E WELL DEF9$ )\T
! NE$ = GRADE #A SYMBOL 9DICATORS4
,3SID] ! FOLL[+ EQUA;N3 #AJ
#A ,G5]AL ,PR9CIPLES A#F
#CX M9US #DY PLUS ;Y SQU>$ EQUALS ;X
SQU>$
,3SID] ! FOLL[+ EQUA;N3
;;;#CX"-#DY"6Y9#B "7 X9#B;'
,NOTE T ? "PICUL> EQUA;N CD AL 2
WRITT5
#CX"-#DY"6Y9#B "7 X;9#B
2C ! LEFT H& SIDE ( ! EQUA;N IS 9 GRADE
#A MODE FOLL[+ ! NUM]IC 9DICATOR "<SEE
,SEC;N #A4B">4
,SIMIL>LY
FRAC;N3 ;X SQU>$ PLUS #BX ALL OV] #A
PLUS ;X SQU>$ CLOSE FRAC;N
C 2 SAFELY WRITT5 Z
;;;(X9#B"6#BX./#A"6X9#B) "7 #A;'
B CD AL 2 WRITT5
;;(X9#B"6#BX./#A"6X9#B) "7 #A
,SEE ,SEC;N #AA4E = M EXAMPLES ( ! USE
( GRADE #A PASSAGE 9DICATORS4
#AA
#A ,G5]AL ,PR9CIPLES B#F
#A4G4D ,IF A COMPLEX ALGEBRAIC
EXPRES.N DOES N 9CLUDE A COMP>ISON SIGN
"<S* Z AN EQUALS SIGN"> !N X IS UNLIKELY
TO 9CLUDE 9T]IOR SPACES 9 BRL "<SEE
,SEC;N #A4A4B">4 ,9 ? CASE A GRADE #A ^W
9DICATOR W 2 5 TO 5SURE T SUP]SCRIPT1
SUBSCRIPT1 FRAC;NS1 RADICALS1 >R[S &
%APE 9DICATORS >E WELL DEF9$ )\T ! NE$ =
GRADE #A SYMBOL 9DICATORS4
,EVALUATE ! SQU>E ROOT ( OP5 R.D
BRACKET ;Y M9US ;X SUP]SCRIPT #B CLOSE
R.D BRACKET4
,EVALUATE ;;%"<Y"-X9#B">+4
,SEE ,SEC;N #G4C = M EXAMPLES ( ! USE
( GRADE #A ^W 9DICATORS4
--------------------------------------#G
#A4G4E ,:5 5TIRE "W$ EXAMPLES OR SETS
( EX]CISES >E 5CLOS$ 9 GRADE #A PASSAGE
9DICATORS1 ! GRADE #A 9DICATORS C 2
PREC$$ BY ! 8USE 9DICATOR0 & PLAC$ ON A
L9E ( _! [N4
#AB
#A ,G5]AL ,PR9CIPLES A#G
,SOLVE ! FOLL[+ QUADRATIC EQUA;NS3
""=;;;
#A4 X9#B"-X"-#B "7 #J
#B4 X9#B"-#DX"-#C "7 #J
#C4 #BX9#B"-X "7 #A
""=;'
#A4G4F ,:5 ONLY A FEW 3TRACT$ ^WS >E
9VOLV$1 ! GRADE #A PASSAGE 9DICATOR C 2
US$ TO 5CLOSE 5TIRE "W$ EXAMPLES & SETS
( EX]CISES4 ,9 ? SITUA;N ANY ^WS O3URR+
9 ! EX]CISES W 2 WRITT5 9 UNCONTRACT$
BRL & ISOLAT$ LRS W N NE$ LR SIGNS4 ,":
"! IS M TEXT 9VOLV$ X IS BETT] TO /AY 9
GRADE #B & USE GRADE #A PASSAGE1 ^W OR
SYMBOL 9DICATORS ONLY Z REQUIR$4
#A4G4G ,9 ! EXAMPLES 9 ? DOCU;T1 GRADE
#B MODE IS ASSUM$ TO 2 9 E6ECT1 & GRADE
#A 9DICATORS H BE5 9CLUD$ AC TO !
GUIDEL9ES 9 ? SEC;N4 ,M9IMIS+ ! NUMB] (
9DICATORS M/ 2 BAL.ED AG/ R$UC+ CLUTT]
)9 ! EXPRES.N XF4 ,A GRADE #A SYMBOL
9DICATOR : O3URS HALF WAY "? AN EXPRES.N
MAY 2 M 4RUPTIVE TO ! R1D] ?AN A #AC
#A ,G5]AL ,PR9CIPLES B#G
^W OR PASSAGE 9DICATOR1 EV5 IF ^! TAKE
UP M CELLS4 ,X IS AL IMPORTANT TO USE A
3SI/5T APPROA* :5 TRANSCRIB+ A "PICUL>
TEXT4 ,OV]ALL ! FOCUS %D 2 ON MA!MATICAL
CL>;Y = ! R1D]4
,FUR!R GUID.E W 2 GIV5 :5 M FE$BACK
HAS BE5 RCVD F /UD5TS4
--------------------------------------#H
#B ,NUMB]S & ,A2REVIA;NS
,REF] TO ,SEC;N #A4B = A SUMM>Y ( !
RULES = ,GRADE #A MODE & ,NUM]IC MODE Z
!Y APPLY TO ! BRAILL+ ( NUMB]S & LRS 9
MA!MATICS4
,! BRL REPRES5TA;N ( NUMB]S S* Z DATES
& "TS %D REFLECT ! PUNCTUA;N US$ 9 PR9T4
#B4A ,:OLE NUMB]S
#DEF
#C1JJJ "<COMMA US$ Z SEP>ATOR 9 PR9T">
#E"JJJ"JJJ "<SPACE US$ Z SEP>ATOR 9
PR9T">
,CALL+ S1T NUMB]S #CJ-#EI4
,9 ! #FJ'S #AD
#B ,NUMB]S & ,A2REVIA;NS A#H
,9 ! #FJS
,9 ! '#FJS
,PH"O #JI-#ECG"JHIA
,= NEGATIVE NUMB]S SEE #D4B4
--------------------------------------#I
#B4B ,DECIMALS
#H4IC
#J4G
#4G
,IS ! NUMB] 9 ! RANGE #B-#E4E8
#4H IS A DECIMAL FRAC;N4
,= RECURR+ DECIMALS SEE ,SEC;N #AB
"<B>S1 DOTS ETC4 OV] & "U">
#B4C ,DATES
#BH-#E-#BJJA
#E-#BH-#JA
#BJJA_/#E_/#BH
#BJJA4E4BH
#BH_/#E-#CA_/#E
-------------------------------------#AJ
#AE
#B ,NUMB]S & ,A2REVIA;NS A#AJ
#B4D ,"T
#E3#CJ PM
#E4CJ
#JH4JJ
#ACJJ
#F-#G A4M4
#F3#AE-#G3#DE
#B4E ,ORD9AL NUMB]S
#AST
#BND OR #B;D
#CRD OR #C;D
#DTH
#A;ER
-------------------------------------#AA
#B4F ,ROMAN ,NUM]ALS
,ROMAN NUM]ALS %D 2 BRLD Z IF !Y 7
NORMAL LRS US+ ! RULES = GRADE #A MODE4
,NOTE T ;V & ;X W H GRADE #A 9DICATORS B
I W N4
,R1D "PS ,I1 ,,II & ;,V4
,ANSW] "QS I1 VI & ;X4 #AF
#B ,NUMB]S & ,A2REVIA;NS A#AA
;,,CD
#B4G ,EMPHASIS ( ,DIGITS
,IF A TYPE=M 9DICATOR APPLIES TO A
DIGIT OR DIGITS )9 A NUMB]1 ! NUM]IC
9DICATOR NE$S REP1T+ AF ANY TYPE=M
9DICATOR4 ,IF ! F/ DIGIT IS A6ECT$ !N !
TYPE=M 9DICATOR %D 2 PLAC$ 2F ! NUM]IC
9DICATOR4
#FG^2#HDE
@.<,! DIGIT #H IS 9 BOLD4@.>
#FG^1#HDE
@.<,! LA/ ?REE DIGITS >E 9
BOLD4@.>
#FG^1#HD^'#E
@.<,DIGITS #H & #D >E 9 BOLD4@.>
_1#FGH_'#DE
@.<,! F/ ?REE DIGITS >E "UL9$4@.>
_2#FGHDE
@.<,DIGIT #F IS "UL9$4@.>
,= RECURR+ DECIMALS SEE ,SEC;N #AB
"<B>S1 DOTS ETC4 OV] & "U"> #AG
#B ,NUMB]S & ,A2REVIA;NS B#AA
#B4H ,ANCI5T ,NUM]A;N SY/EMS
,BRL SYMBOLS TO REPRES5T NUM]ALS F O!R
NUMB] SY/EMS MAY 2 DEVIS$ = EA* SITUA;N
US+ TRANSCRIB] DEF9$ PR9T SYMBOLS4 ,^!
%D 2 DEF9$ EI ON ! SPECIAL SYMBOLS PAGE
OR 9 A TRANSCRIB]'S NOTE4 "<,SEE EXAMPLE
9 ,SEC;N #AA4E4G">
-------------------------------------#AB
#B4I ,HEXADECIMAL NUMB]S
,HEXADECIMAL NUMB]S O3UR 9 A COMPUT]
SETT+ & >E MADE UP ( ! DIGITS #J TO #I &
! LRS ,A TO ;,F4 ,!Y %D 2 TR1T$ ! SAME Z
ANY O!R /R+ ( LRS & NUMB]S4
,FATAL EXCEP;N #J,E HAS O3URR$ AT
#JJBH3,C#JJJHB,,CD
#B4AJ ,A2REVIA;NS
,! FOLL[+ SIGNS >E US$ = SPECIAL PR9T
SYMBOLS3
.=@C C5T
.=@E EURO
.=@F FRANC #AH
#B ,NUMB]S & ,A2REVIA;NS A#AB
.=@L P.D "</]L+">
.=@S DOLL>
.=@Y Y5 "<,JAPAN">
.=@N NAIRA "<,NIG]IA">
.=.0 P]C5T
.=^J DEGREE
.=7 FOOT OR M9UTE "<%[N Z A PRIME SIGN">
.=77 9* OR SECOND "<%[N Z A D\BLE PRIME
SIGN">
.=,^$A ANG/ROM "<,A ) SMALL CIRCLE ABV">
,NOTE T ! ,R& "<,S\? ,AFRICA"> IS
WRITT5 9 PR9T Z A NORMAL CAPITAL ;,R S
WD 2 BRLD Z S*4
,NOTE T ! FOOT OR M9UTE MAY 2 %[N 9
PR9T BY AN APO/ROPHE "<'"> & ! 9* OR
SECOND BY A NONDIREC;NAL D\BLE QUOTE
"<,7">4 ,? USAGE C 2 FOLL[$ 9 BRL4
-------------------------------------#AC
,FOLL[ PR9T = ORD]1 SPAC+1
CAPITALISA;N & PUNCTUA;N ( A2REVIA;NS4
"<,IF X IS UNCLE> 9 PR9T :E!R "! IS A
SPACE 2T A NUMB] & XS UNIT1 OR IF #AI
#B ,NUMB]S & ,A2REVIA;NS A#AC
PR9T SPAC+ IS 9CONSI/5T1 !N X IS
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,": %D ,I WRITE ! DOLL> SIGN1 ,,US@S OR
@S,,US8
#CJ C5TS C 2 WRITT5 Z @S#J4CJ1 #CJ;C OR
#CJ@C4
,9 ,S\? ,AFRICA1 ? WD CO/ ,R#AC4EA4
,2F DECIMALISA;N1 @L#A4GE 0 @L#A #AES S
HALF ( X 0 #AGS #F;D OR #AG_/#F4
,HALF A Y>D IS #A FT #F 9 OR #A7 #F77 :
IS AB #DE CM OR #J4DE ;M4
#A ;,L ( WAT] WEI<S #AJJJ ;G : IS AB #B
LBS #D OZ4
,IS ! SPE$ LIMIT #CJ MPH OR #EJ KM_/H8
,WAT] FREEZES AT #J^J,C OR #CB^J,F4
-------------------------------------#AD
,TO DECR1SE BY #AE.0 MULTIPLY BY #J4HE4
,ADD #A C ( B1NS1 #A ;C ( FL\R1 #B ;,T (
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DEGREE & #FJ77 9 A M9UTE4
,"O COMPLETE ORBIT LA/S #BYR #EM #AE;D
#G;H #AGMIN & #DES4 #BJ
#B ,NUMB]S & ,A2REVIA;NS A#AD
,A #F ;,V BATT]Y W CAUSE A CURR5T ( #C
,A TO FL[ "? A RESI/.E ( #B ,.W4
,! R1D+ 0 #AE M,HZ4
,! PATT]N SAYS K#D P#A SL#A K#A PSSO4
#A ,^$A "7 #A/AJ1JJJ .M
-------------------------------------#AE
#C ,SIGNS ( OP]A;N1 ,COMP>ISON &
,OMIS.N
,OP]A;N SIGNS3
.="6 PLUS
.="- M9US "<:5 4T+UI%$ F HYPH5">
.="8 "TS "<MULTIPLICA;N CROSS">
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.=_6 PLUS OR M9US "<PLUS OV] M9US">
.=_- M9US OR PLUS "<M9US OV] PLUS">
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,COMP>ISON SIGNS3
.="7 EQUALS
.=@< LESS ?AN1 OR OP5+ ANGLE BRACKET
.=@> GRT] ?AN1 OR CLOS+ ANGLE BRACKET
.=_@< LESS ?AN OR EQUAL TO #BA
#C ,OP]A;N & ,COMP>ISON A#AE
.=_@> GRT] ?AN OR EQUAL TO
.="7@: N EQUAL TO "<L9E "? AN EQUALS
SIGN">
.=_9 APPROXIMATELY EQUAL TO "<TILDE OV]
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,LESS COMMON SIGNS ( COMP>ISON3
.=.@< IS M* LESS ?AN
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"<APPROXIMATELY EQUAL">
.=."7 EQUALS SIGN DOTT$ ABV & 2L
"<APPROXIMATELY EQUAL">
.=^"7 EQUALS SIGN ) BUMP 9 TOP B>
"<DI6];E 2T OR APPROXIMATELY EQUAL">
.=_= EQUIVAL5T TO "<?REE HORIZONTAL
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.=_"7 IS PROPOR;NAL TO
,RATIO
.=3 RATIO SIGN "<REPRES5T$ BY A COLON Z
9 PR9T"> #BB
#C ,OP]A;N & ,COMP>ISON #AF
"<SEE AL ,SEC;N #AA = SIGNS ( OP]A;N &
COMP>ISON US$ 9 SET !ORY1 GR\P !ORY &
LOGIC">
#C4A ,EXAMPLES
,9 MO/ ( ! EXAMPLES 2L1 OP]A;N SIGNS
>E UNSPAC$ F PREC$+ & FOLL[+ T]MS B
COMP>ISON SIGNS >E SPAC$4 ,! F/ TWO
EXAMPLES %[ ! USE ( EXTRA SPACE = ! "Y]
LE>N]4 ,FOLL[ ! GUIDEL9ES 9 ,SPAC+
"<,SEC;N #A4A4B">4
#C "6 #E "7 #H
#H "- #E "7 #C
#C"8#E "7 #E"8#C "7 #AE
#B CM"6#D CM "7 #F CM
#BJJ;G"8#E "7 #AKG
#E4GB M"/#AJ "7 #EG4B CM
#AE_6#J4E
,>EA "7 BH "7 #E"4#C "7 #AE
#C4I"8#D4A _9 #AF
#E"-#C "7@: #C"-#E
-------------------------------------#AG
,A SCALE ( #A3#BJJ "<AL? ! RATIO #BC
#C ,OP]A;N & ,COMP>ISON A#AG
SIGN IS US$ TO COMP>E TWO NUMB]S1 X IS
BE/ TR1T$ Z AN OP]A;N SIGN = PURPOSES
( SPAC+">
#B3#D "7 #F3#AB
#C4B ,ALGEBRAIC ,EXAMPLES
,9 ! ALGEBRAIC EXAMPLES 2L NOTE ! USE
( ! GRADE #A 9DICATOR :5"E A LR IS 8/&+
AL"O0 AC TO ! DEF9I;N 9 ,SEC;N #A4B & S
CD R1D Z A 3TRAC;N4 ,9 ! LA/ EXAMPLE1 !
COLON XF CD R1D Z ! CC 3TRAC;N4
,IF ;Y _"7 ;X !N ;Y "7 KX
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;Y "7 X"6#D
#BY "7 #B;C"-#D
D"6AB "7 ;AC
,! RATIO X;3Y
#C4C ,USE ( ! BRL HYPH5
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9DI/+UI%ABLE 9 PR9T !N ! BRL HYPH5 C 2
US$ = BO?4 ,H["E MO/ MA?S & SCI;E TEXTS
%[ ! M9US SIGN Z SLI<TLY L;G] ?AN #BD
#C ,OP]A;N & ,COMP>ISON B#AG
A HYPH5 9 PR9T S ! DOT #E =M IS BE/ US$
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-------------------------------------#AH
9T]E/-RATE"-9FLA;N-RATE
,! TEMP]ATURE 0 #AE-#AG4 "<AMBIGU\S
PR9T">
#C4D ,POSITIVE & NEGATIVE NUMB]S
,"S"TS POSITIVE & NEGATIVE NUMB]S1 Z
OPPOS$ TO ADD$ OR SUBTRACT$ NUMB]S1 >E
%[N 9 PR9T BY A PLUS SIGN OR A M9US SIGN
2+ WRITT5 Z A LEFT SUP]SCRIPT "<T IS 2+
WRITT5 ABV & TO ! LEFT ( ! NUMB]">4 ,? C
2 %[N 9 BRL US+ ! SUP]SCRIPT 9DICATOR4
"<,SEE ,SEC;N #G">
,GRADUATE ! ;X AXIS F "-#D TO "6#E4
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,EVALUATE ;9"-#B"69"-#C
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9 PR9T">
#BE
#C ,OP]A;N & ,COMP>ISON A#AH
#C4E ,CALCULATOR KEYS
,:5 NUMB]S & OP]ATORS >E 5CLOS$ 9
BOXES TO %[ T CALCULATOR KEYS >E TO 2
PRESS$1 X MAY N 2 HELP;L TO USE ! %APE
5CLOSURE 9DICATORS 9 ,SEC;N #AD4C4 ,IF !
BOXES NE$ TO 2 %[N !N AN ALT]NATIVE
APPROA* IS TO USE ! "UL9E 9DICATOR & A
TRANSCRIB]'S NOTE4
#C "6 #D "7 #G
@.<,"UL9+ %[S T ! SYMBOL 0 BOX$ 9
PR9T4@.>
_2#C _2"6 _2#D _2"7 _2#G
-------------------------------------#AI
#C4F ,OMIS.N M>KS 9 MA!MATICAL
EXPRES.NS
.=",- L;G DA%
.=.- "USCORE
.=+ VISIBLE SPACE
.=$#D SQU>E "<SEE ,SEC;N #AD">
#C"6#G "7 ",- "<OMIS.N %[N Z A C5TR$
HORIZONTAL L9E 9 PR9T"> #BF
#C ,OP]A;N & ,COMP>ISON A#AI
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HORIZONTAL L9E 9 PR9T">
#C"6#G "7 ;8 "<OMIS.N %[N Z A "Q M>K 9
PR9T">
#C$#D:#G "7 #AJ "<OMIS.N %[N Z AN EMPTY
SQU>E 9 PR9T">
#C+#G "7 #AJ "<OMIS.N %[N Z AN UNM>K$
GAP 9 PR9T">
#I/AB "7 ;(#C./+) "<OMIS.N %[N )9 A
FRAC;N 9 PR9T">
,NOTE T ! VISIBLE SPACE IS ! SAME
SYMBOL Z ! CLOS+ ROOT 9DICATOR "<DEF9$ 9
SEC;N #AB">4 ,IF ! RADIC& IS TO 2 FILL$
9 !N ! SYMBOL WD 2 REP1T$ TO T]M9ATE !
ROOT4
#E "7 ;;%++
-------------------------------------#BJ
#D ,SPATIAL ,LAY\T & ,DIAGRAMS
#D4A ,SPATIAL CALCULA;NS
.="3 2G9 HORIZONTAL L9E MODE
.=_ V]TICAL L9E SEG;T #BG
#D ,SPATIAL & ,DIAGRAMS A#BJ
.=# SPAC$ NUM]IC 9DICATOR
.=## NUM]IC PASSAGE 9DICATOR
.=#' NUM]IC PASSAGE T]M9ATOR
,": HORIZONTAL L9ES >E NE$$ )9 *N'S
SUMS1 HORIZONTAL L9E MODE %D 2 US$4 ,!
LAY\T ( ! CALCULA;N C FOLL[ PR9T1 ?\<
FE$BACK F T1*]S "W+ ) /UD5TS %D AL 2
TAK5 9TO A3.T4 ,^! GUIDEL9ES C AL 2
APPLI$ = O!R SPATIAL >RANGE;TS S* Z
F9ANCIAL /ATE;TS OR A3.TANCY TEXTS4
,COLUMNS TO 2 ADD$ %D N 3TA9 NUM]IC
9DICATORS OR OP]A;N SIGNS4 ,? C 2 A*IEV$
BY ALIGN+ NUM]IC 9DICATORS V]TICALLY,-A
NUM]IC 9DICATOR FOLL[$ BY A SPACE /
9ITIATES NUM]IC MODE4
,ALT]NATIVELY USE ! NUM]IC PASSAGE
9DICATOR & ! NUM]IC T]M9ATOR : SET
NUM]IC MODE & GRADE #A MODE = ! 5CLOS$
TEXT4 ,9 A NUM]IC PASSAGE NUM]IC
9DICATORS >E N US$1 & ANY L[]CASE LR A
TO ;J IS PREC$$ BY A GRADE #A 9DICATOR4
,! L9E ABV & 2L SPATIAL CALCULA;NS %D
EI 2 BLANK1 OR %D ONLY 3TA9 ! #BH
#D ,SPATIAL & ,DIAGRAMS B#BJ
NUM]IC PASSAGE 9DICATOR OR T]M9ATOR4
#D4A4A ,ADDI;N OR SUBTRAC;N
#DEF
"6# CD
"333
#DIJ
-------------------------------------#BA
,! SECOND V].N 2L ILLU/RATES ! USE ( !
NUM]IC PASSAGE 9DICATOR4 ,! PLACE;T ( !
OP]A;N SIGN DOES N NE$ TO FOLL[ PR9T1 B
C 2 ADJU/$ TO SUIT LOCAL T1*+ PRACTICES4
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! CALCULA;N CLE> ( NUM]IC 9DICATORS4
##
DEF "6
CD
"33
DIJ
#'
#BI
#D ,SPATIAL & ,DIAGRAMS A#BA
#D4A4B ,L;G MULTIPLICA;N
##
ABC "8
AB
"333
BDF
ABC
"333
ADGF
#'
-------------------------------------#BB
#D4A4C ,DIVI.N
,! SPAC$ V]TICAL L9E SEG;T "<DOTS
#DEF"> C 2 US$ TO REPRES5T ! CURV$ OR
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#AE04 ,A S+LE SPACE MAY AL 2 A3EPTABLE4
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"3333
#E _ #AE "7 #C #CJ
#D ,SPATIAL & ,DIAGRAMS A#BB
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LAY\T1 :ILE ! V].N ON ! "R IS AN
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MAKE X EASI] = ! /UD5T TO REPLICATE4
@.<,! V].N ON ! LEFT IS GIV5
F/4@.>
#CA
#D ,SPATIAL & ,DIAGRAMS B#BB
##
IC
"3333
E _ DFE
DE
"33
AE
AE
"33
#'
##
E _ DFE
DE _I
"33
AE _C
AE
"33
#'
,ANS3 #IC
#CB
#D ,SPATIAL & ,DIAGRAMS #BC
#D4A4D ,SPATIAL FRAC;NS = T1*+
PURPOSES
TWO ?IRDS
#B
"3
#C
"O ?IRD PLUS "O HALF EQUALS TWO SIX?S
PLUS ?REE SIX?S EQUALS FIVE SIX?S
##
A A B C E
"3 "6 "3 "7 "3 "6 "3 "7 "3
C B F F F
#'
#CC
#D ,SPATIAL & ,DIAGRAMS A#BC
#D4A4E ,C>RYOV]S
,9S]T SPACES 2T ! DIGITS 9 BRL S T !
SMALL NUMB]S C 2 PLAC$ ON _! [N R[ ABV
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TYPE ( EXAMPLE C 2 ADJU/$ TO SUIT LOCAL
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"<SET \ 9 V]TICAL =M">4
@.<,! SMALL NUMB]S >E A "O TO !
LEFT & ABV ! NUMB] TWO 9 ! TOP L9E1
& A "O TO ! "R & 2L ! NUMB] "O 9 !
SECOND L9E4@.>
##
A
C B "-
A D
A
"333
A H
#'
#CD
#D ,SPATIAL & ,DIAGRAMS #BD
#D4A4F ,C.ELLA;N
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! USUAL BRL =M & EXPLA9 ! PROCESS 9 A
TRANSCRIB]'S NOTE4 ,TAKE ! AGE ( ! /UD5T
9TO A3.T & *ECK ) LOCAL T1*+ PRACTICES4
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APPROA*ES >E %[N4
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#CE
#D ,SPATIAL & ,DIAGRAMS A#BD
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##
A
C@: H H
"333 "8 "333 "7 "33
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C
#'
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RECOMM5D$ T ! NUM]IC 9DICATOR IS AL US$
2F EA* ITEM NUMB]4 ,EX3
#CF
#D ,SPATIAL & ,DIAGRAMS #BE
##
#A #B
AB "6 CF "6
CD EB
"33 "33
#C #D
FC "6 AED "6
AI GF
"33 "33
#'
#D4B ,TALLY M>KS
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____3 ____3 ____3 ___
OR
,EI<TE5
_____ _____ _____ ___
-------------------------------------#BF
#D4C ,TABLES
,HORIZONTAL L9E MODE IS US$ TO "UL9E !
COLUMN H1D+S4 ,TWO SPACES >E LEFT 2T
COLUMNS4 ,IF NUMB]S >E TO 2 ADD$ !Y %D 2
"R ADJU/$4 ,EX3
#CH
#D ,SPATIAL & ,DIAGRAMS A#BF
;;;Y "7 X9#B"-#C;'
;X ;Y
"333 "333
"-#A "-#B
#J "-#C
#A "-#B
#B #A
#C #F
,X IS AL A3EPTABLE TO 9CLUDE A BOX+
L9E ABV & 2L ! TABLE4
-------------------------------------#BG
#D4D ,DIAGRAMS
,? SEC;N DOES N ATTEMPT TO GIVE ADVICE
ON ! DESIGN ( TACTILE DIAGRAMS OR ON !
RANGE ( PRODUC;N ME?ODS AVAILABLE4
#D4D4A ,HORIZONTAL ,L9E ,MODE
,MO/ TRANSCRIB]S H EXP]I;T$ ) ! USE (
BRL SYMBOLS = DRAW+ SIMPLE BOX %APES OR
FAMILY TREES4 ,AL? ^! >E (T5 N Z #CI
#D ,SPATIAL & ,DIAGRAMS A#BG
R1DABLE Z A RAIS$ L9E DRAW+1 !Y / H _!
USES4
,IF AN UNBROK5 BRL L9E 2G9S ) !
HORIZONTAL L9E MODE 9DICATOR "3 !N ALL !
O!R CELLS 9 T L9E W 2 R1D Z %APES R ?AN
SYMBOLS4
,IF V]TICAL L9ES >E AL %[N US+ !
V]TICAL L9E SEG;T _ !N A BACK TRANSLA;N
PROCESS OR A TUN$ 9 R1D] C 2 CLE> T ? IS
A DIAGRAM R ?AN A /R+ ( TEXT4
"3333433334
_ _ _
_ _ _
"3333J3333J
H5RY ;M M>Y
_
"333333W33333334
_ _ _
SUE DAVE PAT
#DJ
#D ,SPATIAL & ,DIAGRAMS B#BG
,REF] TO ! .7,RULES ( ,UNIFI$ ,5GLI%
,BRL1 ,SEC;N #AF3 ,L9E ,MODE1 ,GUIDE
,DOTS.' = SYMBOLS TO USE 9 L9E MODE4
,SPECIAL M5;N IS MADE ( >R[ SYMBOLS : C
2 US$ )\T T]M9AT+ HORIZONTAL L9E MODE4
,! FOLL[+ TWO EXAMPLES ILLU/RATE ! USE (
>R[ SYMBOLS & AL 9CLUDE UNLI/$ DOT
PATT]NS = ! %APES ( ! HOLL[ & SOLID
CIRCLES US$ 9 PR9T TO REPRES5T ! OP5 OR
CLOS$ 5D ( AN 9T]VAL4
-------------------------------------#BH
,9 ! PR9T EXAMPLE 2L A NUMB] L9E IS
GRADUAT$ F "-#B TO #C & AN 9T]VAL IS
M>K$ ABV X />T+ AT #J & F9I%+ ) AN OP5
CIRCLE AT #B4 ,9 ! BRL V].N 2L1 ,,UEB
>R[ MODE IS US$ = ! LEFT & "R PO9T+ >R[S
"<,SEE ,SEC;N #AC"> & HORIZONTAL L9E
MODE IS US$ 2T !M4 ,! 9T]VAL XF IS %[N Z
A L9E F9I%+ ) ! DOT PATT]N .=Y = ! EMPTY
CIRCLE4
"333333333Y
\["33W3333W3333W3333W3333W3333W333\O
"-#B "-#A #J #A #B #C #DA
#D ,SPATIAL & ,DIAGRAMS A#BH
,9 ! SECOND EXAMPLE 2L ! 9T]VAL />TS )
A SOLID CIRCLE AT "-#B & F9I%ES AT #J4
,9 ! BRL V].N1 ! DOT PATT]N .== HAS BE5
US$ = ! SOLID CIRCLE4
=3333333333
\["33W3333W3333W3333W3333W3333W333\O
"-#B "-#A #J #A #B #C
,NOTE T 9 ? SECOND BRL EXAMPLE1 !
HORIZONTAL L9E MODE 9DICATOR .="3 CD N
PREC$E ! CIRCLE SYMBOL )\T OBSCUR+ !
POSI;N ( T />T PO9T ON ! NUMB] L9E4 ,9
ANY DIAGRAM1 TACTUAL CL>;Y MAY 2 DEEM$ M
IMPORTANT ?AN TE*NICAL AMBIGU;Y4
,ALW BE> 9 M9D T A WELL EXECUT$ RAIS$
L9E DRAW+ W G5]ALLY 2 M R1DABLE ?AN A
DIAGRAM CR1T$ US+ BRL CELL PATT]NS4
,SEE ,SEC;N #AF = EXAMPLES ( *EMI/RY
/RUCTURES DRAWN US+ HORIZONTAL L9E MODE
& V]TICAL & DIAGONAL L9E SEG;TS4
#DB
#D ,SPATIAL & ,DIAGRAMS #BI
#D4D4B ,LABELL+ DIAGRAMS
,? SEC;N DOES N ATTEMPT TO GIVE ADVICE
ON H[ TO E6ECTIVELY LABEL A DIAGRAM 9
BRL S T X IS CLE> TO ! R1D] : "P ( !
DIAGRAM EA* LABEL REF]S TO4 ,R1 X LI/S
CODE A3OMMODA;NS : C 2 MADE )9 ! 3T5T (
C]TA9 LABELS :5 SPACE IS AT A PREMIUM4
_4 ,:5 LABELL+ PO9TS1 L9ES ETC4 ) S+LE
LRS1 ^! C 2 3SI/5TLY %[N Z TWO CELLS
BY OMITT+ ! GRADE #A 9DICATOR = ALL
CAPITAL LRS & 9CLUD+ X = ALL L[]CASE
LRS 9CLUD+ A1 I & O4
_4 ,:5 GRADUAT+ AN AXIS ) NEGATIVE
NUMB]S1 ! HYPH5 CD 2 US$ 9/1D ( ! M9US
SIGN "<I4E4 ! DOT #E PREFIX CD 2
OMITT$">4 ,9CLUDE A TRANSCRIB]'S NOTE4
_4 ,:5 WRIT+ COORD9ATES S* Z
"<#B1 -#C">1 ANY ( ! FOLL[+ CD 2
3SID]$3 OMITT+ ! SPACE2 OMITT+ ! DOT
#E PREFIX F ! R.D BRACKETS2 OMITT+ !
DOT #E PREFIX F ! M9US SIGN4 ,9CLUDE A
TRANSCRIB]'S NOTE4 #DC
#D ,SPATIAL & ,DIAGRAMS A#BI
_4 ,:5 LABELL+ A L9E OR CURVE ) XS
EQUA;N1 ! SPACE EI SIDE ( ! EQUALS
SIGN CD 2 OMITT$4
_4 ,:5 LABELL+ ANGLES1 ! DEGREE SIGN CD
2 OMITT$4 ,9CLUDE A TRANSCRIB]'S NOTE4
-------------------------------------#CJ
#E ,GR\P+ ,DEVICES "<,BRACKETS">
.="< OP5+ R.D P>5!SIS
.="> CLOS+ R.D P>5!SIS
.=.< OP5+ SQU>E BRACKET
.=.> CLOS+ SQU>E BRACKET
.=_< OP5+ CURLY BRACE
.=_> CLOS+ CURLY BRACE
.=@< OP5+ ANGLE BRACKET
.=@> CLOS+ ANGLE BRACKET
.=_\ V]TICAL B> "<OP5 OR CLOSE ABSOLUTE
VALUE OR MODULUS">
.=,"< BIG "<MULTI-L9E"> OP5+ R.D P>5!SIS
.=,"> BIG "<MULTI-L9E"> CLOS+ R.D
P>5!SIS
.=,.< BIG "<MULTI-L9E"> OP5+ SQU>E
BRACKET #DD
#E ,GR\P+ ,DEVICES A#CJ
.=,.> BIG "<MULTI-L9E"> CLOS+ SQU>E
BRACKET
.=,_< BIG "<MULTI-L9E"> OP5+ CURLY BRACE
.=,_> BIG "<MULTI-L9E"> CLOS+ CURLY
BRACE
.=,_\ BIG "<MULTI-L9E"> V]TICAL B>
,PR9T BRACKETS >E USUALLY UNSPAC$ F !
ITEMS !Y 5CLOSE & ! SAME %D 2 D"O 9 BRL4
,SEE ,SEC;N #AE = ! LAY\T ( MATRICES &
VECTORS4
,LET ! MIDPO9T ( ,A"<#C1 "-#E"> &
,B"<#J1 #D"> ; ! PO9T "<;X1 ;Y">4
.<#B"<X"6Y">.>"/#D @< #AJ
_\"-#F_\ "7 _\#F_\ "7 #F "<ABSOLUTE
VALUE">
,3SID] ! SEQU;E @<,T;5N@>4 "<;,T SUB ;N
5CLOS$ 9 ANGLE BRACKETS">
#DE
#F ,FRAC;NS #CA
#F ,FRAC;NS
.=/ SIMPLE NUM]IC FRAC;N L9E
.=./ G5]AL FRAC;N L9E
.=( G5]AL FRAC;N OP5 9DICATOR
.=) G5]AL FRAC;N CLOSE 9DICATOR
#F4A ,SIMPLE NUM]IC FRAC;NS
,A SIMPLE NUM]IC FRAC;N IS "O ^:
NUM]ATOR & D5OM9ATOR 3TA9 ONLY DIGITS1
DECIMAL PO9TS1 COMMAS OR SEP>ATOR SPACES
& ^: FRAC;N L9E 9 PR9T IS DRAWN 2T ! TWO
V]TICALLY "<OR NE>LY V]TICALLY"> >RANG$
NUMB]S4 ,9 S* A CASE A NUM]IC FRAC;N L9E
SYMBOL MAY 2 US$ 2T ! NUM]ATOR &
D5OM9ATOR & 3T9UES ! NUM]IC MODE4
,FIVE EI<?S ( ! CLASS >E BOYS4
#E/H ( ! CLASS >E BOYS4
,CALCULATE #E4G OV] #B1JJJ
,CALCULATE #E4G/B1JJJ
#DF
#F ,FRAC;NS A#CA
#F4B ,MIX$ NUMB]S
,MIX$ NUMB]S %D 2 TR1T$ Z TWO UNSPAC$
NUM]IC ITEMS4
,TWO & A HALF CUPS ( SUG>
#B#A/B CUPS ( SUG>
#AGEJ CM "7 "O & ?REE QU>T] ;M
#AGEJ CM "7 #A#C/D ;M
-------------------------------------#CB
#F4C ,FRAC;NS WRITT5 9 L9E> =M 9
PR9T
,! NUM]IC FRAC;N L9E NE$ N 2 US$ :5 !
PR9T IS EXPRESS$ L9E>LY US+ AN ORD9>Y
SLA% "<OBLIQUE /ROKE"> SYMBOL4 ,9 S* A
CASE ! SAME SYMBOLS C 2 US$ Z 9 PR9T4
#C_/#H ( ! CLASS >E GIRLS4
#F4D ,G5]AL FRAC;N 9DICATORS
,IF ! NUM]ATOR OR D5OM9ATOR IS N
5TIRELY NUM]IC Z DEF9$ 9 #F4A1 !N !
G5]AL FRAC;N 9DICATORS %D 2 US$4 #DG
#F ,FRAC;NS A#CB
,AF ! OP5+ 9DICATOR ! NUM]ATOR EXPRES.N
IS WRITT51 !N ! G5]AL FRAC;N L9E SYMBOL1
!N ! D5OM9ATOR EXPRES.N & F9ALLY ! CLOS+
9DICATOR4 ,BO? NUM]ATOR & D5OM9ATOR MAY
2 ANY K9D ( EXPRES.N :AT"E1 9CLUD+
FRAC;NS ( EI SIMPLE NUM]IC OR G5]AL
TYPE4
,9 ! V]BALISA;NS 2L1 QUOTA;N M>KS H
BE5 US$ TO REFLECT ! SPATIAL >RANGE;T
US$ 9 PR9T4
;Y EQUALS 8;X OV] #B0
;;;Y "7 (X./#B);'
TWO & A HALF OV] 8;X PLUS ;Y0
;(#B#A/B./X"6Y)
TWO SLA% ?REE ALL OV] #E
;(#B_/#C./#E)
FRAC;N 8;X OV] #B0 PLUS 8;Y OV] #C0 ALL
OV] 8;X PLUS ;Y0
;;((X./#B)"6(Y./#C)./X"6Y)
#DH
#F ,FRAC;NS #CC
SPE$ EQUALS 4T.E OV] "T1 %[+ ALT]NATIVE
USAGE ( GRADE #A 9DICATORS
;;;SPEED "7 (DISTANCE./TIME);'
SPE$ "7 ;;(DISTANCE./TIME)
SPE$ "7 ;(DI/.E./"T;)
#F4E ,EXTRA ,EXAMPLES
,^! EXAMPLES >E 9CLUD$ TO FUR!R
ILLU/RATE ! DI6];E 9 PR9T 2T SIMPLE
NUM]IC FRAC;NS Z DEF9$ 9 #F4A1 & FRAC;NS
: W NE$ G5]AL FRAC;N 9DICATORS 9 BRL Z
DEF9$ 9 #F4D4
,SIMPLE NUM]IC FRAC;NS3
#C OV] #D
#C/D
#GC OV] #D
#GC/D
#C OV] #D4B
#C/D4B #DI
#F ,FRAC;NS A#CC
#E4C OV] #D1BJJ
#E4C/D1BJJ
#E & ?REE-QU>T]S
#E#C/D
#F & 8#C OV] #AJ"JJJ0 "<SPACE US$ Z
SEP>ATOR 9 PR9T">
#F#C/AJ"JJJ
,FRAC;NS NE$+ G5]AL FRAC;N 9DICATORS
2C !Y 3TA9 LRS3
;X OV] ;Y
;;(X./Y)
;X OV] #D
;(X./#D)
#C OV] ;B
;(#C./B)
A OV] 8;B PLUS ;C0
;;(A./B"6C) #EJ
#F ,FRAC;NS B#CC
#CX OV] #D
;(#CX./#D)
;X OV] #D1JJJ
;(X./#D1JJJ)
,FRAC;NS NE$+ G5]AL FRAC;N 9DICATORS
2C !Y 3TA9 "S?+ M ?AN DIGITS1 COMMAS OR
DECIMAL PO9TS3
#C SQU>$ OV] #F
;(#C9#B./#F)
@S#EE OV] #E
;(@S#EE./#E)
#BDM OV] #C;CM
;(#BDM./#C;CM)
"Q M>K OV] #D
;(8./#D)
#EA
#F ,FRAC;NS C#CC
TWO-?IRDS OV] #D
;(#B/C./#D)
8#B M9US #E0 OV] #C
;(#B"-#E./#C)
-------------------------------------#CD
#G ,SUP]SCRIPTS & SUBSCRIPTS
.=5 LEVEL *ANGE D[N "<SUBSCRIPT">
.=9 LEVEL *ANGE UP "<SUP]SCRIPT OR
EXPON5T">
.=.5 EXPRES.N DIRECTLY 2L
.=.9 EXPRES.N DIRECTLY ABV
.=< BRL GR\P+ OP5
.=> BRL GR\P+ CLOSE
#G4A ,DEF9I;N ( AN ITEM
,! SCOPE ( ANY ( ! F\R LEVEL *ANGE
9DICATORS1 T IS1 ! SYMBOL"<S"> A6ECT$ BY
X1 IS ! NEXT 8ITEM04 ,AN ITEM IS DEF9$ Z
ANY ( ! FOLL[+ GR\P+S IF IMMLY AF !
LEVEL *ANGE 9DICATOR3
#A4 ,AN 5TIRE NUMB]1 I4E4 ! #EB
#G SUP]SCRIPT SUBSCRIPT A#CD
9ITIAT+ NUM]IC SYMBOL & ALL SU3E$+
SYMBOLS )9 ! NUM]IC MODE ?US E/ABLI%$
"<: WD 9CLUDE ANY 9T]IOR DECIMAL
PO9TS1 COMMAS1 SEP>ATOR SPACES1 OR
SIMPLE NUM]IC FRAC;N L9ES">4
#B4 ,AN 5TIRE G5]AL FRAC;N1 5CLOS$ 9
FRAC;N 9DICATORS "<SEE ,SEC;N #F">4
#C4 ,AN 5TIRE RADICAL EXPRES.N1 5CLOS$ 9
RADICAL 9DICATORS "<SEE ,SEC;N #H">4
#D4 ,AN >R[ "<SEE ,SEC;N #AC">4
#E4 ,AN >BITR>Y %APE "<SEE ,SEC;N #AD">4
#F4 ,ANY EXPRES.N 5CLOS$ 9 MAT*+ PAIRS (
R.D P>5!SES1 SQU>E BRACKETS OR CURLY
BRACES "<SEE ,SEC;N #E">4
#G4 ,ANY EXPRES.N 5CLOS$ 9 ! BRL GR\P+
9DICATORS4
#H4 ,IF N"O ( ! =EGO+ APPLY1 ! ITEM IS
SIMPLY ! NEXT 9DIVIDUAL SYMBOL4
-------------------------------------#CE
#G4B ,SUP]SCRIPTS & SUBSCRIPTS )9
LIT]>Y TEXT
,NOTE T IF A SUP]SCRIPT OR SUBSCRIPT
APPE>S )9 A GRADE #B PASSAGE1 X W NE$ A
GRADE #A 9DICATOR4 #EC
#G SUP]SCRIPT SUBSCRIPT A#CE
,! >EA IS #F ;M SQU>$
,! >EA IS #F M;9#B
,! PO9TS ;,P SUB #A & ;,P SUB #B
,! PO9TS ,P;5#A & ,P;5#B
,SMI? WROTE A PAP] SUP] #EF : SAYS 444
"<SUP] #EF 9DICAT+ A FOOTNOTE">
,SMI? WROTE A PAP];9#EF : SAYS 444
,! =MULA = WAT] IS ;,H SUB #B ,O
,! =MULA = WAT] IS ,H;5#B,O
#G4C ,ALGEBRAIC EXPRES.NS 9VOLV+
SUP]SCRIPTS
,REF] TO ! DEF9I;N ( AN ITEM 9 ,SEC;N
#G4A TO DECIDE :5 BRL GR\P+ SYMBOLS >E
REQUIR$4 ,NOTE T ! BRL GR\P+ SYMBOLS
!MVS H AN ALT]NATIVE GRADE #B M1N+4
,9 ! V]BALISA;NS 2L1 QUOTA;N M>KS H
BE5 US$ TO REFLECT ! SPATIAL >RANGE;T
US$ 9 PR9T4
#ED
#G SUP]SCRIPT SUBSCRIPT B#CE
;X SQU>$
X;9#B
;X SQU>$ "TS ;Y
X;9#BY
;X TO ! #BY
;;X9<#BY>
-------------------------------------#CF
8;X TO ! ;Y0 PLUS #A
X;9Y"6#A
;X TO ! 8;Y PLUS #A0
;;X9<Y"6#A>
;X TO ! 8;Y PLUS #A0 PLUS #C
;;X9<Y"6#A>"6#C
;X TO ! TWO ?IRDS
X;9#B/C
;X SQU>$ OV] #C
;;(X9#B./#C) #EE
#G SUP]SCRIPT SUBSCRIPT A#CF
;X TO ! 8HALF ;Y0
;;X9<#A/BY>
8;X TO ! HALF0 ;Y
X;9#A/BY
;X SUP]SCRIPT 8A OV] ;B0 5D SUP]SCRIPT
;Y EQUALS ;X
;;;X9(A./B)Y "7 X;'
,IF FRAC;N 8;X SQU>$ PLUS #BX0 ALL OV]
8#A PLUS ;X SQU>$0 5D FRAC;N EQUALS #A
;X SQU>$ PLUS #BX EQUALS #A PLUS ;X
SQU>$
;X EQUALS #A OV] #B
;;;,IF (X9#B"6#BX./#A"6X9#B) "7 #A
X9#B"6#BX "7 #A"6X9#B
X "7 #A/B;'
-------------------------------------#CG
#G4D ,MULTIPLE LEVELS
,NOTE T A SUP]SCRIPT : XF HAS A
SUP]SCRIPT DOES N FIT ! ABV #EF
#G SUP]SCRIPT SUBSCRIPT A#CG
DEF9I;N ( AN ITEM4 ,9 S* CASES BRL GR\P+
SYMBOLS >E REQUIR$4
;E TO ! ;X SQU>$
;;E9<X9#B>
;E TO ! OP5 P>5!SIS ;X SQU>$ CLOSE
P>5!SIS
;;E9"<X9#B">
;,P ) AN ;X SUB I 9 ! SUBSCRIPT POSI;N
;;,P5<X5I>
#G4E ,NEGATIVE SUP]SCRIPTS
,NEGATIVE SUP]SCRIPTS M/ 2 5CLOS$ 9
BRL GR\P+ SYMBOLS4 "<,? IS 2C A M9US
SIGN C 2 AN ITEM 9 XS [N "R1 Z 9 #G4F
2L4">
#J4JJDE EQUALS #D4E "TS #AJ TO ! M9US #C
#J4JJDE "7 #D4E"8#AJ9<"-#C>
;V EQUALS #FJ MS TO ! M9US #A
;;;V "7 #FJ MS9<"-#A>;' #EG
#G SUP]SCRIPT SUBSCRIPT B#CG
A TO ! M9US #B;B P[]
;;A9<"-#B;B>
-------------------------------------#CH
#G4F ,EXAMPLES F ,*EMI/RY
,SEE ,SEC;N #AF = M EXAMPLES4
,,CH SUB #D PLUS #B ,CL EQUALS ,,CH SUB
#C ,CL PLUS ,H,CL
;;;,C,H5#D"6#B,CL "7 ,C,H5#C,CL"6,H,CL;'
,IONS ;,H SUP] PLUS1 ,CL SUP] M9US & ,CA
SUP] 8#B PLUS0
,IONS ,H;9"61 ,CL;9"- & ;;,CA9<#B"6>
#G4G ,SIMULTANE\S SUP]SCRIPTS &
SUBSCRIPTS
,IF M ?AN "O SUP]SCRIPT OR SUBSCRIPT
APPLY1 "W F BOTTOM TO TOP1 OR LEFT TO
"R4 ,IF ! PR9T 9DICATES BY ! PLAC+ ( !
SUBSCRIPT T X IS 2+ APPLI$ AF !
SUP]SCRIPT !N ! ORD] C 2 REV]S$4
;X SUB #A SQU>$ EQUALS ;Y SUB #B CUB$
;;;X5#A9#B "7 Y5#B9#C;' #EH
#G SUP]SCRIPT SUBSCRIPT A#CH
;X SQU>$ SUB ;K
;;X9#B5K
#G4H ,LEFT-4PLAC$ SUP]SCRIPTS OR
SUBSCRIPTS
,"S"TS 9 PR9T A SUP]SCRIPT OR
SUBSCRIPT IS WRITT5 TO ! LEFT ( ! BASE
SYMBOL 9/1D ( TO ! "R4 ,^! >E H&L$
SIMPLY BY US+ ! CORRESPOND+ ORD9>Y 9DEX
EXPRES.N PRIOR TO ! BASE SYMBOL4
;,U ) #IB WRITT5 2L LEFT & #BCH WRITT5
ABV LEFT
;;5#IB9#BCH,U
M9US #B PLUS M9US #C EQUALS M9US #E "<)
M9US SIGNS 9 ! SUP]SCRIPT POSI;N">
;;;9"-#B"69"-#C "7 9"-#E;'
-------------------------------------#CI
#G4I ,MODIFI]S DIRECTLY ABV OR 2L
,IF "S?+ IS WRITT5 DIRECTLY ABV OR 2L
A T]M R ?AN TO ! "R OR LEFT1 USE !
DIRECTLY ABV 9DICATOR OR DIRECTLY 2L
9DICATOR 9/1D ( ! SUP]SCRIPT OR #EI
#G SUP]SCRIPT SUBSCRIPT A#CI
SUBSCRIPT 9DICATOR4
,COMMON MODIF]S S* Z ! B>1 >R[1 DOT1
TILDE1 HAT OR >C >E TR1T$ SEP>ATELY 9
,SEC;N #AB4
,! SUM F ;X EQUALS #A TO ;N ( ;X SUB I
SQU>$ ) SUMMA;N LIMITS DIRECTLY 2L &
ABV A CAPITAL SIGMA
22,.S.5<X"7#A>.9NX5I9#B
,! LIMIT1 Z ;X T5DS TO A1 ( ;F ( ;X1
EQUALS #A1 ) ;X >R[ A DIRECTLY 2L LIM
;;;LIM.5<X\OA>F"<X"> "7 #A;'
;X ) B> OV] X
X;: "<SEE ,SEC;N #AB4A">
-------------------------------------#DJ
#H ,SQU>E ,ROOTS & O!R RADICALS
.=% OP5 RADICAL "<ROOT">
.=+ CLOSE RADICAL "<ROOT">
.="% SQU>E ROOT SIGN )\T V9CULUM
#FJ
#H ,SQU>E ,ROOTS A#DJ
#H4A ,SQU>E ROOTS
,! EXPRES.N 9SIDE ! SQU>E ROOT SIGN 9
PR9T "<! RADIC&"> %D 2 PREC$$ BY ! OP5
RADICAL SIGN & FOLL[$ BY ! CLOSE RADICAL
SIGN4 ,! RADIC& XF MAY 2 ANY EXPRES.N
:ATSO"E1 & MAY "!=E 3TA9 RADICALS Z WELL
Z O!R MA!MATICAL /RUCTURES4 ,NOTE T BO?
! OP5+ & CLOS+ RADICAL SIGNS H
ALT]NATIVE GRADE #B M1N+S S MAY NE$
GRADE #A 9DICATORS4
! SQU>E ROOT ( #I EQUALS #C
;%#I+ "7 #C
"<,NOTE T ! SQU>E ROOT ( A SIMPLE
NUMB] / NE$S TO 2 T]M9AT$ BY ! CLOSE
RADICAL SIGN4">
;R EQUALS ! SQU>E ROOT ( ;X SQU>$ PLUS
;Y SQU>$ 5D ROOT
;;;R "7 %X9#B"6Y9#B+;'
#FA
#H ,SQU>E ,ROOTS B#DJ
! SQU>E ROOT ( FRAC;N #GHC4B "TS #F4EDG
OV] #J4DFBH 5D FRAC;N 5D ROOT
;;%(#GHC4B"8#F4EDG./#J4DFBH)+
;X EQUALS ! FRAC;N3 M9US ;B PLUS-OR-M9US
! SQU>E ROOT ( ;B SQU>$ M9US #D;AC 5D
ROOT ALL OV] #B;A
;;;X "7 ("-B_6%B9#B"-#D;AC+./#B;A);'
-------------------------------------#DA
#H4B ,CUBE ROOTS ETC
,9 PR9T ! RADICAL 9DEX1 IF PRES5T1 IS
PR9T$ ABV & TO ! LEFT ( ! RADICAL SIGN4
,? 9DEX IS PLAC$ 9 BRL Z A SUP]SCRIPT
EXPRES.N IMMLY FOLL[+ ! OP5+ RADICAL
SYMBOL4
! CUBE ROOT ( #H EQUALS #B
;;%9#C#H+ "7 #B
;Q EQUALS ! CUBE ROOT ( ;X CUB$ PLUS ;Y
CUB$ PLUS ;Z CUB$ 5D ROOT
;;;Q "7 %9#CX9#C"6Y9#C"6Z9#C+;'
#FB
#H ,SQU>E ,ROOTS A#DA
! MN-TH ROOT ( XY
;;%9<MN>XY+
#HA TO ! ?REE-QU>T]S EQUALS "<! F\R?
ROOT ( #HA"> CUB$ EQUALS "<! SQU>E
ROOT ( ! SQU>E ROOT ( #HA"> CUB$
EQUALS "<! SQU>E ROOT ( #I"> CUB$
EQUALS ?REE CUB$ EQUALS #BG
;;;#HA9#C/D "7 "<%9#D#HA+">9#C
"7 "<%%#HA++">9#C "7 "<%#I+">9#C
"7 #C9#C "7 #BG;'
#H4C ,SQU>E ROOT SIGN ON XS [N
,"S"TS PR9T OMITS ! HORIZONTAL L9E
"<V9CULUM"> ABV ! RADIC&4 ,9 OLD] BOOKS1
@.<? PR9T SYMBOL FOLL[$ BY #D@.> CD M1N
! SQU>E ROOT ( #D1 B 9 "S MOD]N 3TEXTS
CD 2 J A SEQU;E ( SYMBOLS4
,:5 9TRODUC+ SIMPLE ROOTS TO "Y] *N1 X
IS BETT] TO 9TRODUCE ! 3CEPT ( OP5+ &
CLOS+ 9DICATORS F ! />T4 ,IF X IS CLE> F
! 3TEXT T ? IS A SQU>E ROOT1 !N X IS BE/
TO ASSUME AN 9T5D$ V9CULUM & TO USE !
/&>D =M4 ,IF1 H["E1 ! SQU>E ROOT #FC
#H ,SQU>E ,ROOTS B#DA
SIGN IS 2+ US$ Z AN ISOLAT$ GRAPHIC
SYMBOL !N ! DOT #E =M C 2 US$4 ,EX4
#D )9 SQU>E ROOT ) V9CULUM EQUALS #B
;%#D+ "7 #B
SQU>E ROOT SIGN )\T V9CULUM FOLL[$ BY #D
EQUALS #B
;%#D+ "7 #B
,! 8SQU>E ROOT )\T V9CULUM0 SIGN
,! "% SIGN
-------------------------------------#DB
#I ,FUNC;NS
,IF A FUNC;N "N IS PREC$$ OR FOLL[$ BY
A LR ( ! SAME FONT & ALPHABET1 X MAY N 2
CLE> ": ! FUNC;N "N 2G9S OR 5DS4 ,9 PR9T
? IS CL>IFI$ US+ A V>IETY ( TE*NIQUES4
,TAKE ! EXAMPLE A "TS ! COS9E ( ;T4 ,9
PR9T ? IS (T5 WRITT5 .7A COS ;T.'4
,NOTICE T "H ! .2A & .2;T >E WRITT5 9
ITALICS TO %[ !Y >E V>IABLES1 & A SMALL
SPACE IS 9CLUD$ EI SIDE ( ! FUNC;N #FD
#I ,FUNC;NS A#DB
"N4 ,? TE*NIQUE IS AL US$ 9 PR9T TO
4T+UI% ^W FRAG;TS O!R ?AN FUNC;NS4
,! EXAMPLES 9 ? SEC;N >E TAK5 F
SECOND>Y S*OOL SCI;E & MA!MATICS B !
SAME RULES %D 2 FOLL[$ :5 D1L+ ) FUNC;NS
)9 ANY SUBJECT >EA4
#I4A ,SPELL+ & CAPITALISA;N
,FOLL[ PR9T = ! SPELL+ & CAPITALISA;N
( FUNC;N "NS4
,F9D ! VALUE ( ,COS9E ;,B4
#I4B ,ITALICS
,": LRS 2F OR AF ! FUNC;N "N >E WRITT5
9 ITALICS TO 9DICATE !Y >E V>IABLES1 !
ITALICS %D 2 OMITT$ 9 A3ORD.E ) RULE
#A4C4
-------------------------------------#DC
#I4C ,SPAC+
,": ! FUNC;N "N IS PREC$$ OR FOLL[$ BY
A LR1 A SPACE MAY 2 NE$$1 TO REMOVE
AMBIGU;Y Z TO ": ! FUNC;N "N 2G9S & 5DS4
,! SPACE IS N NE$$ IF ! FUNC;N "N #FE
#I ,FUNC;NS A#DC
IS ALR SEP>AT$ BY A BRACKET OR BY A BRL
9DICATOR S* Z A CAPITALISA;N 9DICATOR1 A
,GREEK LR 9DICATOR OR A FRAC;N 9DICATOR4
,C>E %D 2 TAK5 :5 A CAPITAL LR PREC$ES A
FUNC;N "N4 ,? IS SUMM>IS$ 2L4
#I4C4A ,IF A FUNC;N "N IS DIRECTLY
PREC$$ OR FOLL[$ BY A NUMB]1 !N ! NUMB]
%D 2 WRITT5 UNSPAC$ F ! FUNC;N "N4
,S9#CJ
#CTAN#DE^J "<#C TAN #DE DEGREES">
#D;COS#EX
#I4C4B ,9S]T A SPACE IF A FUNC;N "N IS
FOLL[$ DIRECTLY BY A L[] CASE ,LAT9 LR )
NO 9T]V5+ BRL 9DICATORS OR BRACKETS4
LOG ;Y
S9.? "<S9 !TA">
,SEC,A
LOG"<X"6Y">
;;,LIM(X./#B) "<,LIM 8;X OV] #B0">
-------------------------------------#DD
#I4C4C ,9S]T A SPACE IF A FUNC;N #FF
#I ,FUNC;NS A#DD
"N IS PREC$$ DIRECTLY BY A L[] OR UPP]
CASE ,LAT9 LR ) NO 9T]V5+ BRL 9DICATORS
OR BRACKETS4 ,NOTE T LRS ISOLAT$ BY ^!
EXTRA SPACES MAY NE$ GRADE #A 9DICATORS4
;X S9#FJ
X,S9#FJ
;,X LOG ;Y
X,LOG ;Y
S9 "<,A PLUS ;,B"> EQUALS S9 ,A COS ;,B
PLUS COS ,A S9 ;,B
S9"<,A"6,B">
"7 S9,A COS,B"6COS,A S9,B
S9 #B 2TA EQUALS #B S9 2TA COS 2TA
S9#B.B "7 #BSIN.BCOS.B
#I4D ,TRIGONOMETRIC FUNC;NS
,COMMON TRIGONOMETRIC FUNC;NS >E ,S9E1
,COS9E1 ,TANG5T1 ,SECANT1 ,COSECANT &
,COTANG5T1 USUALLY A2REVIAT$ 9 PR9T TO
S91 COS1 TAN1 SEC1 COSEC & COT4
,_! 9V]SES MAY 2 WRITT5 9 PR9T Z #FG
#I ,FUNC;NS B#DD
S9 SUP] M9US #A1 COS SUP] M9US #A1 TAN
SUP] M9US #A1 SEC SUP] M9US #A1 COSEC
SUP] M9US #A & COT SUP] M9US #A1 OR LESS
COMMONLY Z >CS91 >CCOS1 >CTAN1 >CSEC1
>CCOSEC & >CCOT4
,Y MAY AL MEET ! ASSOCIAT$ HYP]BOLIC
FUNC;NS S9H1 CO%1 TANH1 SE*1 COSE* & CO?
& _! 9V]SES4
,IF S9 !TA EQUALS O OV] ;H EQUALS #B OV]
#D !N !TA EQUALS S9 TO ! M9US #A (
#J4E EQUALS #CJ DEGREES
,IF ;;;SIN.? "7 (O./H) "7 #B/D;' !N
;;;.? "7 SIN9<"-#A>#J4E "7 #CJ^J;'
-------------------------------------#DE
,PROVE SEC SQU>$ ;X EQUALS #A PLUS TAN
SQU>$ ;X
,PROVE ;;;SEC9#BX "7 #A"6TAN9#BX;'
,F9D ! D]IVATIVE ( ;F ( ;X EQUALS FRAC;N
CO% ;X OV] S9H ;X SQU>$ 5D FRAC;N
,F9D ! D]IVATIVE (
;;;F"<X"> "7 (COSH X./SINH X9#B);'
#FH
#I ,FUNC;NS A#DE
#I4E ,LOG>I?MIC FUNC;NS
,! LOG>I?MIC FUNC;N IS USUALLY WRITT5
LOG OR ,LOG & MAY 2 FOLL[$ BY A
SUBSCRIPT 9DICAT+ ! BASE4 ,A LOG>I?M TO
BASE ;E IS CALL$ A NATURAL LOG & IS (T5
A2REVIAT$ TO LN4
#C LOG ;X
#CLOG ;X
LOG BASE #B ( #H EQUALS #C
;;LOG5#B#H "7 #C
! NATURAL LOG ( ;E EQUALS #A
LN ;E "7 #A
,LOG A PLUS ,LOG ;B EQUALS ,LOG ;AB
;;;,LOG A"6,LOG B "7 ,LOG AB;'
,! 9TEGRAL ( TAN ;X ) RESPECT TO ;X
EQUALS ! NATURAL LOG ( COS ;X PLUS A
3/ANT
;;;!TAN XDX "7 LN COS X"6C;' #FI
#I ,FUNC;NS B#DE
,! LOG BASE ;B ( ;X EQUALS ! LOG BASE A
( ;X OV] ! LOG BASE A ( ;B
;;;LOG5BX "7 (LOG5AX./LOG5AB);'
-------------------------------------#DF
#I4F ,! ,LIMIT FUNC;N
,LIMIT1 LIM1 LM1 LT >E ALL US$ TO
9DICATE LIMIT1 "S"TS ) CAPITALS1 "S"TS
)\T4
,NOTE3 SEE ,SEC;N #AC = ! REPRES5TA;N
( >R[S4
,! LIMIT1 Z ;X T5DS TO A1 ( ;F ( ;X1
EQUALS #A1 ) ;X >R[ A DIRECTLY 2L ,LIM
;;,LIM.5<X\OA>F"<X"> "7 #A
,! LIMIT1 Z !TA T5DS TO #J1 ( S9 !TA OV]
!TA1 EQUALS #A
;;LIMIT.5<.?\O#J>(SIN.?./.?) "7 #A
#I4G ,/ATI/ICAL FUNC;NS
,PROBABIL;Y IS %[N 9 PR9T 9 _M WAYS1
9CLUD+ ;,P1 ,PROB OR ,PR4 ,O!R /ATI/ICAL
FUNC;NS 9CLUDE EXPECTA;N : MAY 2 %[N 9
PR9T Z ;,E OR ,EXP4 #GJ
#I ,FUNC;NS A#DF
PROBABIL;Y ( ,A & ;,B EQUALS PROBABIL;Y
( ,A PLUS PROBABIL;Y ( ;,B
,PR"<,A & ;,B"> "7 ,PR,A"6,PR,B
,! EXPECTA;N ( ;,R EQUALS 8;N OV] #B0
PLUS #A
;;;,EXP"<,R"> "7 (N./#B)"6#A;'
-------------------------------------#DG
#I4H ,COMPLEX NUMB]S
,FUNC;NS US$ 9 COMPLEX NUMB] !ORY
9CLUDE >G "<>GU;T">1 ,RE "<R1L "P">1 ,IM
"<IMAG9>Y "P"> & CIS4
>G ( ;Z SUB #A ;Z SUB #B EQUALS >G ;Z
SUB #A PLUS >G ;Z SUB #B
;;;ARG"<Z5#AZ5#B">
"7 ARG Z5#A"6ARG Z5#B;'
;Z EQUALS ;R CIS !TA EQUALS ;R COS !TA
PLUS IR S9 !TA
;;;Z "7 R CIS.?
"7 R COS.?"6IR SIN.?;'
#GA
#AJ ,SET ,!ORY #DH
#AJ ,SET ,!ORY1 ,GR\P ,!ORY &
,LOGIC
.=.6 UNION "<UP"R ;,U %APE">
.=.8 9T]SEC;N "<9V]T$ ;,U %APE">
.=@J NULL SET "<SLA%$ Z]O">
.=7 COMPLE;T "<PRIME SIGN">
.=^E IS AN ELE;T ( "<V>IANT EPSILON">
.=@^E 3TA9S Z AN ELE;T "<REV]SE V>IANT
EPSILON">
.=^< 3TA9$ IN1 IS A SUBSET ( "<;,U OP5
TO "R">
.=^> 3TA9S1 IS A SUP]SET ( "<;,U OP5 TO
LEFT">
.=_^< 3TA9$ 9 OR EQUAL TO
.=_^> 3TA9S OR EQUAL TO
.=.^< 3TA9$ IN1 B N EQUAL TO "<PROP]
SUBSET">
.=.^> 3TA9S1 B IS N EQUAL TO "<PROP]
SUP]SET">
.=@_< IS A NORMAL SUBGR\P ( "<CLOS$
8LESS ?AN0"> #GB
#AJ ,SET ,!ORY A#DH
.=@_> 9V]SE 8IS NORMAL SUBGR\P0 "<CLOS$
8GRT] ?AN0">
.=__< IS NORMAL SUBGR\P ( OR EQUAL
"<CLOS$ 8LESS ?AN01 L9E "U">
.=__> 9V]SE 8NORMAL SUBGR\P OR EQUAL0
"<CLOS$ 8GRT] ?AN01 L9E "U">
.=._< NORMAL SUBGR\P B N EQUAL "<CLOS$
8LESS ?AN01 C.ELL$ L9E "U">
.=._> 9V]SE 8NORMAL SUBGR\P B N EQUAL0
"<CLOS$ 8GRT] ?AN01 C.ELL$ L9E "U">
.=@6 OR "<UP"R ;V %APE">
.=@8 & "<9V]T$ ;V %APE">
.=@? 8N0 SIGN "<L9E HORIZONTAL1 !N D[N
AT "R">
.=_3 ASS];N "<8IS A !OREM0 SIGN2 8;,T0
LY+ ON LEFT SIDE">
.=@_3 REV]SE ASS];N "<8;,T0 LY+ ON "R
SIDE">
.=^_3 8IS VALID0 SIGN "<ASS];N ) D\BLE
/EM ON 8;,T0">
.=._3 REV]SE 8IS VALID0 SIGN
#GC
#AJ ,SET ,!ORY #DI
,IF ,A EQUALS ! SET #A1 #B1 #C1 #D
& ;,B EQUALS ! SET #B1 #D1 #E1 #H
IS #C AN ELE;T ( ,A 9T]SEC;N ;,B
& IS ,A 9T]SEC;N ;,B A SUBSET ( ,A UNION
;,B8
,IF ,A "7 _<#A1 #B1 #C1 #D_>
& ;,B "7 _<#B1 #D1 #E1 #H_>
IS #C ^E ,A.8,B
& IS ,A.8,B ^< ,A.6,B8
! UNION ( ,A COMPLE;T & ;,B COMPLE;T
EQUALS ! COMPLE;T ( ! 9T]SEC;N ( ,A &
;,B
;;;,A7.6,B7 "7 "<,A.8,B">7;'
,= ! /ATE;TS ;P & ;Q
EI ;P OR ;Q2 & N ;P2 "!=E1 ;Q
,= ! /ATE;TS ;P & ;Q
;;;.<"<P@6Q">@8@?P.> _3 Q;'
#GD
#AA MISCELLANE\S SYMBOLS #EJ
#AA ,MISCELLANE\S ,SYMBOLS
.=! 9TEGRAL SIGN
.=@! CLOS$ L9E 9TEGRAL "<SMALL CIRCLE
HALFWAY UP">
.=@D "PIAL D]IVATIVE "<CURLY ;D">
.=^D DEL1 NABLA "<9V]T$ CAPITAL DELTA">
.=7 PRIME "<:5 4T+UI%$ F APO/ROPHE 9
PR9T">
.=_"7 IS PROPOR;NAL TO "<V>IES Z">
.=@9 TILDE "<SWUNG DA%">
.=@5 C>ET "<HAT">
.="9 A/]ISK
.="0 HOLL[ DOT
.=_\ V]TICAL B>
.=#= 9F9;Y
.=6 FACTORIAL SIGN "<EXCLAMA;N M>K 9
PR9T">
.=_[ ANGLE SIGN
.=._[ M1SUR$ ANGLE SIGN
.=#_[ M1SUR$ "R ANGLE SIGN
.=#L P>ALLEL TO
.=#- P]P5DICUL> TO #GE
#AA MISCELLANE\S SYMBOLS A#EJ
.=,* 8"!=E0 "<?REE DOTS 9 UP"R PYRAMID">
.=@/ 8S9CE0 "<?REE DOTS 9 9V]T$
PYRAMID">
.=^5 8"! EXI/S0 "<REV]SE ;,E">
.=^A 8= ALL0 "<9V]T$ ,A">
.=@A AT SIGN
.=_* BACKSLA%
.=.- "USCORE
.=_? CROSS HAT*
.=@& AMP]S&
.=.\ BROK5 V]TICAL B>
-------------------------------------#EA
.=? F/ TRANSCRIB]-DEF9$ PR9T SYMBOL
.=#? SECOND TRANSCRIB]-DEF9$ PR9T SYMBOL
.=@#? ?IRD TRANSCRIB]-DEF9$ PR9T SYMBOL
.=^#? F\R? TRANSCRIB]-DEF9$ PR9T SYMBOL
.=_#? FIF? TRANSCRIB]-DEF9$ PR9T SYMBOL
.="#? SIX? TRANSCRIB]-DEF9$ PR9T SYMBOL
.=.#? SEV5? TRANSCRIB]-DEF9$ PR9T SYMBOL
#AA4A ,SPAC+
,9 G5]AL1 ! SPAC+ ( SYMBOLS C FOLL[
PR9T4 ,H["E IF A SYMBOL IS CLE>LY 2+ US$
Z A SIGN ( OP]A;N OR COMP>ISON1 #GF
#AA MISCELLANE\S SYMBOLS A#EA
FOLL[ ! GUIDEL9ES 9 ,SEC;N #A4A4
#AA4B ,UNUSUAL ,PR9T SYMBOLS
,IF A PR9T SYMBOL IS N DEF9$ 9 ,,UEB1
X C 2 REPRES5T$ EI US+ "O ( ! SEV5
TRANSCRIB]-DEF9$ PR9T SYMBOLS ABV1 OR BY
US+ ! TRANSCRIB]-DEF9$ %APE SYMBOLS 9
,SEC;N #AD4 "<,SEE EXAMPLE #AA4E4G 2L">
#AA4C ,GRADE #A 9DICATORS
,IF ! BRL V].N ( A PR9T SYMBOL AL HAS
A GRADE #B M1N+1 & GRADE #A MODE IS N
ALR 9 =CE1 !N GRADE #A 9DICATORS W 2
NE$$4 ,SYMBOLS 9 ! LI/ ABV = : ? APPLIES
>E ! 9TEGRAL SIGN1 ! PRIME SIGN & ! "!=E
SIGN4
#AA4D ,SYMBOLS : H M ?AN "O M1N+ 9
PR9T
,"O ( ! "ULY+ DESIGN F1TURES ( ,,UEB
IS T EA* PR9T SYMBOL %D H "O & ONLY "O
BRL EQUIVAL5T4 ,= EXAMPLE ! V]TICAL B>
IS US$ 9 PR9T TO REPRES5T ABSOLUTE
VALUE1 3DI;NAL PROBABIL;Y & ! ^WS #GG
#AA MISCELLANE\S SYMBOLS B#EA
8S* T01 TO GIVE J ?REE EXAMPLES4 ,! SAME
BRL SYMBOL %D 2 US$ 9 ALL ^! CASES4
-------------------------------------#EB
#AA4E ,EXAMPLES
#AA4E4A
,IF ;Y "7 F"<X"> !N ! D]IVATIVE IS DY
OV] DX OR ;F DA% ;X & ! "PIAL D]IVATIVE
IS CURLY ;D ;Y OV] CURLY ;D ;X
,IF ;Y "7 F"<X"> !N ! D]IVATIVE IS
;;(DY./DX) OR ;;F7"<X"> & ! "PIAL
D]IVATIVE IS ;;(@DY./@DX)4
#GH
#AA MISCELLANE\S SYMBOLS A#EB
#AA4E4B
! 9TEGRAL F #B TO #C ( "<#BX PLUS #A">
DX
EQUALS .<;X SQU>$ PLUS ;X.> SUB #B SUP]
#C
EQUALS "<#C SQU>$ PLUS #C"> M9US "<#B
SQU>$ PLUS #B">
EQUALS #AB M9US #F EQUALS #F
;;;!5#B9#C"<#BX"6#A">DX
"7 .<X9#B"6X.>5#B9#C
"7 "<#C9#B"6#C">"-"<#B9#B"6#B">
"7 #AB"-#F "7 #F;'
,NOTE3 ! SPAC+ ( ! 9TEGRAL SIGN 9 PR9T
C 2 UNCLE> OR 9CONSI/5T4 ,9 BRL X IS BE/
TO H ! 9TEGRAL SIGN UNSPAC$ F ! FUNC;N &
TR1T XS LIMITS Z SUBSCRIPTS &
SUP]SCRIPTS4 ,! DX AT ! 5D M1NS
89TEGRATE ) RESPECT TO ;X01 & C AL 2
WRITT5 UNSPAC$4
#GI
#AA MISCELLANE\S SYMBOLS #EC
#AA4E4C
"<SUP] ;N CAPITAL ;,C SUB ;R EQUALS
5L>G$ BRACKETS 5CLOS+ ;N AT ! TOP & ;R
AT ! BOTTOM
EQUALS FRAC;N ;N FACTORIAL OV] ;R
FACTORIAL "TS "<;N M9US ;R"> FACTORIAL
5D FRAC;N">
;;;9N,C5R "7 "<N]R">
"7 (N6./R6"<N"-R">6);'
,NOTE3 ! B9OMIAL COE6ICI5T "WS BETT] Z
A %APE ?AN A VECTOR "<REF] TO #AD4C4C">4
#AA4E4D
,A/]ISK IS 4TRIBUTIVE OV] HOLL[ DOT IF
A A/]ISK "<;B HOLL[ DOT ;C"> EQUALS "<A
A/]ISK ;B"> HOLL[ DOT "<A A/]ISK ;C">
"9 IS 4TRIBUTIVE OV] "0 IF
A"9"<B"0C"> "7 "<A"9B">"0"<A"9C">
,NOTE3 ! HOLL[ DOT %D N 2 US$ TO #HJ
#AA MISCELLANE\S SYMBOLS A#EC
REPRES5T ! A2REVIA;N = DEGREES1 : IS
COV]$ 9 ,SEC;N #C4
-------------------------------------#ED
#AA4E4E
,IF ;F F ;,X TO ;,Y IS A FUNC;N !N !
RELA;N ;F SUP] M9US #A F ;,Y TO ;,X IS
XF A FUNC;N IF & ONLY IF = ALL ;Y 9 ;,Y
"! EXI/S ;X 9 ;,X S* T ;F ( ;X EQUALS ;Y
,IF ;;;F3 ,X \O ,Y;' IS A FUNC;N !N !
RELA;N ;;;F9<"-#A>3 ,Y \O ,X;' IS XF A
FUNC;N IF & ONLY IF
;;;^AY ^E ,Y ^5X ^E ,X;' S* T
;;;F"<X"> "7 Y;'
,NOTE3 SEE ,SEC;N #AC = ! REPRES5TA;N
( >R[S4
#AA4E4F
,! SET ( "<;X1 ;Y"> S* T ;X PLUS ;Y
EQUALS #F
;;;_<"<X1 Y"> _\ X"6Y "7 #F_>;'
#HA
#AA MISCELLANE\S SYMBOLS A#ED
#AA4E4G
,BABYLONIAN NUM]ALS USE TWO SYMBOLS1
;? M1NS #A & #? M1NS #AJ4
,NOTE3 ! TWO TRANSCRIB] DEF9$ SYMBOLS
WD 2 DEF9$ EI ON ! SPECIAL SYMBOLS PAGE
OR 9 A TRANSCRIB]'S NOTE4
-------------------------------------#EE
#AA4F ,EMBELLI%$ CAPITAL LRS
,EMBELLI%$ CAPITAL LRS >E (T5 US$ TO
"N COMMON SETS S* Z ! UNIV]SAL SET ;,E1
! SET ( R1L NUMB]S ;,R OR 9TEG]S ,I4 ,^!
V>Y 9 PR9T F BOOK TO BOOK B C 2
REPRES5T$ 9 BRL BY ! SCRIPT TYPE=M
9DICATORS4
,! SET ( R1L NUMB]S @2;,R
-------------------------------------#EF
#AA4G ,GREEK LRS
,GREEK LRS >E US$ H1VILY 9 ,MA!MATICS4
,! ALPHABET IS LI/$ 2L4 ,REF] AL TO !
,RULE ON ,LRS & _! MODIFI]S4
#HB
#AA MISCELLANE\S SYMBOLS A#EF
.=.A ,GREEK ALPHA
.=.B ,GREEK 2TA
.=.G ,GREEK GAMMA
.=.D ,GREEK DELTA
.=.E ,GREEK EPSILON
.=.Z ,GREEK ZETA
.=.: ,GREEK ETA
.=.? ,GREEK !TA
.=.I ,GREEK IOTA
.=.K ,GREEK KAPPA
.=.L ,GREEK LAMBDA
.=.M ,GREEK MU
.=.N ,GREEK NU
.=.X ,GREEK XI
.=.O ,GREEK OMICRON
.=.P ,GREEK PI
.=.R ,GREEK RHO
.=.S OR .S ,GREEK SIGMA
.=.T ,GREEK TAU
.=.U ,GREEK UPSILON
.=.F ,GREEK PHI
.=.& ,GREEK *I
.=.Y ,GREEK PSI
.=.W ,GREEK OMEGA #HC
#AA MISCELLANE\S SYMBOLS B#EF
.=,.A CAPITAL ,GREEK ALPHA
.=,.B CAPITAL ,GREEK 2TA
.=,.G CAPITAL ,GREEK GAMMA
.=,.D CAPITAL ,GREEK DELTA
.=,.E CAPITAL ,GREEK EPSILON
.=,.Z CAPITAL ,GREEK ZETA
.=,.: CAPITAL ,GREEK ETA
.=,.? CAPITAL ,GREEK !TA
.=,.I CAPITAL ,GREEK IOTA
.=,.K CAPITAL ,GREEK KAPPA
.=,.L CAPITAL ,GREEK LAMBDA
.=,.M CAPITAL ,GREEK MU
.=,.N CAPITAL ,GREEK NU
.=,.X CAPITAL ,GREEK XI
.=,.O CAPITAL ,GREEK OMICRON
.=,.P CAPITAL ,GREEK PI
.=,.R CAPITAL ,GREEK RHO
.=,.S CAPITAL ,GREEK SIGMA
.=,.T CAPITAL ,GREEK TAU
.=,.U CAPITAL ,GREEK UPSILON
.=,.F CAPITAL ,GREEK PHI
.=,.& CAPITAL ,GREEK *I
.=,.Y CAPITAL ,GREEK PSI
.=,.W CAPITAL ,GREEK OMEGA #HD
#AB ,B>S & DOTS OV] & "U #EG
#AB ,B>S & DOTS ETC4 OV] & "U
.=: B> OV] PREVI\S ITEM
.=,: B> "U PREVI\S ITEM
.=@: L9E "? PREVI\S ITEM "<C.ELLA;N1
8N0">
.=^: SIMPLE "R PO9T+ >R[ OV] PREVI\S
ITEM
.=,^: SIMPLE "R PO9T+ >R[ "U PREVI\S
ITEM
.=^4 DOT OV] PREVI\S ITEM
.=,^4 DOT "U PREVI\S ITEM
.=_: TILDE OV] PREVI\S ITEM
.=,_: TILDE "U PREVI\S ITEM
.=": HAT OV] PREVI\S ITEM
.=,": HAT "U PREVI\S ITEM
.=._: >C OV] PREVI\S ITEM
#AB4A ,! DEF9I;N ( AN ITEM
,! DEF9I;N ( AN ITEM 2L IS ! SAME Z T
GIV5 = SUP]SCRIPTS & SUBSCRIPTS 9 ,SEC;N
#G4A4
#HE
#AB ,B>S & DOTS OV] & "U A#EG
,Z 9 ,SEC;N #G1 AN ITEM IS DEF9$ Z ANY
( ! FOLL[+ GR\P+S3
#A4 ,AN 5TIRE NUMB]1 I4E4 ! 9ITIAT+
NUM]IC SYMBOL & ALL SU3E$+ SYMBOLS )9
! NUM]IC MODE ?US E/ABLI%$ "<: WD
9CLUDE ANY 9T]IOR DECIMAL PO9TS1
COMMAS1 SEP>ATOR SPACES1 OR SIMPLE
NUM]IC FRAC;N L9ES">4
#B4 ,AN 5TIRE G5]AL FRAC;N1 5CLOS$ 9
FRAC;N 9DICATORS "<SEE ,SEC;N #F">4
#C4 ,AN 5TIRE RADICAL EXPRES.N1 5CLOS$ 9
RADICAL 9DICATORS "<SEE ,SEC;N #H">4
#D4 ,AN >R[ "<SEE ,SEC;N #AC">4
#E4 ,AN >BITR>Y %APE "<SEE ,SEC;N #AD">4
#F4 ,ANY EXPRES.N 5CLOS$ 9 MAT*+ PAIRS (
R.D P>5!SES1 SQU>E BRACKETS OR CURLY
BRACES "<SEE ,SEC;N #E">4
#G4 ,ANY EXPRES.N 5CLOS$ 9 ! BRL GR\P+
9DICATORS4
#H4 ,IF N"O ( ! =EGO+ APPLY1 ! ITEM IS
SIMPLY ! PREVI\S 9DIVIDUAL SYMBOL4
#HF
#AB ,B>S & DOTS OV] & "U #EH
,EXAMPLES3
;X B> EQUALS #AJ PLUS #AA PLUS #AB ALL
OV] #C ": ;X B> IS ! >I?METIC M1N4
;;;X: "7 (#AJ"6#AA"6#AB./#C);' ": X;: IS
! >I?METIC M1N4
,NOTE3 ! SECOND O3UR;E ( ;X B> NE$S A
GRADE #A SYMBOL 9DICATOR 2C X IS )9 A
GRADE #B PASSAGE4
;X PLUS ;Y ALL ) A B> "U
22<X"6Y>,:
EQUALS SIGN ) A L9E "? X "<N EQUALS">
"7@:
#J4C ) A DOT OV] ! #C,-! RECURR+ DECIMAL
#J4CCCCC444
#J4<#C>^4
,NOTE3 BRL GR\P+ SIGNS >E NE$$ "H
O!RWISE ! DOT WD REF] TO ! 5TIRE NUMB]4
#HG
#AB ,B>S & DOTS OV] & "U A#EH
,! RECURR+ DECIMAL #J4EFABCABCABC 444
#J4EF<#A>^4#B<#C>^4
,2C ! DIGITS #ABC >E 2+ IMPLICITLY GR\P$
9 PR9T ? C AL 2 WRITT5 Z3
#J4EF<#ABC>^4
D]IVATIVES ;X DOT & ;X D\BLE DOT
D]IVATIVES X^4 & ;;X.9<44>
,ANGLE ,,ABC ) A HAT OR C>ET OV] ! ;,B
,ANGLE ;;,A,B":,C
-------------------------------------#EI
#AB4B ,TWO 9DICATORS APPLI$ TO !
SAME ITEM
,IF TWO 9DICATORS APPLY TO ! SAME
ITEM1 !N BRL GR\P+ SYMBOLS M/ 2 US$ TO
%[ : APPLIES F/4
;X TO ! 8;Y B>0
;;X9<Y:>
;X TO ! ;Y ) A B> OV] ! :OLE EXPRES.N
;;<X9Y>: #HH
#AC ,>R[S #FJ
#AC ,>R[S
#AC4A ,SIMPLE >R[S
.=\ >R[ 9DICATOR
.=^\ BOLD >R[ 9DICATOR
.=\O SIMPLE "R PO9T+ >R[ "<EA/">
.=\[ SIMPLE LEFT PO9T+ >R[ "<WE/">
.=\+ SIMPLE UP PO9T+ >R[ "<NOR?">
.=\% SIMPLE D[N PO9T+ >R[ "<S\?">
.=\S SIMPLE UP & "R PO9T+ >R[ "<NOR?1/">
.=\< SIMPLE D[N & "R PO9T+ >R[ "<S\?1/">
.=\: SIMPLE UP & LEFT PO9T+ >R[
"<NOR?WE/">
.=\> SIMPLE D[N & LEFT PO9T+ >R[
"<S\?WE/">
,A SIMPLE >R[ HAS A /&>D B>B$ TIP AT
"O 5D "<L A ;V ON XS SIDE1 PO9T+ AWAY F
! %AFT">4 ,! %AFT IS /RAI<T & XS L5G? &
?ICK;S >E N SIGNIFICANT4 ,^! >R[S >E
REPRES5T$ BY AN OP5+ >R[ 9DICATOR & !
APPROPRIATE CLOS+ >R[ 9DICATOR4 ,NOTICE
T ALL ^! T]M9AT+ SYMBOLS H ?REE #HI
#AC ,>R[S A#FJ
DOTS1 >RANG$ 9 A 3SI/5T PATT]N T BE/
DESCRIBES ! DIREC;N4
,NOTE T UN.S Y >E ALR 9 GRADE #A MODE1
! >R[ 9DICATOR W NE$ A GRADE #A SYMBOL
9DICATOR4 ,! BOLD >R[ 9DICATOR W N NE$
"O1 Z ? TWO CELL SYMBOL DOES N H A GRADE
#B M1N+4 ,BO? >R[ 9DICATORS SET >R[ MODE
S NO FUR!R GRADE #A 9DICATORS W 2 NE$$4
,>R[S >E SIGNS ( COMP>ISON S %D
USUALLY 2 SPAC$4 ,AN EXCEP;N IS :5 !Y >E
WRITT5 2L ! LIMIT FUNC;N "<SEE ,SEC;N
#I4F">4
,D N USE >R[ 9DICATORS :5 A SIMPLE "R
PO9T+ >R[ IS ! ONLY MODIFI] ABV OR 2L AN
ITEM4 ,SEE 8>R[ OV] PREVI\S ITEM0 & 8>R[
2L PREVI\S ITEM0 9 ,SEC;N #AB4
-------------------------------------#FA
;N "R >R[ #J,-;N T5DS TO Z]O
;;;N \O #J;'
9PUT 8"R >R[0 PROCESS 8BOLD "R >R[0
\TPUT
9PUT ;\O PROCESS ^\O \TPUT #IJ
#AC ,>R[S A#FA
#AC4B ,>R[S ) UNUSUAL %AFTS & A /&>D
B>B$ TIP
,%AFT SYMBOLS3
.=3 %ORT S+LE /RAI<T L9E
.=33 M$IUM S+LE /RAI<T L9E
.=333 L;G S+LE /RAI<T L9E
.=7 D\BLE1 %ORT
.=111 DOTT$1 L;G
.=9 CURV$ OR B5T TO ! LEFT
"<ANTICLOCKWISE 9 L9E ( DIREC;N">
.=5 CURV$ OR B5T TO ! "R "<CLOCKWISE 9
L9E ( DIREC;N">
.=4 %>P TURN TO ! "R "<9 L9E ( DIREC;N">
.=0 %>P TURN TO ! LEFT "<9 L9E (
DIREC;N">
,ALL %AFT SYMBOLS C 2 EL;GAT$ BY
REPETI;N4 ,! %AFT SYMBOLS >E PLAC$ 2T !
OP5+ & CLOS+ >R[ 9DICATORS4 ,>R[ L5G?
ONLY NE$S TO 2 9DICAT$ 9 BRL :5 9 PR9T
>R[S ( DI6]5T L5G?S H DI6]5T M1N+S4
,^! EXAMPLES / H /&>D B>B$ TIPS4 #IA
#AC ,>R[S B#FA
D\BLE %AFT$ M$IUM L5G? "R PO9T+ >R[
;\77O
M$IUM >R[ PO9T+ UP ) A %>P TURN TO ! "R
;\44+
M$IUM L5G? "R PO9T+ >R[ B5D+ CLOCKWISE
;\55O
-------------------------------------#FB
L;G1 BROK5 LEFT PO9T+ >R[
;\111[
%ORT1 M$IUM & L;G D[N PO9T+ SIMPLE >R[S
;\3% ;\% ;\333%
#AC4C ,>R[S ) UNUSUAL TIPS
,B>B SYMBOLS3
.=R REGUL> B>B1 FULL1 9 L9E ( DIREC;N
.=W REGUL> B>B1 FULL1 C.T] TO L9E (
DIREC;N
.=@W REGUL> B>B1 UPP] HALF1 C.T]
.=,W REGUL> B>B1 L[] HALF1 C.T] #IB
#AC ,>R[S A#FB
.=@R REGUL> B>B1 UPP] HALF1 9 L9E
.=,R REGUL> B>B1 L[] HALF1 9 L9E
.=& CURV$1 FULL1 C.T]
.=Y CURV$1 FULL1 9 L9E
.=@& CURV$1 UPP] HALF1 C.T]
.=,& CURV$1 L[] HALF1 C.T]
.=@Y CURV$1 UPP] HALF1 9 L9E
.=,Y CURV$1 L[] HALF1 9 L9E
.=\ /RAI<T1 FULL1 "<DIREC;N.S">
.=@\ /RAI<T1 UPP] HALF1 "<DIREC;N.S">
.=,\ /RAI<T1 L[] HALF1 "<DIREC;N.S">
,IF AN >R[ HAS UNUSUAL TIPS1 DECIDE :
IS ! H1D 2F Y *OOSE ! DIREC;N ( YR CLOS+
9DICATOR4 ,! COMPLETE RULES = DECID+ >R[
DIREC;N >E3
#A4 ,IF "! >E DIREC;NAL TIPS1 & ALL L1D
9 ! SAME DIREC;N1 ! H1D IS ! 5D T LIES
9 T DIREC;N4
#B4 ,IF "! >E NO DIREC;NAL TIPS1 B "O 5D
HAS A TIP & ! O!R DOES N1 ! 5D ) ! TIP
IS ! H1D4
#C4 ,9 ALL O!R CASES1 ! H1D ( ! #IC
#AC ,>R[S B#FB
>R[ IS DEEM$ TO 2 ! 5D AT ! "R1 OR 9 !
CASE ( /RICTLY V]TICAL >R[S1 AT ! TOP4
-------------------------------------#FC
,! TIP"<S"> & %AFT SEG;T"<S"> >E
TRANSCRIB$ 2T ! OP5+ & CLOS+ 9DICATORS4
,^! ITEMS >E EXPRESS$ 9 LOGICAL ORD]1 T
IS />T+ ) ! >R[ TAIL & PROGRESS+ T[>DS !
H1D1 EV5 IF T RUNS C.T] TO ! PHYSICAL
ORD] "<Z 9 ! CASE ( A LEFT PO9T+ >R[">4
,C]TA9 ELE;TS >E OMITT$1 CORRESPOND+ TO
^! R1D] RULES3
#A4 ,IF NO TIP IS TRANSCRIB$1 X IS
"U/OOD T AN ORD9>Y FULL B>B$ TIP O3URS
AT ! >R[ H1D1 & "! IS NO O!R TIP4
#B4 ,IF NO %AFT IS TRANSCRIB$1 X IS
"U/OOD T ! %AFT IS A /RAI<T L9E (
M$IUM L5G?4 ,9 ? CASE1 IF NO TIP IS
TRANSCRIB$1 RULE "<#A"> AL APPLIES2 IF
"O TIP IS TRANSCRIB$1 X IS AT ! H1D2
IF TWO TIPS >E TRANSCRIB$1 ! F/ IS AT
! TAIL & ! SECOND AT ! H1D4
#ID
#AC ,>R[S A#FC
O!RWISE ORD9>Y "R >R[1 ) CURV$ H1D
;\YO
COMMON HORIZONTAL BIDIREC;NAL >R[
;\WRO
HORIZONTAL BIDIREC;NAL >R[1 TILT$ F L[]
LEFT TO UPP] "R
;\WRS
O!RWISE ORD9>Y "R >R[1 ) TAIL & H1D TIPS
;\RRO
O!RWISE ORD9>Y "R >R[1 ) TWO H1D TIPS
;\33RRO
O!RWISE ORD9>Y LEFT >R[1 ) TAIL & H1D
TIPS
;\RR[
O!RWISE ORD9>Y LEFT >R[1 ) TWO H1D TIPS
;\33RR[
#IE
#AC ,>R[S #FD
"R >R[ ) A /RAI<T TAIL TIP & A NORMAL
H1D TIP
;\\RO
BOLD >R[ UP1 FOLL[$ BY ORD9>Y >R[ D[N
^\+\%
COMMON "R >R[ OV] COMMON LEFT >R[,-SEE
,SEC;N #AD4C4C REG>D+ V]TICAL
JUXTAPOSI;N
;\O]\[
HALF B>B$ LEFT >R[ OV] HALF B>B$ "R
>R[,-REV]SIBLE *EMICAL REAC;N
;\@R[]\,RO
,SEE ,SEC;N #AF = EQUILIBRIUM >R[S T
O3UR 9 ,*EMI/RY4
#IF
#AD ,%APE ,SYMBOLS #FE
#AD ,%APE ,SYMBOLS & ,COMPOSITE
,SYMBOLS
,LI/+ ( %APE 9DICATORS3
.=$ %APE 9DICATOR
.=_$ FILL$ "<SOLID"> %APE 9DICATOR
.=.$ %AD$ %APE 9DICATOR
.=@$ TRANSCRIB]-ASSIGN$ %APE 9DICATOR
.=@_$ TRANSCRIB]-ASSIGN$ FILL$ "<SOLID">
%APE 9DICATOR
.=@.$ TRANSCRIB]-ASSIGN$ %AD$ %APE
9DICATOR
.=: %APE T]M9ATOR
,LI/+ ( SPECIFIC %APES3
.=$#C REGUL> "<EQUILAT]AL"> TRIANGLE
.=$#D SQU>E
.=$#E REGUL> P5TAGON
.=$#F REGUL> HEXAGON
.=$#G REGUL> HEPTAGON
.=$#H REGUL> OCTAGON "<ETC4 = ALL REGUL>
POLYGONS">
.=$= CIRCLE #IG
#AD ,%APE ,SYMBOLS A#FE
.=$@#D P>ALLELOGRAM
,COMPOSITE ,SYMBOLS3
.=& SUP]POSI;N 9DICATOR
.== HORIZONTAL JUXTAPOSI;N 9DICATOR
.=] V]TICAL JUXTAPOSI;N 9DICATOR
.=[ PHYSICAL 5CLOSURE 9DICATOR
-------------------------------------#FF
#AD4A ,USE ( ! %APE T]M9A;N 9DICATOR
#AD4A4A ,IF A %APE IS FOLL[$ BY A
SPACE !N NO T]M9A;N SYMBOL IS NE$$4
TRIANGLE SYMBOL SPACE ,,ABC
;$#C ,,ABC
#AD4A4B ,IF ! %APE SYMBOL IS FOLL[$ BY
PUNCTUA;N1 OR UNSPAC$ F A FOLL[+ SYMBOL1
!N ! %APE T]M9ATOR M/ 2 US$4
TRIANGLE ,,ABC UNSPAC$
;$#C:,,ABC
#IH
#AD ,%APE ,SYMBOLS A#FF
,:AT IS ! NEXT %APE8
OP5 BRACE SQU>E1 %AD$ CIRCLE1 SOLID
TRIANGLE1 %AD$ SQU>E 444 CLOSE BRACE
,:AT IS ! NEXT %APE8
_<;$#D:1 .$=:1 _$#C:1 .$#D 444_>
,NOTE T UN.S Y >E ALR 9 GRADE #A MODE1
A GRADE #A SYMBOL 9DICATOR W 2 NE$$ 2F !
%APE 9DICATOR4 ,? DOES N H["E APPLY TO !
%AD$ & FILL$ %APE 9DICATORS 2C ^! TWO
CELL SYMBOLS D N H A GRADE #B M1N+4 ,ALL
! 9ITIAL %APE 9DICATORS 9ITIATE %APE
MODE S NO FUR!R GRADE #A 9DICATORS W 2
NE$$4
#AD4B ,TRANSCRIB] DEF9$ %APES
,! DESCRIP;N )9 TRANSCRIB] DEF9$ %APES
%D 2 A %ORT S]IES ( 9ITIALS OR A S+LE
GRADE #A ^W4 ,!Y %D N 2 US$ IF ! PR9T
SYMBOL IS ALR COV]$ ELSE": 9 ! CODE4 ,!
DEF9I;NS ( ALL %APE SYMBOLS %D 2
AVAILABLE TO ! R1D] 9 EI A TRANSCRIB]'S
NOTE OR ON A SPECIAL SYMBOLS PAGE4 #II
#AD ,%APE ,SYMBOLS B#FF
,= EXAMPLE1 A SMIL+ FACE US$ Z AN ICON
"?\T A BOOK CD 2 DEF9$ Z
@$SF OR @$SMILE
R ?AN
@$<SMILING FACE>
: IS TOO L;G & REQUIRES BRL GR\P+
SYMBOLS TO /OP ! SPACE T]M9AT+ ! %APE4
-------------------------------------#FG
#AD4C ,COMB9$ %APES
,IF TWO PR9T SYMBOLS H BE5 COMB9$ TO
=M A NEW PREVI\SLY UNDEF9$ SYMBOL1 !N X
M/ 2 DECID$ :E!R ! SECOND SYMBOL IS
5CLOS$1 SUP]IMPOS$1 COMB9$ ON ! "R OR
COMB9$ 2L4 ,EA* ( ! F\R COMPOSITE SYMBOL
9DICATORS SIGNALS A COMB9+ ( ! ITEM J
PRIOR ) ! ITEM IMMLY FOLL[+ X1 ": 8ITEM0
IS Z DEF9$ 9 ,SEC;N #G4
,EA* COMPOSITE SYMBOL 9DICATOR W NE$ A
GRADE #A SYMBOL 9DICATOR UN.S ! :OLE
EXPRES.N IS ALR 9 GRADE #A MODE4
#AD4C4A ,PHYSICAL ,5CLOSURE1 ;[
CIRCLE 5CLOS+ A PLUS SIGN
;$=["6 #AJJ
#AD ,%APE ,SYMBOLS A#FG
,9 ! EXAMPLE 2L1 ! CIRCLE 5CLOS+ A
PLUS SIGN IS 2+ US$ Z AN OP]A;N SIGN4 ,9
! F/ V].N ! OP]A;N SIGN IS UNSPAC$ S A
T]M9A;N SIGN IS NE$$ B NO GRADE
9DICATORS >E NE$$ 2C ! NUMB] SIGNS
9ITIATE NUM]IC MODE4 ,9 ! SECOND V].N !
OP]A;N SIGN IS SPAC$ = CL>;Y S NO
T]M9A;N SIGN IS NE$$ B ! %APE SYMBOLS D
NE$ GRADE #A 9DICATORS4
#B 8CIRCL$ PLUS0 #C EQUALS #C 8CIRCL$
PLUS0 #B
#B$=["6:#C "7 #C$=["6:#B
#B 8CIRCL$ PLUS0 #C EQUALS #C 8CIRCL$
PLUS0 #B
#B ;$=["6 #C "7 #C ;$=["6 #B
,NOTE3 ,SEE ,SEC;N #C4E IF NUMB]S >E
5CLOS$ 9 A SQU>E TO REPRES5T A
CALCULATOR KEY4
#AJA
#AD ,%APE ,SYMBOLS #FH
#AD4C4B ,SUP]POSI;N1 ;&
,NOTE T ? /RUCTURE %D N 2 US$ =
NEGA;N4 ,SEE 8L9E "? PREVI\S ITEM0 9
,SEC;N #AB4
;,R ) SUP]IMPOS$ ;X,-PRESCRIP;N SYMBOL
,R;&X
9TEGRAL SIGN ) A SMALL CIRCLE SUP]IMPOS$
HALF WAY UP,-CLOS$ 9TEGRAL DEF9$ 9
,SEC;N #AA4A
@!
9TEGRAL SIGN ) A SMALL SQU>E SUP]IMPOS$
HALF WAY UP,-! T]M9A;N CD 2 OMITT$ IF
"! 0 A FOLL[+ SPACE
;;!&$#D:
#AD4C4C ,V]TICAL ,JUXTAPOSI;N1 ;]
,! UPP] SYMBOL %D 2 GIV5 F/1 FOLL[$ BY
! V]TICAL JUXTAPOSI;N 9DICATOR1 !N ! L[]
SYMBOL4
,? /RUCTURE %D N 2 US$ = B>S1 #AJB
#AD ,%APE ,SYMBOLS A#FH
>R[S1 DOTS1 TILDES OR HATS OV] OR "U O!R
SYMBOLS "<SEE ,SEC;N #AB">4 ,NEI %D X 2
US$ = SUP]SCRIPTS OR SUBSCRIPTS WRITT5
DIRECTLY OV] OR "U "<SEE ,SEC;N #G">4
A HOLL[ DOT ) AN EQUAL SIGN "UN1?
"0;]"7
B9OMIAL COE6ICI5T "<REF] ,SEC;N
#AA4E4C">
"<N;]R">
#AD4C4D ,HORIZONTAL ,JUXTAPOSI;N1 ;=
8,HORIZONTAL JUXTAPOSI;N0 IS TO 2
9VOK$ ONLY :5 TWO SYMBOLS >E WRITT5 9
CLOSE PROXIM;Y & X IS CLE> F ! USAGE T A
NEW S+LE SYMBOL1 4T9CT F ! ELE;T>Y
SYMBOLS 3SID]$ 9 SEQU;E1 IS 9T5D$4
,O!RWISE1 SYMBOLS WRITT5 "O AF ! O!R %D
SIMPLY 2 BRLD ACLY4
#AJC
#AE ,MATRICES & VECTORS #FI
#AE ,MATRICES & VECTORS
#AE4A ,5L>G$ GR\P+ SYMBOLS
,:5 5L>G$ BRACKETS >E US$ 9 PR9T =
VECTORS1 MATRICES1 SY/EMS ( EQUA;NS1
FUNC;N DEF9I;NS ETC41 ! APPROPRIATE
5L>G$ GR\P+ SYMBOLS %D 2 US$ 9 BRL4 ,^!
>E ! USUAL GR\P+ SYMBOLS PREC$$ BY A DOT
#F4 ,SEE ! FULL LI/ 9 ,SEC;N #E4 ,^! %D
2 PLAC$ DIRECTLY "U EA* O!R4 ,BLANK L9ES
2F & AF S* >RANGE;TS MAY 2 NE$$ = CL>;Y4
#AE4B ,MATRICES
,! COLUMNS %D 2 LEFT ADJU/$ EXCEPT =
M9US SIGNS : %D 2 BRLD TO /& \4 ,"O
COLUMN ( BLANK CELLS %D 2 LEFT 2T
COLUMNS4 ,MAT]IAL \TSIDE ! MATRIX1 S* Z
SIGNS ( OP]A;N & COMP>ISON1 %D 2 PLAC$
ON ! TOP L9E1 EV5 IF !Y >E C5T]$ 9 PR9T4
,I "7 ,"<#A #J,">
,"<#J #A,">
#AJD
#AE ,MATRICES & VECTORS A#FI
,.<#A #B #C,.>,.< #A #B,.>
,.<#D #E #F,.>,.<"-#C #D,.>
,.< #E "-#F,.>
,"< A "-B,">
,"<"-C D,">
-------------------------------------#GJ
#AE4C ,DET]M9ANTS
,^! H ! SAME /RUCTURE Z MATRICES B >E
NORMALLY 5CLOS$ 9 PR9T ) 5L>G$ V]TICAL
B>S4
;;;_\,P_\ "7 ,_\A B_\ "7 AD"-BC;'
,_\C D_\
#AE4D ,OMIS.N DOTS
,! PLACE;T ( DOTS US$ TO 9DICATE !
OMIS.N ( "O OR M R[S OR COLUMNS C FOLL[
! PR9T4
#AJE
#AE ,MATRICES & VECTORS A#GJ
;;;,_\A5#AA A5#AB 444 A5<#AN>,_\
,_\A5#BA A5#BB 444 A5<#BN>,_\
,_\ 4 4 444 4 ,_\
,_\A5<M#A> A5<M#B> 444 A5<MN> ,_\;'
-------------------------------------#GA
#AE4E ,D1L+ ) WIDE MATRICES
,IF A MATRIX OR DET]M9ANT IS TOO WIDE
= ! BRL PAGE1 RUNOV]S )9 5TRIES MAY 2
NEC4 ,IF "! IS N ROOM TO 9D5T ^!
RUNOV]S1 !Y C 2 BLOCK$ & A BLANK L9E
LEFT 2T R[S4
#AJF
#AE ,MATRICES & VECTORS A#GA
""=;;;
,"<A5#AX5#A A5#AY5#A A5#AZ5#A ,">
,"<"6B5#AX5#B "6B5#AY5#B "6B5#AZ5#B,">
,"<"6C5#AX5#C "6C5#AY5#C "6C5#AZ5#C,">
,"< ,">
,"<A5#BX5#A A5#BY5#A A5#BZ5#A ,">
,"<"6B5#BX5#B "6B5#BY5#B "6B5#BZ5#B,">
,"<"6C5#BX5#C "6C5#BY5#C "6C5#BZ5#C,">
,"< ,">
,"<A5#CX5#A A5#CY5#A A5#CZ5#A ,">
,"<"6B5#CX5#B "6B5#CY5#B "6B5#CZ5#B,">
,"<"6C5#CX5#C "6C5#CY5#C "6C5#CZ5#C,">
""=;'
,ANO!R APPROA* IS TO COMPLETE ! F/
COLUMN )\T OV]RUNS & !N TO PLACE ! NEXT
COLUMN 2L ?1 9D5T$ TWO CELLS4
#AJG
#AE ,MATRICES & VECTORS B#GA
""=;;;
,"<A5#AX5#A"6B5#AX5#B"6C5#AX5#C
,"<A5#BX5#A"6B5#BX5#B"6C5#BX5#C
,"<A5#CX5#A"6B5#CX5#B"6C5#CX5#C
A5#AY5#A"6B5#AY5#B"6C5#AY5#C
A5#BY5#A"6B5#BY5#B"6C5#BY5#C
A5#CY5#A"6B5#CY5#B"6C5#CY5#C
A5#AZ5#A"6B5#AZ5#B"6C5#AZ5#C,">
A5#BZ5#A"6B5#BZ5#B"6C5#BZ5#C,">
A5#CZ5#A"6B5#CZ5#B"6C5#CZ5#C,">
""=;'
,NOTICE T 9 ! F/ EXAMPLE ! /RUCTURE (
! MATRIX IS CLE>] B 9 ! SECOND EXAMPLE !
9DIVIDUAL 5TRIES >E EASI] TO R1D4
,NOTICE AL ! DI6]5T PLACE;T ( ! 5L>G$
GR\P+ SIGNS 9 ! TWO EXAMPLES4
-------------------------------------#GB
,A TRANSCRIB]'S NOTE %D 2 9CLUD$ 9 EI
CASE TO EXPLA9 ! PLACE;T ( EA* 5TRY4
#AJH
#AE ,MATRICES & VECTORS A#GB
#AE4F ,VECTORS
,LRS REPRES5T+ VECTORS >E (T5 PR9T$ 9
BOLD FONT & MAY H >R[S OR B>S ABV OR 2L4
,BOLDFACE ONLY NE$S TO 2 %[N 9 BRL IF X
IS ! ONLY ME?OD US$4 ,= >R[S & B>S ABV &
2L SEE ,SEC;N #AB4
,IF ! VECTOR 8#B OV] M9US #A0 0 CALL$
;P & W5T F PO9T ,A TO PO9T ;,B "H >E "S
( ! MO/ LIKELY =MS3
;P ) A B> "U EQUALS 5L>G$ R.D BRACKETS1
#B AT ! TOP & M9US #A AT ! BOTTOM
P,: "7 ,"< #B,">
,"<"-#A,">
;P ) A B> OV]
P;:
BOLD ;P
^2;P
#AJI
#AE ,MATRICES & VECTORS B#GB
;,,AB ) AN >R[ OV]
;;<,,AB>^:
;,,AB ) A B> "U
;;<,,AB>,:
-------------------------------------#GC
#AE4G ,GR\P+ ( EQUA;NS
,OP5+ 5L>G$ CURLY BRACES >E (T5 US$ TO
GR\P EQUA;NS4 ,PR9T SPAC+ %D 2 FOLL[$ ":
POSSIBLE4
,SOLVE3
,_< X"6#BY "7 #G
,_<#BX"- ;Y "7 "-#D
;;;F"<X"> "7 ,_<#J IF X @< #J
,_<X9#B IF X @> #J;'
#AAJ
#AF ,*EMI/RY #GD
#AF ,*EMI/RY
.=@_7 EQUILIBRIUM >R[1 TR5D TO ! LEFT
.="_7 EQUILIBRIUM >R[1 TR5D TO ! "R
.=^_7 EQUILIBRIUM >R[ "<H>POONS">
.=,- DA% OR S+LE L9E BOND
.=.,- D\BLE L9E BOND
.=_,- TRIPLE L9E BOND
.=@_,- QUADRUPLE L9E BOND
.="5 S+LE DA%$ L9E
.=."5 D\BLE DA%$ L9E
.=_"5 TRIPLE DA%$ L9E
.=@_"5 QUADRUPLE DA%$ L9E
.="4 S+LE DOT
.=."4 D\BLE DOT
.=_"4 TRIPLE DOT
.=@_"4 QUADRUPLE DOT
.="8 S+LE CROSS
.=."8 D\BLE CROSS #AAA
#AF ,*EMI/RY A#GD
.=_"8 TRIPLE CROSS
.=@_"8 QUADRUPLE CROSS
.="0 S+LE SMALL CIRCLE
.=."0 D\BLE SMALL CIRCLE
.=_"0 TRIPLE SMALL CIRCLE
.=@_"0 QUADRUPLE SMALL CIRCLE
-------------------------------------#GE
#AF4A ,*EMICAL "NS
CYCLOHEXANE ) ! F/ FIVE LRS 9 ITALIC
.1CYCLO.'HEXANE
#B COMMA #B HYPH5 ,DIME?YLPROPANE
#B1B-,DIME?YLPROPANE
#A HYPH5 ,HYDROXYL COMMA #B HYPH5
NITROB5Z5E
#A-,HYDROXYL1#B-NITROB5Z5E
,COPP] OP5 P>5!SIS ,ROMAN CAPITAL TWO
CLOSE P>5!SIS SPACE ,SULPHATE
,COPP]"<,,II"> ,SULPHATE
#AAB
#AF ,*EMI/RY A#GE
,COPP] SUP]SCRIPT ,ROMAN CAPITAL TWO
SPACE ,SULPHATE
,COPP];;9<,,II> ,SULPHATE
#AF4B ,*EMICAL =MULAE
,USE ( CAPITAL 9DICATORS & T]M9ATORS3
,! G5]AL ,,UEB PR9CIPLES ON ! *OICE (
S+LE LR1 ^W OR PASSAGE MODE APPLY2 9
"PICUL>1 A CAPITAL T]M9ATOR %D N 2 US$
)9 A TWO-LR *EMICAL ELE;T SYMBOL4 ,US+
S+LE CAPITAL 9DICATORS = *EMICAL =MULAE
PROVIDES A UNI=M APPE>.E TO ! BRL2
N"E!.S1 "! MAY 2 A CLE> ADVANTAGE 9 US+
CAPITAL PASSAGE MODE 9 "S CASES4 ,LRS
REPRES5T+ *EMICAL ELE;TS %D N"E 2
3TRACT$4
,O
,FE
,N,O
,NA,CL
;,AL OP5 P>5!SIS ,O CLOSE P>5!SIS ,CL
,AL"<,O">,CL #AAC
#AF ,*EMI/RY B#GE
,O SUB #B
,O;5#B
-------------------------------------#GF
;,H SUB #B ,O
,H;5#B,O
,CA "<,O ;,H"> SUB #B
,CA"<,O,H">;5#B
,CU ;,S ,O SUB #D DOT #E ;,H SUB #B ,O
,CU,S,O;5#D"4#E,H5#B,O
;,H SUP] PLUS
,H;9"6
;,S ,O SUB #D SUP] M9US M9US
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,AN EXAMPLE ( US+ CAPITAL PASSAGE
MODE3
#AAD
#AF ,*EMI/RY A#GF
;,R DOT ;,,,C ;H "<O ;H,'"> DOT ;,C ;,H
SUB #B DOT ;,C ;,H SUB #B DOT ;,C ,O
SUB #B ;,H
,,,R"4CH"<OH">"4CH;5#B"4CH5#B
"4CO;5#B;H,'
,NOTE T ,ROMAN NUM]ALS >E CAPITALIS$ Z
A S+LE GR\P3
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;;,FE9<,,III>,CL5#C
#AF4C ,ATOMIC MASS NUMB]S
,^! >E (T5 %[N ) LEFT-H& SUP]SCRIPT &
SUBSCRIPT NUMB]S3
SUB #IB SUP] #BCH ;,U
;5#IB9#BCH,U
#AF4D ,ELECTRONIC 3FIGURA;N
,S* NOTA;N C 9VOLVE LEFT-H&
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PR9T NE$S TO 2 RETA9$ 9 ! BRL TO MAKE !
ATTA*;T CLE>4 #AAE
#AF ,*EMI/RY B#GF
,AR #AS SUP] #B #BS SUP] #B #BP SUP] #F
#CS SUP] #B #CP SUP] #F
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SUP] #A ;,S SUB #J
;9#A,S5#J
#D;D SUP] #I #ES SUP] #B SUP] #B ;,D SUB
#E_/#B
#D;D9#I#ES9#B ;9#B,D5<#E_/#B>
-------------------------------------#GG
#AF4E ,*EMICAL ,EQUA;NS
#B ,NA ,O ;,H PLUS ;,H SUB #B ;,S ,O SUB
#D "R >R[ ,NA SUB #B ;,S ,O SUB #D
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;;;#B,NA,O,H"6,H5#B,S,O5#D
\O ,NA5#B,S,O5#D"6#B,H5#B,O;'
#AAF
#AF ,*EMI/RY A#GG
;,N SUB #B "R >R[ ;,N ;,H SUB #C1 ) AN
;,H SUB #B WRITT5 ABV ! >R[ & 8,HAB]
PROCESS0 WRITT5 2L ! >R[
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OR
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"<G">;'
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-------------------------------------#GH
#AF4F ,ELECTRONS
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""=;;;
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_
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_
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_
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_
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""=;'
-------------------------------------#GI
"37
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> <
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#ABJ
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,H
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_
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_ _
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_ __ _
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-------------------------------------#HC
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-------------------------------------#HD
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=
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