,,! ,,NEME? ,,BRL ,,CODE ,,=
,,MA!MATICS ,,& ,,SCI;E ,,NOT,N
#AIGB ,,REVI.N
,-PIL$ ,"U ! ,AU?OR;Y (!
,AM]ICAN ,ASSOCI,N ( ,"W]S =! ,BL
,ASSOCI,N = ,$UC,N (! ,VISU,Y
,H&ICAPP$ &! ,N,NAL ,BRL ,ASSOCI,N
,ADOPT$ ,JANU>Y #BJ1 #AIGB1 0!
,,AAWB-AEVH-NBA ,ADVISORY ,C\NCIL
6! ,BRL ,AU?OR;Y
,PUBLI%$1 #AIHH1 BY
,AM]ICAN ,PR9T+ ,H\SE =! ,BL
,L\ISVILLE1 ,K5TUCKY #DJBJF-JJHE
,EMBOSS$ 9 ,FIVE ,VOLUMES
,VOLUME ,,II
,BRL PAGES P#A-P#D & #ACI-BHJ
,PR9T PAGES C#DA-AAJ P#A
,NEME? ,MA?4 ,CODE1 #1972
,,3T5TS
,PAGE
,,RULES (,3T4)
,,RULE ,,VI--,,PUNCTU,N
,,SIGNS ,,& ,,SYMBOLS
(@,S#36-@,S#45) """"""""""""" C#41
,,RULE ,,VII--,,REF];E ,,SIGNS
,,& ,,SYMBOLS (@,S#46-
@,S#48) """""""""""""""""""""" #52
,,RULE ,,VIII--,,A2REVI,NS
@,S#49-@,S#54) """"""""""""""" #54
,,RULE ,,IX--,,3TRAC;NS ,,&
,,%ORT-=M ,,^WS @,S#55-
@,S#56) """""""""""""""""""""" #62
,,RULE ;,X--,,OMIS.NS (@,S#57-
@,S#59) """""""""""""""""""""" #70
,,RULE ,,XI--,,C.ELL,N (@,S#60) #73
,,RULE ,,XII--,,FRAC;NS
(@,S#61-@,S#70) """""""""""""" #75
,,RULE ,,XIII--,,SUP]SCRIPTS ,,&
,,SUBSCRIPTS (@,S#71-@,S#84) #82
,,RULE ,,XIV--,,MODIFI]S
P#C
(@,S#85-@,S#102) """"""""""""" #97
,,RULE ,,XV--,,RADICALS
(@,S#103-@,S#105) """"""""""" #108
,,! ,,NEME? ,,BRL ,,CODE C#DA
,,= ,,MA!MATICS
,,& ,,SCI;E ,,NOT,N
#1972 ,,REVI.N
,,RULES (,3T9U$)
,RULE ,,VI--,PUNCTU,N ,SIGNS
& ,SYMBOLS
,PUNCTU,N ,9DICATOR _
,PUNCTU,N ,M>KS
,APO/ROPHE '
,COLON 3
-----------------------------------#DB
,-MA
,LIT]>Y 1
,MA!MATICAL ,
,DA%
,%ORT --
#ACI
,NEME? ,MA?4 ,CODE1 #1972 A#DB
,L;G ----
,ELLIPSIS '''
,EXCLAM,N ,PO9T 6
,HYPH5 -
,P]IOD 4
,"Q ,M>K 8
,QUOT,N ,M>KS
,LEFT 9N] ,8
,LEFT \T] 8
,"R 9N] 0'
,"R \T] 0
,SEMICOLON 2
#ADA
,NEME? ,MA?4 ,CODE1 #1972 B#DB
@,S#36_4 .,MODES .( .,PUNCTU,N3
,S9CE NUM]ALS >E REPRES5T$ 0SYMBOLS 9
! L[] "P (! CELL1 & S9CE ^! SYMBOLS AL
S]VE Z PUNCTU,N M>KS1 X IS NEC 6=MU-
LATE RULES 3C]N+ PUNCTU,N S T ! M1N+S
( S* SYMBOLS >E UNAMBIGU\S4 ,? ,CODE
EMPLOYS TWO MODES ( PUNCTU,N--MA!MATI-
CAL & LIT]>Y4
@,S#37_4 ..,USE (! ,PUNCTU,N .,9DI-
CATOR3 ,SUBJECT 6! PROVI.NS ( @,S#38,
! PUNCTU,N 9DICATOR M/ 2 US$ .2F A
PUNCTU,N M>K & .AF ANY SYMBOL (! TYPE
LI/$ 2L4 ,9 ALL ^! CIRCUM/.ES1 ! MODE
( PUNCTU,N IS 3SID]$ 6BE MA!MATICAL4
;I_4 ,AF ANY BRL 9DICATOR4
(1) ?1/2#, ?3/4#_4
-----------------------------------#DC
(2) "VELOCITY<:]_4
(B> OV] 8VELOC;Y0)
;II_4 ,AF ANY NUM]IC SYMBOL
WRITT5 Z 9 ! ,NEME? ,CODE4 #ADC
(1) #0_4 A#DC
(2) 8#49_0
;III_4 ,AF A ,ROMAN NUM]AL4
(1) ;,I, ,,II, ,,III_4
;IV_4 ,AF A DA% OR ELLIPSIS1 :5
^! O3UR 9 A MA!MATICAL 3TEXT4 ,:5 !
NATURE (! 3TEXT IS 9 D\BT1 ! PUNCTU,N
9DICATOR M/ 2 US$4
(1) #24 .K #6+ ----_4
(2) #1, #3, '''_4
;V_4 ,AF ANY REF];E SYMBOL4
(1) NOTE @#_4
;VI_4 ,AF ! G5]AL OMIS.N SYMBOL4
(1) #5@*3 .K =_4
;VII_4 ,AF A 8S+LE LR40
,NEME? ,MA?4 ,CODE1 #1972 B#DC
(1) ;A, ;B, ;C_4
;VIII_4 ,AF A SEQU;E ( M ?AN "O
LR 9 : EA* LR HAS A SEP>ATE ID5T;Y1
PROVID$ T S* A SEQU;E IS N AN A2REVIA-
;N4
(1) $T ,A,B,C_4
;IX_4 ,AF ORD9AL1 PLURAL1 OR
POSSESSIVE 5D+S : >E JO9$ 6NUM]ALS1
LRS1 OR O!R MA!MATICAL EXPRES.NS4
(1) #1ST, #2ND, #3RD, #4TH_4
-----------------------------------#DD
(2) ;,X_'S, ;,Y_'S, & ;,Z_'S_4
;X_4 ,AF ANY ^W OR A2REVI,N : IS
N ON ! BASE L9E1 IF ! PUNCTU,N : FOL-
L[S IS ON ! BASE L9E4
(1) #13;SEVEN_4
#ADE
;XI_4 ,AF ANY MODIFI$ A#DD
EXPRES.N4
(1) X:_4
;XII_4 ,AF ! RADICAL SYMBOL4
(1) 8>_0 M1NS 8SQU>E ROOT40
;XIII_4 ,AF ANY SYMBOL ( %APE OR
%APE MODIFIC,N1 OP],N1 OR -P>ISON4
(1) 8$4+$C .K $T_0
(2) ,! 8+_0 IS US$ = A4I;N4
(3) ,R1L NUMB]S MAY 2 -P>$ BY
8"K_0, 8.K_0, OR 8.1_04
;XIV_4 ,AF ANY SYMBOL ( GR\P+
:E!R BRLD OR DRAWN IN4
(1) (_8#8_0)_4
(2) (,,LCD)_4
-----------------------------------#DE
;XV_4 ,AF ANY A2REVIAT$ FUNC;N
"N OR UNA2REVIAT$ FUNC;N "N1 PROVID$ T
,NEME? ,MA?4 ,CODE1 #1972 A#DE
! LATT] O3URS 9 A MA!MATICAL 3TEXT4
(1) 8SIN_0 & 8COS_0 >E CIRCUL>
FUNC;NS4
;XVI_4 ,AF ANY (! MISCELLANE\S
SYMBOLS ( ,RULE ,,XXII_4
(1) #100@0_4
;XVII_4 ,AF A -MA1 HYPH51 OR
DA%1 PROVID$ T IF ^! 7 REMOV$ &! SPACE
: !Y O3UPY 7 N PRES5T1 "O (! 3DI;NS
;I-;XVI WD APPLY4
(1) #0,_0
(2) (--_8#1_0)
@,S#38_4 ..,NON-,USE (! ,PUNCTU,N
.,9DICATOR3 ,X M/ N 2 ASSUM$ T 2C A
PUNCTU,N M>K O3URS T ! PUNCTU,N 9DI-
CATOR M/ 2 US$4 ,! PUNCTU,N 9DICATOR
M/ N 2 US$ "U ANY (! CIRCUM/.ES LI/$
2L4 ,9 ALL ^! CIRCUM/.ES1 ! #ADG
MODE ( PUNCTU,N IS 3SID]$ 6BE B#DE
LIT]>Y4
;I_4 ,AT ! 2G9N+ (A BRL L9E OR
AF A SPACE4
(1) 8#24 IS A TWO-DIGIT NUM]AL40
(2) '#49
;II_4 ,AF ANY NUM]IC SYMBOL
WRITT5 Z 9 ,5GLI% ,BRL4
(1) ,COPY"R #AIGJ4
(ITEM ON A TITLE PAGE)
;III_4 ,AF A DA% OR ELLIPSIS1 :5
^! O3UR 9 A LIT]>Y 3TEXT4
(1) ,! F\R FUNDA;TAL OP],NS >E
----1 ----1 ----1 & ----4
(2) FIVE & ?REE >E '''4
-----------------------------------#DF
;IV_4 ,AF A ^W OR A2REVI,N PRO-
VID$ T ! PUNCTU,N IS AT ! SAME LEVEL Z
T ^W OR A2REVI,N4
,NEME? ,MA?4 ,CODE1 #1972 A#DF
(1) E4G4
(2) P4 #27_4
(3) ,,LCD4
(4) MI4_/MIN4
(5) .(,W$41 ,?URS41 ,FRI4.)
(6) $T;REG4 ;POLYGON
(7) #2 QU>TS4
(8) (_8?REE0)
(9) #5-C5T4
(10) ;X-9T]CEPT4
(11) ?1/2#-(F4
(12) RATE@*TIME4
;V_4 ,AF ANY UNA2REVIAT$ FUNC;N
"N : O3URS 9 A LIT]>Y 3TEXT4
(1) ,! PR9CIPAL TRIGONOMETRIC
FUNC;NS >E 8S9E01 8TANG5T01
& 8SECANT04
;VI_4 ,2F A -MA1 HYPH51 DA%1 OR
ELLIPSIS4
#ADI
(1) #0, #1, #2 B#DF
(2) #1_'S, #2_'S, & #3_'S_4
-----------------------------------#DG
(3) (1), (2), (3)_4
(4) .(P5NIES1 NICKELS1 '''1 HALF-
DOLL>S.)
(5) ,TRANSCRIB$1 #AIGJ1 BY
(ITEM ON A TITLE PAGE)
(6) XY-COORD9ATES4
(7) ,EX]CISES #30-40_4
(8) #1-, #2-, & #3-DIM5.NAL
SPACES4
(9) ,"O-1 TWO-1 ?REE-DIM5.NAL
SPACES4
(10) #65--#75
(11) ,USE ONLY #0_'S & #1_'S--USE
! B9>Y SY/EM4
;VII_4 ,2F ANY EXCEPT ! F/
PUNCTU,N M>K 9 A SEQU;E ( PUNCTU,N
M>KS : REQUIRES ! USE (! PUNCTU,N 9DI-
CATOR4
(1) ,PROBABIL;Y--8#0_04
(2) #0_40
,NEME? ,MA?4 ,CODE1 #1972 A#DG
@,S#39_4 ..,PLURAL & ,POSSESSIVE
.,5D+S3 ,! APO/ROPHE-;S -B9,N MAY 2
JO9$ 6NUM]ALS1 LRS1 & O!R MA!MATICAL
EXPRES.NS 6=M _! PLURALS OR POSSES-
SIVES4 ,:51 9 9K PR9T1 ! APO/ROPHE HAS
BE5 OMITT$1 X LIKEWISE M/ 2 OMITT$ 9 !
TRANSCRIP;N4 ,! *OICE 2T ! S+UL> &
PLURAL =M (A ^W IS "S"TS %[N 05CLOS+
AN ;S )9 P>5!SES4
(1) #0_'S_4
-----------------------------------#DH
(2) ;,A_'S, ;,B_'S, & ;,C_'S_4
(3) #1S, #2S, & #3S_4
(4) "X<@:]_'S
(THE PLURAL ( ;X )A SUP]SCRIB$
TILDE)
(5) X:_'S
(THE PLURAL ( ;X )A SUP]SCRIB$ HORI-
ZONTAL B>)
(6) $[S #1 & #2
(7) $TS ,A,B,C & ,D,E,F
#AEA
(8) X^2_'S A#DH
(THE PLURAL ( ;X SQU>$)
(9) C;I_'S
(THE PLURAL ( ;C SUB ;I)
(10) ,! C1_'S, C2_'S, ''',
C;N_'S_4
(THE PLURALS ( ;C SUB #1, ;C SUB #2,
''', ;C SUB ;N)
(11) PR9CIPLE(S)
@,S#40_4 .,COLON3 ,X M/ N 2 ASSUM$ T
! COLON M/ 2 FOLL[$ 0A SPACE Z IS G5-
],Y ! CASE 9 ,5GLI% ,BRL4
(1) #3_3#30
(2) ;F_3(X, Y)
@,S#41_4 .,-MA3
;A_4 ,:5 A -MA IS US$ Z A M>K (
PUNCTU,N 9 A SITU,N 9 : ! MODE (
PUNCTU,N IS MA!MATICAL1 ! -MA IS RE-
F]R$ TO Z ! .MA!MATICAL .-MA4 ,O!R-
WISE1 ! LIT]>Y -MA M/ 2 US$4
(1) #1, #3, #5, & #7_4
,NEME? ,MA?4 ,CODE1 #1972 B#DH
(2) 8;X, ;Y_0
(3) 8#3_0,
-----------------------------------#DI
(4) #4-, #5-, & #6-SID$ POLYGONS4
(NO SPACE AF -MA 9 9K PR9T2 9 BRL1
SPACE REQUIR$ AF A -MA US$ Z A
PUNCTU,N M>K)
(5) #4-SID$1 #5-SID$1 & #6-SID$
POLYGONS4
(6) I4E41
(7) (X, Y)
(8) (-3, 2)
(NO SPACE AF -MA 9 9K PR9T2 9 BRL1
SPACE REQUIR$ AF A -MA US$ Z A
PUNCTU,N M>K)
;B_4 ,NO SPACE M/ 2 LEFT AF ! -MA
: IS US$ Z A NUM]IC SYMBOL EXCEPT =!
PURPOSE ( A*IEV+ ALIGN;T4
(1) #1,000,000
(2) #947, 147, 592
,MILLIONS ,?\S&S ,"OS OR ,UNIT
#AEC
@,S#42_4 .,DA% (.,L;G)_3 ,! A#DI
L;G DA% M/ 2 PREC$$ & FOLL[$ 0A SPACE4
,H["E1 NO SPACE MAY 2 LEFT 2T ! L;G
DA% & ANY (! ITEMS LI/$ 2L1 PROVID$ ^!
ITEMS APPLY 6! L;G DA%4
;I_4 ,SYMBOLS ( PUNCTU,N O!R ?AN
! HYPH54
;II_4 ,BRL 9DICATORS4
;III_4 ,SYMBOLS ( GR\P+4
;IV_4 ,! SYMBOLS = DECIMAL1
DOLL>S1 C5TS1 P]C5T1 P.DS (/]L+), &
PRIMES4
(1) ,! OPPOSITE ( ---- IS
MULTIPLIC,N4
-----------------------------------#EJ
(2) #5- ---- .K #3
(THE M9US IS N "O (! LI/$ ITEMS)
(3) ,! OPPOSITE ( A4I;N IS ----4
(4) ?---- /15# .K #2/3#
(THE OP5+ FRAC;N 9DICATOR APPLIES 6!
DA%2 ! FRAC;N L9E IS N "O (! LI/$
ITEMS)
(5) (----, 4, 6, 8, ----)
(THE OP5+ & CLOS+ GR\P+ SYMBOLS AP-
,NEME? ,MA?4 ,CODE1 #1972 A#EJ
PLY 6_! RESPECTIVE DA%ES)
(6) @S2+@S3 .K @S----
(THE DOLL> SIGN APPLIES 6! DA%)
(7) #2@C+3@C .K ----@C
(THE C5T SIGN APPLIES 6! DA%)
(8) #2@0+3@0 .K ----@0
(THE P]C5T SIGN APPLIES 6! DA%)
(9) @L2+@L3 .K @L----
(THE P.D (/]L+) SIGN APPLIES 6! DA%)
(10) ,A #3-\NCE &A #5-\NCE WEI<T C
2 REPLAC$ 0A ---- -\NCE
WEI<T4
(11) #4@0 .K .1----
(THE DECIMAL PO9T APPLIES 6! DA%)
(12) #12, .K ----''
(THE D\# PRIME APPLIES 6! DA%)
@,S#43_4 .,ELLIPSIS3
;A_4 ,ANY DOT OR S]IES ( DOTS 9
PR9T : REPRES5TS AN OMITT$ T]M1 5TRY1
OR L9E IS AN ELLIPSIS4 ,X M/ 2 REPRE-
S5T$ 9 BRL 0A M9IMUM ( ?REE DOTS4
(1) #1, #3, #5, ''', #15_4 #AEE
(2) ,M>Y1 ,S,Y1 '''4 #EA
(3) ;A, AR, AR^2, '''_4
(4) @,(A11 A12 ''' A;1N"@,)
@,(A21 A22 ''' @;2N"@,)
@,('''''''''''''''''''@,)
@,(A;N1 A;N2 ''' A;NN"@,)
;B_4 ,! ELLIPSIS IS SUBJECT 6!
SAME SPAC+ RULES Z ! L;G DA%4 ,SEE
@,S#42_4
(1) X+Y+ '''_4
(THE P]IOD APPLIES 6! ELLIPSIS)
(2) #1, #3, #5, ''', #15_4
(THE -MA APPLIES 6! ELLIPSIS)
(3) P1^.A^;1"''' P;R^.A^;R
(THE BASE-L9E 9DICATOR APPLIES 6!
ELLIPSIS)
(4) (''', -1, 0, 1, ''')
(THE OP5+ & CLOS+ GR\P+ SYMBOLS AP-
PLY 6_! RESPECTIVE ELLIPSES)
(5) #12@C+14@C .K '''@C
(THE C5T SIGN APPLIES 6! ELLIPSIS)
@,S#44_4 .,EXCLAM,N .,PO9T3 ,! EX-
,NEME? ,MA?4 ,CODE1 #1972 A#EA
CLAM,N PO9T IS REPRES5T$ 0! SAME SIGN
( 9K PR9T Z ! FACTORIAL SIGN4 ,! 3TEXT
IS USU,Y SU6ICI5TLY CLE> 9 REG>D 6?
4T9C;N S T ! POSSIBIL;Y ( D\BT 9 *OOS+
! PROP] SYMBOL IS SMALL4
@,S#45_4 .,HYPH53 ,! HYPH5 IS REPRE-
S5T$ 0! SAME SIGN ( 9K PR9T Z ! M9US
SIGN4 ,S9CE ! CORRESPOND+ BRL SYMBOLS
AL CO9CIDE1 A M9IMUM ( DECI.N-MAK+ 9 ?
REG>D IS REQUIR$ (! TRANSCRIB]4 ,A
SPACE M/ 2 LEFT 2T A HYPH5 & AN ADJA-
C5T DA%4
-----------------------------------#EB
,RULE ,,VII--,REF];E ,SIGNS
& ,SYMBOLS
,G5]AL ,REF];E ,9DICATOR @]
,A/]ISK @#
,DA7]
,S+LE _]
#AEG
,NEME? ,MA?4 ,CODE1 #1972 A#EB
,D\# __]
,P>AGRAPH ,M>K @,P
,SEC;N ,M>K
,S+LE @,S
,D\# @,S@,S
,/> $S
@,S#46_4 ..,REF];E ,SIGNS & .,SYM-
BOLS3 ,! REF];E SIGNS ( ? SEC- #AEI
,NEME? ,MA?4 ,CODE1 #1972 B#EB
;N M/ 2 REPRES5T$ 0! SYMBOLS LI/$ ABV
& ,5GLI% ,BRL SYMBOLS M/ N 2 US$4 ,"S
( ^! SIGNS >E AL US$ Z SIGNS ( OP],N &
9 T CASE ! RULES GOV]N+ SIGNS ( OP],N
APPLY (SEE ,RULE ,,XIX)_4 ,:5 X IS C]-
TA9 T A SYMBOL 9 ! ABV LI/ IS 6BE US$
= REF];E PURPOSES1 ! SUP]SCRIPT POSI-
;N1 IF 9DICAT$ 9 9K PR9T1 M/ 2 IGNOR$
9 ! TRANSCRIP;N4
,:5 A REF];E SIGN O3URS = : NO PRO-
VI.N EXI/S 9 ? ,CODE1 S* Z D>TS1 PIC-
TURES1 ETC41 ! TRANSCRIB] M/ DEVISE A
SUITA# SYMBOL ) AN EXPLANATORY TRAN-
SCRIB]'S NOTE4 ,:E!R A REF];E SYMBOL
EXI/S 9 ? ,CODE OR HAS BE5 DEVIS$ 0!
TRANSCRIB]1 S* SYMBOLS >E SUBJECT 6!
RULES = SIGNS & SYMBOLS ( REF];E4
(1) ,A ,CANTOR @# SET IS '''
(THE A/]ISK IS A REF];E SIGN1 & XS
SUP]SCRIPT POSI;N M/ N 2 9DICAT$)
(2) F@#G
(THE A/]ISK IS A SIGN ( OP],N)
#AFA
@,S#47_4 .,G5]AL .,REF];E C#EB
.,9DICATOR3 ,:5 REF];E 6A FOOTNOTE IS
DENOT$ 0A NUM]AL1 USU,Y 9 ! SUP]SCRIPT
POSI;N1 & NO O!R REF];E SIGN IS EM-
PLOY$1 ! G5]AL REF];E 9DICATOR IMMLY
FOLL[$ 0! NUM]AL (! PR9T$ TEXT M/ 2
US$ 9 ! TRANSCRIP;N &! NUM]AL M/ N 2
REPRES5T$ Z 2+ 9 ! SUP]SCRIPT POSI;N4
(1) ,F9D ! 9DEX @]#1 (! RADICAL4
(IN 9K PR9T1 A #1 APPE>S 9 ! SUP]-
SCRIPT POSI;N AF 89DEX02 X REF]S 6A
FOOTNOTE)
----------------------------------#EC
@,S#48_4 ..,SPAC+ ) ,SYMBOLS (
.,REF];E3
;A_4 ,:5 A REF];E SIGN : CALLS AT-
T5;N TO OR 9TRODUCES A FOOTNOTE IS AT-
TA*$ 6A ^W OR MA!MATICAL EXPRES.N1 !
REF];E SYMBOL M/ FOLL[ T ^W OR EXPRES-
.N )A SPACE 2T4 ,IF S* A REF];E SIGN
IS UNATTA*$ XS POSI;N RELATIVE 6XS
SURR.D+ MAT]IAL M/ 2 PRES]V$1 &A SPACE
M/ 2 LEFT ON EI SIDE (! REF];E SYMBOL4
,H["E1 IF "! IS PUNCTU,N : APPLIES 6S*
,NEME? ,MA?4 ,CODE1 #1972 A#EC
A REF];E1 NO SPACE %D 2 LEFT 2T ! REF-
];E SYMBOL &! PUNCTU,N M>K : APPLIES
6X4
(1) ,IRR,NAL @# NUMB]S '''
(IN 9K PR9T1 ! A/]ISK PREC$ES 8IRRA-
;NAL0 & IS UNSPAC$ F IT)
(2) ,IRR,NAL @# NUMB]S '''
(IN 9K PR9T1 ! A/]ISK FOLL[S 8IRRA-
;NAL0 & IS UNSPAC$ F IT)
(3) @# ,IRR,NAL NUMB]S '''
(IN 9K PR9T1 ! A/]ISK PREC$ES 8IRRA-
;NAL0 & IS SPAC$ AWAY F IT)
(4) ''' SETS4 @#
(IN 9K PR9T1 ! A/]ISK FOLL[S ! P]IOD
B ! P]IOD DOES N APPLY 6! A/]ISK)
(5) ''' SETS @#_4
(IN 9K PR9T1 ! A/]ISK FOLL[S 8SETS0
& IS UNSPAC$ F X2 ! P]IOD APPLIES 6!
A/]ISK)
;B_4 ,! SYMBOLS =! SEC;N M>K1 P>A-
GRAPH M>K1 G5]AL REF];E 9DICATOR1 />1
A/]ISK1 & DA7]1 :5 ! A/]ISK OR #AFC
DA7] DOES N CALL ATT5;N TO OR B#EC
9TRODUCE A FOOTNOTE1 M/ 2 UNSPAC$ F !
LR OR NUM]AL : APPLIES 6!M4 ,ALL REF-
];E SYMBOLS1 :E!R OR N !Y CALL ATT5;N
TO OR 9TRODUCE A FOOTNOTE1 M/ 2 SPAC$
AWAY F ! ^WS 6: !Y APPLY4
(1) @,S#1, @,PA
(2) ,A ,CANTOR @]#1 SET IS '''
(IN 9K PR9T1 A #1 APPE>S 9 ! SUP]-
SCRIPT POSI;N AF 8,CANTOR02 X REF]S
6A FOOTNOTE)
(3) @##10_4
(A/]ISK DENOTES A PRO#M = EXTRA /UDY
& IS FOLL[$ 0! PRO#M NUMB])
(4) #1@#_4
(A/]ISK DENOTES A PRO#M = EXTRA /UDY
& FOLL[S ! PRO#M NUMB])
-----------------------------------#ED
(5) #1_4@#
(A/]ISK DENOTES A PRO#M = EXTRA /UDY
& FOLL[S ! P]IOD2 ! NUM]AL #1 AP-
PLIES 6! A/]ISK EV5 ?\< "! IS AN
9T]V5+ P]IOD)
,NEME? ,MA?4 ,CODE1 #1972 A#ED
(6) @# ,= EXTRA CR$IT4
(THIS IS A FOOTNOTE)
;C_4 ,! PLACE;T & 9D5T,N ( FOOT-
NOTES >E SUBJECT 6! RULES = FOOTNOTES
9 ! 8,CODE ( ,BRL ,TEXTBOOK ,=MATS &
,TE*NIQUES40
,RULE ,,VIII--,A2REVI,NS
@,S#49_4 .,A2REVI,NS3
;A_4 ,A2REVI,NS M/ 2 REG>D$ 9 A
BROAD S5SE 69CLUDE ! FOLL[+ ITEMS3
;I_4 ,UNIV]SAL LIT]>Y A2REVI,NS
(! TYPE -MONLY LI/$ 9 A DIC;N>Y4
(1) #11 ,A4,M4
(2) ,! YE> #1 ,A4,D4
(3) #1_/X VS4 ;,T
(4) VIZ4
(5) ,PH4,D4
(6) ,OCT41 ,NOV41 ,DEC4
(7) ,MON41 ,TUES41 ,W$4
(8) #110 ,W4 #110TH ,/4 #AFE
;II_4 ,A2REVI,NS ( M1S- B#ED
URE;T4
(1) #1 YD4
(2) ,C4 /&S = ,C5TIGRADE
(3) #980 G4
-----------------------------------#EE
(4) #1000 ;M
(5) #1 LI<T-;YR
(6) #25 SQ4 FT4
(7) #100 M4P4H4
(8) #60 MI4_/HR4
(9) #6 FT4-LBS4
;III_4 ,ACRONYMS4
(1) ,,=TRAN
(2) ,,ASCAP
;IV_4 ,P]SONAL OR GEOGRAPHIC
9ITIALS4
(1) ,I SAW ,MR4 ,M4 & ,MR4 ,N4
(2) ,G4 ,B4 ,%AW
(3) ,WA%+TON1 ,D4,C4
,NEME? ,MA?4 ,CODE1 #1972 A#EE
;V_4 ,9ITIALS ( AG5CIES1 ORGANI-
Z,NS1 ETC4
(1) ,,RCA
(2) ,B4,V4,D4
(3) ,! ;,B & ;,O ,RAILROAD4
;VI_4 ,SPECIAL A2REVI,NS 3F9$ 6A
"PICUL> FIELD OR EV5 6A "PICUL> BOOK4
(1) LCD
(M1NS 8L1/ -MON DENOM9ATOR0)
(2) ,L4,U4,B4
(M1NS 8L1/ UPP] B.D0)
-----------------------------------#EF
(3) ,,SAS
(M1NS 8SIDE-ANGLE-SIDE0)
(4) ,CL,,PA
(M1NS 8,CLOSURE ,PROP]TY = ,A4I;N0)
;VII_4 ,A2REVI,NS =M$ 0! USE (
9ITIAL OR PR9CIPAL LRS (A ^W1 PHRASE1
OR "N4 #AFG
(1) ,VA4 A#EF
(M1NS 8,VIRG9IA0)
(2) ;N ST4 ANGLES
(8ST40 M1NS 8/RAI<T0)
(3) ,,PL_/,I
(M1NS 8,PROGRAM ,LANGUAGE";,I_0)
(4) ;,I_/;,O
(M1NS 8,9PUT-,\TPUT0)
(5) ;D-;C
(M1NS 8DIRECT CURR5T0)
;B_4 ,:5 A LR OR SEQU;E ( LRS DOES
N REPRES5T A ^W OR PHRASE1 X M/ N 2
3SID]$ Z AN A2REVI,N & M/ 2 TRANSCRIB$
AC 6O!R RULES ( ? ,CODE4 ,A2REVIAT$
FUNC;N "NS1 Z WELL Z MODEL NUMB]S1
S]IAL NUMB]S1 ETC4 M/ AL N 2 3SID]$
A2REVI,NS & M/ 2 TRANSCRIB$ AC 6O!R
RULES ( ? ,CODE4 ,:5 "! IS D\BT Z TO
:E!R OR N A 3/RUC;N IS AN A2REVI,N1 X
M/ 2 TR1T$ Z IF X 7 N AN A2REVI,N4
(1) ,VITAM9 ;,A_4
(2) ,BLOOD TYPES >E ;,A, ;,B,
,A,B, & ;,O_4
,NEME? ,MA?4 ,CODE1 #1972 B#EF
(3) SIN X
(4) ,S]IAL NO4 ,G,E"96,F"12_4
-----------------------------------#EG
@,S#50_4 .,CAPITALIZ,N .) .,A2REVIA-
;NS3 ,9 AN A2REVI,N1 :5"E LRS >E CAPI-
TALIZ$ 9 9K PR9T1 A S+LE LR M/ 2 PRE-
C$$ 0! S+LE CAPITALIZ,N 9DICATOR1 &A
SEQU;E ( M ?AN "O LR M/ 2 PREC$$ 0!
D\# CAPITALIZ,N 9DICATOR4
(1) ,P4,M4
(2) ,,EST
(M1NS 8,EA/]N ,/&>D ,"T0)
@,S#51_4 ..,5GLI%-,LR ,9DICATOR )
.,A2REVI,NS3
;A_4 ,:5 A P]IOD FOLL[S AN A2REVI-
,N1 "! >E F\R POSSIBILITIES 63SID]3
;I_4 ,! P]IOD APPLIES 6! A2REVI-
,N B DOES N 5D A S5T;E4
;II_4 ,! P]IOD 5DS A S5T;E B DOES
N APPLY 6! A2REVI,N4
;III_4 ,! P]IOD BO? APPLIES 6!
A2REVI,N & 5DS A S5T;E4 #AFI
;IV_4 ,X IS D\BT;L :E!R A#EG
! P]IOD APPLIES 6! A2REVI,N4
,9 ! CASE ( ;II, ! ,5GLI%-LR 9DICA-
TOR M/ 2 US$ OR M/ N 2 US$ Z IF !
P]IOD 7 N PRES5T4 ,9 CASE ;IV, ! P]IOD
%D 2 3SID]$ Z APPLY+ 6! A2REVI,N &!
APPROPRIATE RULE M/ !N 2 APPLI$4
,! USE OR NON-USE (! ,5GLI%-LR 9DI-
CATOR ) A2REVI,NS DOES N DEP5D ^U !
BRL SYMBOLS ) : ! A2REVI,N MAY HAPP5
6BE 9 3TACT1 S* Z GR\P+ SYMBOLS1 BRL
9DICATORS1 FRAC;N L9ES1 ! HYPH51 OR !
SLA%4
;B_4 ,! ,5GLI%-LR 9DICATOR M/ 2
US$ 2F AN A2REVI,N : 3SI/S ( "O LR OR
(A -B9,N ( LRS CORRESPOND+ 6A %ORT-=M
^W PROVID$ ! A2REVI,N IS N FOLL[$ 0A
P]IOD : APPLIES 6X4
(1) #10 ;G +10 ;G .K #20 ;G
(NO P]IODS APPLY 6^! A2REVI,NS)
(2) ,WE "K #32^.* ;,F .K #0^.*
;,C4
(THE P]IOD 5DS A S5T;E & DOES N AP-
PLY 6! A2REVI,N)
,NEME? ,MA?4 ,CODE1 #1972 B#EG
(3) #1 LI<T-;YR
(NO P]IOD APPLIES TO 8;YR0 ^: LRS
CORRESPOND 6A %ORT-=M ^W)
(4) LAT4 #30^.*"20' ;,N
(NO P]IOD APPLIES 6! ;,N)
-----------------------------------#EH
(5) $4 ;L .K #1000 CC
(NO P]IOD APPLIES 6! ;L)
(6) (;M)
(THE PRES;E (! P>5!SES HAS NO E6ECT
^U ! DECI.N T ! ,5GLI% LR 9DICATOR
M/ 2 US$)
(7) $4 ;M^2 .K #100 CM^2
(NO P]IOD APPLIES 6! ;M_2 ! PRES;E
(! SUP]SCRIPT 9DICATOR HAS NO E6ECT
^U ! DECI.N T ! ,5GLI%-LR 9DICATOR
M/ 2 US$)
(8) ?;M/CM#@*?CM/MM#
(NO P]IOD APPLIES 6! ;M_2 ! PRES;E
(! FRAC;N 9DICATOR & FRAC;N L9E HAS
NO E6ECT ^U ! DECI.N T ! ,5GLI%-LR
9DICATOR M/ 2 UP$)
(9) ;,I_/;,O M1NS 8,INPUT_/
,OUTPUT40 #AGA
(NO P]IOD APPLIES 6EI AB- A#EH
BREVI,N2 ! PRES;E (! SLA% HAS NO EF-
FECT ^U ! DECI.N T ! ,5GLI%-LR 9DI-
CATOR M/ 2 US$)
;C_4 ,! ,5GLI%-LR 9DICATOR M/ N 2
US$ 2F AN A2REVI,N : 3SI/S ( "O LR OR
(A -B9,N ( LRS CORRESPOND+ 6A %ORT-=M
^W PROVID$ ! A2REVI,N IS FOLL[$ 0A
P]IOD : APPLIES 6X4 ,! ,5GLI%-LR 9DI-
CATOR M/ AL N 2 US$ 2F AN A2REVI,N ^:
LRS D N CORRESPOND 6A %ORT-=M ^W4 ,9 ?
CASE1 :E!R A P]IOD APPLIES 6! A2REVI,N
OR N HAS NO E6ECT ON ! RULE =! NON-USE
(! ,5GLI%-LR 9DICATOR4
(1) #100^.* ,C4 .K #212^.* ,F4
(THE P]IODS APPLY 6! A2REVI,NS)
(2) ,DOES #1 KM4 .K #1000 M48
(THE P]IOD APPLIES 6! ;M)
(3) #1 LI<T-YR4
(THE P]IOD APPLIES TO 8;YR0 ^: LRS
CORRESPOND 6A %ORT-=M ^W)
(4) (M4)
(THE PRES;E (! P>5!SES HAS NO E6ECT
,NEME? ,MA?4 ,CODE1 #1972 B#EH
^U ! DECI.N T ! ,5GLI%-LR 9DICATOR
M/ N 2 US$)
(5) ?W4/V4#
(THE P]IODS APPLY 6! A2REVI,NS2 !
PRES;E (! FRAC;N 9DICATORS & FRAC;N
L9E HAS NO E6ECT ^U ! DECI.N T !
,5GLI%-LR 9DICATOR M/ N 2 US$)
-----------------------------------#EI
(6) #1 KM .K #1000 ;M
(THE A2REVI,N 8KM0 DOES N CORRESPOND
6A %ORT-=M ^W)
(7) #100 CM4^2 .K = M4^2
(THE P]IODS APPLY 6! A2REVI,NS)
@,S#52_4 .,PUNCTU,N .) .,A2REVI,NS3
,A2REVI,NS M/ 2 PUNCTUAT$ 9 ! LIT]>Y
MODE1 PROVID$ T ! PUNCTU,N IS AT !
SAME LEVEL Z T A2REVI,N4 (,SEE
@,S@,S#37 & #38_4)
(1) I4E41
(2) (E4G41 ''')
(3) #9 FT4^2
(4) #3 GAL41 #2 QT41 #1 PT4 #AGC
(5) $T;REG4 ;POLYGON A#EI
(6) #3 GAL1 #2 QT1 #1 PT4
(7) (A4 .K A4)
(A2REVIAT$ =M = ANGLE .K ANGLE)
(8) ,TURN 6,*AP4 ,,IV, (P4 #27)_4
(9) ,REF] 6,VOL4 ;,I, PP4 #30-35_4
(10) #60 MPH .K #88 FT4_/SEC4
@,S#53_4 .,3TRAC;NS .9 .,A2REVI,NS3
,NO 3TRAC;NS MAY 2 US$ 9 AN A2REVI,N :
IS 9 DIRECT 3TACT ) ANY (! ITEMS 9
@,S#55A_4 ,! A2REVI,N 8IN40 OR 8IN0
USU,Y M1N+ 89*ES0 M/ N"E 2 3TRACT$4
,! 8ST0 3TRAC;N MAY ONLY 2 US$ = A2RE-
VIAT+ 8/REET0 OR 8SA9T04 ,X M/ N 2 US$
= ANY O!R A2REVI,N1 S* Z 8ST40 =
8/RAI<T04
(1) #15 IN4
-----------------------------------#FJ
(2) #15-IN
(3) #23 M94
(4) ;N ST4 $[_'S
(5) ,R1D ,*AP4 ;,V_4
(6) .(,>IZ41 ,>K41 ,3N4.)
,NEME? ,MA?4 ,CODE1 #1972 A#FJ
(7) .(,W$41 ,?URS41 ,FRI4.)
(8) ;X-M94
(9) #6 M94_/360 SEC4
(10) ?1 HR4/60 M94#
(11) #6 MIN_/360 SEC
(12) ?1 HR/60 MIN#
(13) ,S;PART4 ;SUM
(P]IOD AT ! SUBSCRIPT LEVEL)
(14) /ATVOLT-CM_/STATAMP-OER/$
@,S#54_4 .,SPAC+ .) .,A2REVI,NS3
;A_4 ,9 TRANSCRIB+ A2REVI,NS1 !
,5GLI% ,BRL TE*NIQUES ( TRANSPOSI;N
(WRIT+ AN A2REVI,N 9 FRONT ( XS NUMB])
& 3D5S,N (US+ BRL A2REVI,NS %ORT] ?AN
_! 9K PR9T C.T]"PS) M/ N 2 EMPLOY$4
(1) ,TURN 6,*AP4 ,,IV, (P4 #27)_4
(2) ,REF] 6,VOL4 ,,II, PP4 #30-35_
;B_4 ,NO SPACE %D 2 LEFT 2T AN
A2REVI,N & XS P]IOD1 IF PRES5T1 &A
SLA% L9E OR ANY SYMBOL ( GR\P+1 9DI-
CATOR1 PUNCTU,N1 OR FRAC;N L9E #AGE
: APPLIES 6! A2REVI,N4 ,A SPACE B#FJ
M/ AL N 2 LEFT 2T TWO -PO-
-----------------------------------#FA
N5TS ( AN A2REVI,N :5 NO SPACE APPE>S
9 9K PR9T4 ,A SPACE M/ 2 LEFT ON EI
SIDE ( AN A2REVI,N 9 ALL O!R SITU,NS4
(1) #60 MPH .K #88 FT_/SEC
(2) (3 YD)^2 .K #9 YD^2
(3) >60 FT]
(4) ?1 HR4/60 M94#
(5) $T;REG4 ;POLYGON
(6) #60 M4P4H4
(7) #42 FT4 B4M4
(8) #3 FT4 .K #1 YD4
(9) ,Q1 .K ,U1- P4E4
(10) #6 YDS -#2 FT
(11) ,A RECTANGLE ;H FT L;G BY ;K
FT HI<4
(12) #4 YD #2 FT #1 IN
(13) ;,N #35^.* ;,W
(14) (2X-3Y) MI4
(THE CLOS+ P>5!SIS DOES N APPLY 6!
A2REVI,N2 ! A2REVI,N M/ 2 PREC$$ 0A
SPACE)
,NEME? ,MA?4 ,CODE1 #1972 A#FA
(15) ?1/2# HR
(THE CLOS+ FRAC;N 9DICATOR DOES N
APPLY 6! A2REVI,N2 ! A2REVI,N M/ 2
PREC$$ 0A SPACE)
(16) #25 ,SQ4 ,FT4
-----------------------------------#FB
,RULE ,,IX--,3TRAC;NS
& ,%ORT-,=M ,^WS
@,S#55_4 ..,NON-,USE ( ,3TRAC;NS &
,%ORT-,=M .,^WS3
;A_4 ,3TRAC;NS & %ORT-=M ^WS M/ N
2 US$ 9 A ^W1 "P (A ^W1 OR A2REVI,N :5
X IS 9 DIRECT 3TACT ) ANY ITEM LI/$
2L4 ,!Y M/ AL N 2 US$ 2F OR AF ! SPACE
: IMMLY PREC$ES OR FOLL[S A SIGN (
-P>ISON4 ,9 ! CASE ": TRANSI;N 6ANO!R
BRL L9E HAS BE5 MADE1 3TRAC;NS M/ N 2
US$ IF !Y CD N H BE5 US$ )\T ! TRANSI-
;N4 ,9 ! CASE ( AN EXPRES.N 3TA9+ A
HYPH5 OR DA%1 ONLY T POR;N 2T ! HYPH5
OR DA% &! ITEM ) : DIRECT 3TACT IS
MADE IS SUBJECT 6? RULE4
;I_4 ,ANY BRL 9DICATOR #AGG
O!R ?AN CAPITALIZ,N 9DICATORS A#FB
OR ! ITALIC SIGN ( ,5GLI% ,BRL4
(1) #13;SEVEN
(2) #4-_AMPERE
(3) "VELOCITY<:]
(4) $CH
(IN 9K PR9T1 A PICTURE (A *UR*)
(5) $C_$ONE]
(6) 9*-POUND^2
(7) ?DISTANCE/TIME# .K RATE
;II_4 ,ANY NUM]IC SYMBOL WRITT5
Z 9 ! ,NEME? ,CODE4
(1) COS @(2,ARC CSC (-?29/21#)@)
;III_4 ,! G5]AL OMIS.N SYMBOL4
(1) TEN=FOUR .K SIX
;IV_4 ,A S+LE LR4
(1) AARC SIN X+BARC TAN Y
,NEME? ,MA?4 ,CODE1 #1972 #FC
;V_4 ,ANY SEQU;E ( M ?AN "O LR 9
: EA* LR HAS A SEP>ATE ID5T;Y4
(1) XYSINE Z
;VI_4 ,ANY MODIFI] SYMBOL4
(1) "$33O<HEAT]
;VII_4 ,! RADICAL SYMBOL4
(1) >FOUR]
;VIII_4 ,ANY OP],N SYMBOL4
(1) NINE-SEVEN .K TWO
(2) ERGS_/CM^3
(3) #60 M94_/HOUR
(4) /ATVOLT-CM_/STATAMP-OER/$
(5) ?DISTANCE/TIME# .K RATE
(6) SEVEN+THREE
(7) P :O TRAVEL BY BUS
_/PEOPLE :O TRAVEL 0C> #AGI
;IX_4 ,ANY -P>ISON SYM- A#FC
BOL1 EV5 ?\< "! IS A SPACE 2T X &! ^W1
"P ^W1 OR A2REVI,N4
(1) #1 HOUR .K #60 M9UTES
(2) ,LET #3X .K THE L>G] NUMB]
(3) SEVEN+THREE .K TEN
-----------------------------------#FD
(4) ,COPY & REPLACE $4 BY .K OR
/.K TO MAKE A TRUE S5T;E4
(5) ,X IS A FUNDA;TAL PR9CIPLE
THAT .K_'S A4$ TO .K_'S ARE
.K_4
;B_4 ,3TRAC;NS M/ N 2 US$ 9 AB-
BREVIAT$ FUNC;N "NS 9 ANY 3TEXT4 ,9
A4I;N1 3TRAC;NS M/ N 2 US$ 9 UNA2REVI-
AT$ FUNC;N "NS : APPE> 9 A MA!MATICAL
3TEXT4 ,9 "PICUL>1 ! ^W 8>C0 M/ N 2
3TRACT$ :5 IMMLY PREC$$ OR FOLL[$ BY
MA!MATICAL SYMBOLS1 :E!R SPAC$ OR UN-
SPAC$4
(1) SIN X
(2) COSH X
,NEME? ,MA?4 ,CODE1 #1972 A#FD
(3) SINE X+SINE Y
(4) COS (ARC ,TAN X+?.P/3#)
(5) #2ARCSIN X
(6) ,ARC ,SINE X
(7) ,ARC ,A,C,B IS A MAJOR >C4
;C_4 ,! 3TRAC;NS = .TO1 .9TO1 &
.BY M/ N 2 US$ 2F ANY (! ITEMS LI/$
2L4 ,:5 ! 3TRAC;N = .9TO MAY N 2 US$1
! 3TRAC;N = 8IN0 MAY N"E!.S 2 US$ 9
89TO0 UN.S O!RWISE PROHIBIT$4
;I_4 ,2F ANY (! ITEMS 9 ;A ABV4
(1) ,F ;A TO ;Z_4
(3TRAC;N N US$ AC TO ;A (;I))
(2) ,F ?1/4# TO ?1/2#_4
(3TRAC;N N US$ AC TO ;A (;I))
-----------------------------------#FE
(3) ,DECOMPOSE $R ,A,B,C,D
9TO $TS ,A,B,C & ,D,B,C_4
(3TRAC;N N US$ AC TO ;A (;I))
(4) ,F .A TO .W_4
(3TRAC;N N US$ AC TO ;A (;I))
#AHA
(5) ,F -#10 TO #10_4 A#FE
(3TRAC;N N US$ AC TO ;A (;II))
(6) #22@*3 IS EQUAL TO =@*11
(3TRAC;N N US$ AC TO ;A (;III))
(7) ! IMAG9>Y "P DENOT$
BY I .K (0, 1)
(3TRAC;N N US$ AC TO ;A (;IV))
(8) ,A,B IS P>ALLEL TO ,C,D
(3TRAC;N N US$ AC TO ;A (;V))
(9) ,DIVIDE BY >3]_4
(3TRAC;N N US$ AC TO ;A (;VII))
(10) ,F -#10 TO +10_4
(3TRAC;N N US$ AC TO ;A (;VIII))
(11) ,IF .K_'S >E DIVID$ BY .K_'S,
! RESULTS ARE .K_4
(3TRAC;N N US$ AC TO ;A (;IX))
-----------------------------------#FF
;II_4 ,2F ANY A2REVI,N : 3SI/S (
"O LR OR A -B9,N ( LRS CORRESPOND+ 6A
%ORT-=M ^W4
(1) ,TURN TO P4 #27_4
(2) ,3V]T MM TO ;M4
(3) ,3V]T "DS TO YRS4
,NEME? ,MA?4 ,CODE1 #1972 A#FF
;III_4 ,2F ANY ,ROMAN NUM]AL4
(1) ,*APT]S ;,I TO ,,VII_4
;IV_4 ,2F A DA% OR ELLIPSIS4
(1) #20 A4$ TO ---- EQUALS #30
;V_4 ,2F ANY REF];E SYMBOL4
(1) @,S#25 TO @,S#27_4
;VI_4 ,2F A 8S+LE LR04
(1) ! IMAG9>Y "P DENOT$
BY I .K (0, 1)
;VII_4 ,2F A SEQU;E ( M ?AN "O
LR 9 : EA* LR HAS A SEP>ATE ID5T;Y4
(1) ,A,B IS P>ALLEL TO ,C,D
;VIII_4 ,2F ANY ^W1 "P #AHC
^W1 OR A2REVI,N 9 SITU,NS 9 : B#FF
3TRAC;NS >E N P]MITT$ AC 6ANY (! O!R
RULES ( ? SEC;N4
(1) P :O G BY CAR+PEOPLE :O G
0TRA9
-----------------------------------#FG
(2) ! NUMB] ( P :O TRAVEL BY CAR
+THE NUMB] ( P :O G 0TRA9
(_8BY0 _C 2 3TRACT$ 2F 8C>0 2C ! .>
3TRAC;N 9 8C>0 _C 2 US$)
(3) ,! >EA IS DIVID$ 9TO IN^2_4
;IX_4 ,2F ANY MODIFI$ EXPRES.N4
(1) ,! *ANGE F X: TO Y:_4
;X_4 ,2F ANY A2REVIAT$ FUNC;N "N
OR UNA2REVIAT$ FUNC;N "N1 PROVID$ T !
LATT] O3URS 9 A MA!MATICAL 3TEXT4
(1) ;Y IS PROPOR;NAL TO LOG X_4
;XI_4 ,2F ANY GR\P+ SYMBOL4
,NEME? ,MA?4 ,CODE1 #1972 A#FG
(1) ,F (1) TO (5)_4
;XII_4 ,2F ANY (! MISCELLANE\S
SYMBOLS ( ,RULE ,,XXII_4
(1) ,*ANGE TO @0_4
(2) ,*ANGE ! M"OY 9TO @S10 BILLS4
;D_4 ,! .ST & .TH 3TRAC;NS M/ N 2
US$ = ORD9AL 5D+S :5 ^! >E ATTA*$ TO
NUM]ALS1 LRS1 OR O!R MA!MATICAL EX-
PRES.NS4 ,IF AN ORD9AL 5D+ IS -POS$ (
ONLY "O LR1 FOLL[ ! 9K PR9T4
(1) #1ST, #2ND, #3RD, #4TH_4
(2) ;ITH, ;JTH, ;KTH, ''',
(N-1)TH_4
-----------------------------------#FH
(3) #2NTH
(4) #1ST & #2D_4
;E_4 ,! "O-CELL :OLE ^W ALPHABET
3TRAC;NS = .B1 .C1 '''1 .Y1 .Z #AHE
&! "O-CELL L[]-SIGN :OLE-^W A#FH
3TRAC;NS = ..BE1 5\<1 W]E1 HIS1 IN1
.WAS1 :E!R CAPITALIZ$1 ITALICIZ$1 OR
NEI1 M/ N 2 US$ :5 ^! ^WS >E 9 DIRECT
3TACT ) ANY GR\P+ SYMBOL4 ,! 3TRAC;NS1
:OLE-^W OR "P-^W1 = ..&1 =1 (1 !1 .)1
:E!R CAPITALIZ$1 ITALICIZ$1 OR NEI1 M/
AL N 2 US$ :5 9 DIRECT 3TACT ) ANY
GR\P+ SYMBOL4 ,IF ANY PUNCTU,N 9T]V5ES
2T A GR\P+ SYMBOL & ANY 3TRAC;N (!
TYPES M5;N$ ABV1 ! RULE / APPLIES4 ,:5
? RULE PRECLUDES ! USE (A 3TRAC;N 9 "O
"P (A ^W1 NO "P (! ^W MAY 2 3TRACT$4
(1) (AND1 9 A4I;N)
(2) (THAT IS)
(3) (NOT-;P)
(4) ,! ;X (IN ! EXAMPLE ABV)
REPRES5TS AN 9TEG]4
(5) (.OF C\RSE)
(6) (,FORMALLY A POLYNOMIAL)
(7) (OFFICIALLY )DRAWN)
(8) (_8,CAN Y F9D ! ANSW]80)
(9) (_8IN A4I;N1 X /.K #0_0)
(10) (_8,OF C\RSE NOT60)
,NEME? ,MA?4 ,CODE1 #1972 B#FH
(11) (,GIVE ! COMMAND)
;F_4 ,3TRAC;NS M/ N 2 US$ :5 !Y >E
LIKELY 6BE MISTAK5 = MA!MATICAL EX-
PRES.NS4
(1) ,USE THE ! 6F9D ! VOLUME4
-----------------------------------#FI
(2) ,CAN ,C .K #100_8
(3) A .K B, BUT B /.K C_4
(4) ,WE SEE THAT C .K D_4
(5) ,LET .THIS 2 AN ANGLE 9 /&>D
POSI;N4
(_8THIS0 IS SPELL$ \ 2C 9 IMMLY SUR-
R.D+ TEXT 9 9K PR9T ! ,GREEK LR
8!TA0 ( ) AL APPE>$)
@,S#56_4 ..,USE ( ,3TRAC;NS & ,%ORT-
,=M .,^WS3 ,SUBJECT 6! 3DI;NS (
@,S#55, ! USE ( 3TRAC;NS & %ORT-=M ^WS
( ,5GLI% ,BRL M/ 2 US$4
(1) #1 LI<T-YE> #AHG
,NEME? ,MA?4 ,CODE1 #1972 A#FI
(2) N-;P
(3) ;X-9T]CEPT
(4) UN>Y1 B9>Y1 '''1 ;M->Y4
(5) ?1/2#-(F SALE
(6) #9-9*
(7) HYDROG5-#3
(8) #360^.*"-9T]VAL
(9) =-\NCE
(THE OMIS.N SYMBOL REPLACES A "Q M>K
9 9K PR9T)
(10) (FT4-P.D)
(11) 9*-POUND^2
(12) #60 M94_/HOUR
(13) (RATE)@*("T) .K (4T.E)
-----------------------------------#GJ
(14) ENERGY .K MASS@*(SPE$ (
LI<T)^2
(15) (PEOPLE :O TRAVEL 0C>)_/(100
*N)
(16) .(,M>Y1 ,S,Y1 ,J1N.)
(17) .(,W$41 ,?URS4.)
(18) ,! HYPO!SIS AB ;AB IS UNT5A#4
(19) ,! A2REVI,N = 8S9E0 IS
8SIN_40 #AHI
(20) ,TURN TO P4 #92_4 A#GJ
,RULE ;,X--,OMIS.NS
,G5]AL ,OMIS.N ,SYMBOL =
@,S#57_4 .,OMIS.NS3 ,A L>GE NUMB] (
SIGNS >E EMPLOY$ 9 9K PR9T 6DENOTE
OMITT$ MA!MATICAL OR LIT]>Y MAT]IAL4
,:5 A "Q M>K1 EI 0XF OR 9 -B9,N ) HY-
PH5S OR DA%ES1 OR A BLANK SPACE IS EM-
PLOY$ 9 9K PR9T 6DENOTE OMIS.N1 ! G5-
]AL OMIS.N SYMBOL M/ 2 US$ 9 ! TRAN-
SCRIP;N4 ,! NUMB] ( G5]AL OMIS.N SYM-
BOLS 6BE US$ M/ 2 ! SAME Z ! OMIS.N
SIGNS 9 9K PR9T4 ,:5 A DA% IS US$ 6DE-
NOTE OMIS.N 9 9K PR9T1 ! L;G DA% M/ 2
US$ 9 ! TRANSCRIP;N4 ,IF AN OMIS.N
SIGN IS US$ 9 9K PR9T = : ? ,CODE
PROVIDES NO REPRES5T,N1 ? SIGN MAY 2
REPRES5T$ 0DRAW+ X IN1 OR ! TRANSCRIB]
MAY DEVISE A BRL SYMBOL 6REPRES5T X4
,9 ALL O!R CASES1 ! OMIS.N SYMBOL : IS
US$ M/ CORRESPOND 6! SIGN : APPE>S 9
9K PR9T4 ,ALL ( ^! RULES APPLY UN.S "W
,NEME? ,MA?4 ,CODE1 #1972 B#GJ
IS SPATI,Y >RANG$ = -PUT,N1 9 : CASE
SEE @,S#58_4
(1) (=)^3 .K #27
("Q M>K O3URS 9 9K PR9T)
(2) #92 IN4 .K = FT4 = IN4
("Q M>K O3URS 9 9K PR9T)
(3) #7@*2 = #14
("Q M>K O3URS 9 9K PR9T)
(4) =+= .K #10
("Q M>K O3URS 9 9K PR9T)
(5) #7-= .K #5
(A DA% O3URS 2N A "Q M>K 9 9K PR9T)
-----------------------------------#GA
(6) #9-5 .K =
(A "Q M>K IS PREC$$ & FOLL[$ 0A HY-
PH5 9 9K PR9T)
(7) (5, =)+(=, 15) .K (7, 13)
(BLANK SPACES O3UR 9 9K PR9T)
(8) #5@*25 .K =
(A BLANK SPACE O3URS 9 9K PR9T)
(9) FIVE@* ---- .K FIFTEEN
(A DA% O3URS 9 9K PR9T)
#AIA
(10) #2, #4, #6, ''', #10_4 A#GA
(AN ELLIPSIS O3URS 9 9K PR9T)
(11) ,! QK BR[N FOX '''4
(AN ELLIPSIS O3URS 9 9K PR9T)
(12) $4+$C .K #5
(A SQU>E & CIRCLE O3UR 9 9K PR9T)
(13) #5@*7 @,: #35
(AN EXT5D$ TILDE O3URS 9 9K PR9T)
@,S#58_4 ..,OMIS.NS 9 ,"W ,>RANG$
,SPATI,Y = .,-PUT,N3 ,9 "W >RANG$
SPATI,Y = -PUT,N1 ONLY ! G5]AL OMIS.N
SYMBOL MAY 2 US$ 9 BRL REG>D.S ( H[ !
OMIS.N IS DENOT$ 9 9K PR9T4 ,9 A4I;N1
! NUMB] ( G5]AL OMIS.N SYMBOLS 6BE US$
M/ 2 ! SAME Z ! NUMB] ( OMIS.N SIGNS :
O3UR 9 9K PR9T4
(1) 40
+70
-----
===
("Q M>KS >E %[N 9 9K PR9T)
,NEME? ,MA?4 ,CODE1 #1972 #GB
(2) 642
-===
------
453
("Q M>KS >E %[N 9 9K PR9T)
(3) 300
+500
------
=
(A "Q M>K IS %[N 9 9K PR9T)
(4) 651
@*252
--------
===2
===5
===2
CCCCCCCC
=====2
(DOTS >E %[N 9 9K PR9T) #AIC
(5) 144 #GC
-----
6O864
=
-----
==
==
-----
==
==
-----
(DOTS >E %[N 9 9K PR9T)
@,S#59_4 .,SPAC+ .) .,OMIS.NS3 ,!
G5]AL OMIS.N SYMBOL %D 2 SPAC$ 9 !
SAME MANN] Z ! MAT]IAL : X REPLACES4
,O!R OMIS.N SYMBOLS M/ 2 SPAC$ 9 AC-
CORD.E )! RULES GOV]N+ ! SPAC+ ( ^?
SYMBOLS4
,NEME? ,MA?4 ,CODE1 #1972 A#GC
,RULE ,,XI--,C.ELL,N
,C.ELL,N ,9DICATORS
,OP5+ [
,CLOS+ ]
@,S#60_4 .,C.ELL,N .,9DICATORS3 ,!
C.ELL,N 9DICATORS M/ 2 US$ 6%[ ! EXT5T
(A MA!MATICAL EXPRES.N : HAS BE5 C.EL$
9 9K PR9T4 ,A SPATIAL >RANGE;T M/ 2
US$ :5 C.ELL,N IS REPRES5T$ 9 BRL4
,:5"E A FRAC;N OR ANY ( XS "PS IS
C.EL$1 A SPATIAL >RANGE;T M/ 2 US$ = T
FRAC;N4 ,ITEMS : >E 9DIVIDU,Y C.EL$ 9
9K PR9T M/ 2 REPRES5T$ Z 9DIVIDU,Y
C.EL$ 9 ! TRANSCRIP;N4
-----------------------------------#GD
(1) #1
[X]
?3333#
[X]Y
;Y #AIE
(2) #1 A#GD
[5]
?3333# .K ?1/5#
[25]
#5
(3) [(X+Y)]
?333333333333# .K ?1/Y+Z#
[(X+Y)](Y+Z)
(4) H 9 9 12
[9][0][0][ 2]
- 3 6 9 3
----------------
5 3 0 9
(5) [X][Y]
?3333333#
[X][Y]Z
,NEME? ,MA?4 ,CODE1 #1972 #GE
,RULE ,,XII--,FRAC;NS
,FRAC;N ,9DICATORS
,SIMPLE
,OP5+ ?
,CLOS+ #
,-PLEX
,OP5+ ,?
,CLOS+ ,#
,HYP]COMPLEX
,OP5+ ,,?
,CLOS+ ,,#
,FRAC;NAL ,"P (A ,MIX$ ,NUMB]
,OP5+ _?
,CLOS+ _#
,FRAC;N ,L9ES
,US$ ) ,SIMPLE-,FRAC;N ,9DICATORS
,DIAGONAL L9E OR SLA% _/
,HORIZONTAL /
#AIG
,NEME? ,MA?4 ,CODE1 #1972 A#GE
,US$ )! ,FRAC;NAL ,"P (A ,MIX$
,NUMB]
,DIAGONAL L9E OR SLA% _/
,HORIZONTAL /
,US$ ) ,-PLEX-,FRAC;N ,9DICATORS
,DIAGONAL L9E OR SLA% ,_/
,HORIZONTAL ,/
-----------------------------------#GF
,US$ ) ,HYP]COMPLEX-,FRAC;N ,9DI-
CATORS
,HORIZONTAL ,,/
,US$ )A ,SPATIAL ,>RANGE;T
,HORIZONTAL (V>Y+ 9 L5G?) 33333
@,S#61_4 .,SIMPLE .,FRAC;NS3 ,=!
PURPOSES ( ? ,CODE1 A .SIMPLE #AII
,NEME? ,MA?4 ,CODE1 #1972 A#GF
.FRAC;N IS "O ^: NUM]ATOR & DENOM9ATOR
3TA9 NO FRAC;NS EXCEPT POSSIBLY AT !
SUP]SCRIPT OR SUBSCRIPT LEVEL4
@,S#62_4 ..,USE ( ,SIMPLE-,FRAC;N
.,9DICATORS3
;A_4 ,SIMPLE-FRAC;N 9DICATORS M/ 2
US$1 EXCEPT 9 ! CASE ( MIX$ NUMB]S1
65CLOSE A SIMPLE FRAC;N ^: NUM]ATOR &
DENOM9ATOR >E SEP>AT$ 0A HORIZONTAL
FRAC;N L9E 9 9K PR9T4
(1) ?1/3#
(2) X^?1/2#
(3) ?A+B/C#
(4) ?X^?1/2#"/2#
(5) RATE .K ?DISTANCE/TIME#
;B_4 ,SIMPLE-FRAC;N 9DICATORS M/ 2
US$ 65CLOSE A SIMPLE FRAC;N ^: NUM]A-
TOR & DENOM9ATOR >E SEP>AT$ 0A DIAGO-
NAL L9E 9 9K PR9T1 :5 ! EXPRES.NS ON E
SIDE (! DIAGONAL L9E APPE> AT DI6]5T
LEVELS RELATIVE 6X1 OR 9 DI6]5T TYPE
SIZE ?AN IS NORMAL =! PURPOSE #BJA
= : ^! EXPRES.NS >E US$4 B#GF
(1) ?A+B_/C+D#
(IN 9K PR9T1 ! NUM]ATOR IS WRITT5
NE> ! TOP (! DIAGONAL L9E &! DENOMI-
NATOR IS WRITT5 NE> ! BOTTOM)
(2) #3?X_/Y#
(IN 9K PR9T ! #3, ;X, & ;Y >E AT !
SAME LEVEL1 B ! ;X & ;Y >E 9 SMALL]
TYPE ?AN ! #3)
@,S#63_4 ..,NON-,USE ( ,SIMPLE-
,FRAC;N .,9DICATORS3
;A_4 ,SIMPLE-FRAC;N 9DICATORS M/
N 2 US$ 65CLOSE ! FRAC;NAL "P (A MIX$
NUMB]4
(1) #4_?3/8_#
(2) #2_?3_/4_#X
;B_4 ,SIMPLE-FRAC;N 9DICATORS M/ N
2 US$ 65CLOSE A SIMPLE FRAC;N ^:
NUM]ATOR & DENOM9ATOR >E SEP>AT$ 0A
DIAGONAL L9E 9 9K PR9T :5 ! EXPRES.NS
ON EI SIDE (! DIAGONAL L9E APPE> AT !
,NEME? ,MA?4 ,CODE1 #1972 C#GF
SAME LEVEL RELATIVE 6X1 OR >E (!
-----------------------------------#GG
SAME TYPE SIZE Z ! SURR.D+ MA!MATICAL
TEXT4 ,"S"TS ! EXPRES.NS ON EI SIDE (!
DIAGONAL L9E >E N ! T]MS (A FRAC;N AT
ALL4 ,EV5 :5 !Y >E1 ! TRANSCRIB] _C
ALW 2 C]TA9 ( ": ! FRAC;N 2G9S OR 5DS4
,ACLY1 X IS BETT] 6AVOID ! USE ( 9DI-
CATORS ALT 9 ^! CASES & P]MIT ! BRL
R1D] 6MAKE A JUDG;T BAS$ ON ! SAME IN-
=M,N T IS AVAILA# 6! SI<T$ R1D]4
(1) #1_/3
(IN 9K PR9T1 #1 & #3 >E AT ! SAME
LEVEL)
(2) X^1_/2
(IN 9K PR9T1 #1 & #2 >E AT ! SAME
LEVEL2 AL? ! #1 & #2 >E 9 SMALL]
TYPE1 !Y >E ( NORMAL SIZE = PR9T+
SUP]SCRIPTS)
(3) X^?1/2# _/2
(IN 9K PR9T1 ! ;X & #2 >E AT ! SAME
LEVEL & >E ( NORMAL SIZE = PR9T+
BASE-L9E SIGNS) #BJC
(4) X^1_/2 1_/7 A#GG
(IN 9K PR9T1 #1 & #2 >E AT ! SAME
LEVEL & ;X & #7 >E AT ! SAME LEVEL2
EA* PAIR ( SIGNS IS ( NORMAL SIZE =
PR9T+ AT XS RESPECTIVE LEVEL)
(5) A+B_/C+D
(IN 9K PR9T1 ALL LRS >E ( NORMAL
SIZE & AT ! SAME LEVEL ON EI SIDE (A
DIAGONAL L9E)
(6) (A+B)_/(C+D)
(IN 9K PR9T1 ALL LRS >E ( NORMAL
SIZE & AT ! SAME LEVEL ON EI SIDE (A
DIAGONAL L9E)
(7) ,I_/(N)
(THE EXPRES.NS ON EI SIDE (! DIAGONAL
L9E >E N ! T]MS (A FRAC;N)
(8) #1_/31_/70
(THE EXPRES.N REPRES5TS A DATE)
@,S#64_4 .,MIX$ .,NUMB]S3 ,=! PUR-
POSES ( ? ,CODE1 A MIX$ NUMB] IS AN
EXPRES.N : ;G9S )A NUM]AL & IS FOLL[$1
USU,Y 9 SMALL] TYPE1 0A SIMPLE FRAC;N
^: NUM]ATOR & DENOM9ATOR >E BO? NU-
M]ALS4 ,! FRAC;N L9E ( ? SIMPLE FRAC;N
,NEME? ,MA?4 ,CODE1 #1972 B#GG
MAY 2 EI HORIZONTAL OR DIAGONAL 9 9K
PR9T4 ,! MIX$-NUMB] 9DICATORS M/ 2 US$
65CLOSE ! FRAC;NAL "P (A MIX$ NUMB]4
,AN EXPRES.N IS N A MIX$ NUMB] IF X
3TA9S ANY LR1 EV5 ?\< S* AN EXPRES.N
IS (! SAME =M Z A MIX$ NUMB] 9 E O!R
RESPECT4
(1) #4_?3/8_#
(2) #4_?3_/8_#
(3) X?3/8#
(THIS IS N A MIX$ NUMB]2 FRAC;N 9
SMALL] TYPE ?AN ! ;X)
(4) X?3_/8#
(THIS IS N A MIX$ NUMB]2 FRAC;N 9
SMALL] TYPE ?AN ! ;X)
-----------------------------------#GH
(5) #3?1_/Y#
(THIS IS N A MIX$ NUMB]2 FRAC;N 9
SMALL] TYPE ?AN ! #3)
@,S#65_4 .,-PLEX .,FRAC;NS3 ,=! PUR-
POSES ( ? ,CODE1 A -PLEX FRAC;N #BJE
IS "O ^: NUM]ATOR1 DENOM9A- A#GH
TOR1 OR BO?1 3TA9S AT L1/ "O SIMPLE
FRAC;N4 ,A FRAC;N IS N A -PLEX FRAC;N
IF ! ONLY SIMPLE FRAC;NS X 3TA9S >E AT
! SUP]SCRIPT OR SUBSCRIPT LEVEL4
@,S#66_4 ..,USE ( ,-PLEX-,FRAC;N
.,9DICATORS3 ,-PLEX-FRAC;N 9DICATORS
M/ 2 US$ 65CLOSE A -PLEX FRAC;N4
(1) ,??3/8#,/5,#
(2) ,?1_/2,/2_?2/3_#,#
(3) ,?2_/3,/3_/2,#
(4) ,?5,/4_?3/8_#,#
(5) ,?3_/4,/5,#
(6) ,??1/2#,_/?3/4#,#
@,S#67_4 .,HYP]COMPLEX .,FRAC;NS3 ,=
! PURPOSES ( ? ,CODE1 A HYP]COMPLEX
FRAC;N IS "O ^: NUM]ATOR1 DENOM9ATOR1
OR BO?1 3TA9 AT L1/ "O -PLEX FRAC;N4
,A FRAC;N IS N A HYP]COMPLEX FRAC;N IF
! ONLY -PLEX FRAC;NS X 3TA9S >E AT !
SUP]SCRIPT OR SUBSCRIPT LEVEL4
,NEME? ,MA?4 ,CODE1 #1972 B#GH
(1) ?A/B^,??3/4#,/?5/6#,#"#
(THIS IS N A HYP]COMPLEX FRAC;N)
@,S#68_4 ..,USE ( ,HYP]COMPLEX-
,FRAC;N .,9DICATORS3
;A_4 ,HYP]COMPLEX-FRAC;N 9DICATORS
M/ 2 US$ 65CLOSE A HYP]COMPLEX FRAC;N4
,! USE (A L9E> >RANGE;T )9 A SPATIAL
>RANGE;T IS PREF]A# 6AN >RANGE;T : IS
5TIRELY L9E> OR 5TIRELY SPATIAL4
(1) ,?1_?1/4_#,/1_?3/5_#,#
,,?3333333333333333333333,,#
#5
(PREF]R$ ME?OD ( TRANSCRIB+ A HYP]-
-PLEX FRAC;N)
#BJG
(2) ,?(1-X)?D/DX#(2X) #GI
-#2X?D/DX#(1-X)
,/(1-X)^2",#
,,?33333333333333333,,#
#1+(?2X/1-X#)^2
(PREF]R$ ME?OD ( TRANSCRIB+ A HYP]-
-PLEX FRAC;N)
(3) ;D ;D
(1-X)?33#(2X)-2X?33#(1-X)
DX DX
,?3333333333333333333333333,#
(1-X)^2
,,?33333333333333333333333333333,,#
#2X
#1+(?3333#)^2
#1-X
(-PLETE SPATIAL >RANGE;T)
,NEME? ,MA?4 ,CODE1 #1972 #HJ
(4)
,,?,?(1-X)?D/DX#(2X)-2X?D/DX#(1-X)
,/(1-X)^2",#,,/1+(?2X/1-X#)^2",,#
(-PLETE L9E> >RANGE;T)
;B_4 ,HYP]COMPLEX FRAC;NS ( HI<]
ORD] MAY 2 TRANSCRIB$ 9 ! MANN] SU7E/$
BY ;A ABV4 ,X IS ONLY NEC 6USE DOT #6
! PROP] NUMB] ( "TS 2F ! FRAC;N 9DICA-
TORS & _! MAT*+ FRAC;N L9E4
@,S#69_4 .,3T9U$ .,FRAC;NS3 ,A 3T9U$
FRAC;N IS "O 9 : EA* DENOM9ATOR1 EX-
CEPT POSSIBLY ! LA/ "O1 IS ! SUM (A
:OLE NUMB] &A FRAC;N4 ,A SPATIAL >-
RANGE;T M/ 2 US$ = EA* FRAC;N4 ,9 ?
CASE1 EA* FRAC;N L9E M/ H PROPOR;NATE-
LY ! L5G? %[N 9 PR9T1 & FRAC;N 9DICA-
TORS ( ANY K9D M/ N 2 US$ )A 3T9U$
FRAC;N4
#BJI
(1) #1 A#HJ
>2] .K #1+333333333333
#1
#2+333333333
#1
#2+333333
#1
#2+333
#2+ '''_4
@,S#70_4 ..,SPATIAL ,>RANGE;T =
.,FRAC;NS3
;A_4 ,:5"E A FRAC;N IS TRANSCRIB$
SPATI,Y1 ALL FRAC;N 9DICATORS1 EXCEPT
= 3T9U$ FRAC;NS1 M/ 2 %[N1 & EA* FRAC-
;N L9E M/ H PRECISELY ! L5G? NEC 6COV]
! L;GE/ EXPRES.N 6: X APPLIES1 &! T]MS
(! FRAC;N
-----------------------------------#HA
M/ 2 C5T]$ ON _! FRAC;N L9ES4 ,! RUN-
OV] ( AN EXPRES.N : IS TOO L;G 6BE C5-
T]$ ON ! FRAC;N L9E : APPLIES 6X MAY 2
E6ECT$ AT SUITA# PLACES 9 A3ORD.E )!
RULES = RUNOV]S1 B EA* POR;N (! DIVID$
EXPRES.N M/ 2 C5T]$ ON ! FRAC;N L9E 6:
,NEME? ,MA?4 ,CODE1 #1972 A#HA
! EXPRES.N1 Z A :OLE1 APPLIES4
;B_4 ,9 G5]AL1 ! L9E> >RANGE;T =
FRAC;NS M/ 2 US$ :5 N EXPRESSLY =BI45
9 ! CASE ( 3T9U$ FRAC;NS4 ,H["E1 :5
FRAC;NAL NOT,N IS F/ PRES5T$ 6! R1D]1
Z 9 ! L[] GRADES1 OR :5 "! IS ANY O!R
SPECIAL NE$1 ANY FRAC;N MAY 2 REPRE-
S5T$ SPATI,Y4
(1) #1+2
?3333#
#2+4
(2) ;X
?33#
;Y
(3) 4T.E
RATE .K ?3333#
"T
#BAA
(4) B#HA
#5280 FT #60 MI #1 HR #1 M9
?33333333#?333333#?333333#?3333333#
#1 MI #1 HR #60 M9 #60 SEC
#88 FT
.K ?333333# .K #88 FT_/SEC4
#1 SEC
;C_4 ,= SPATIAL >RANGE;T ( FRAC;NS
9 3NEC;N ) C.ELL,N SEE @,S#60_4
-----------------------------------#HB
;D_4 ,= SPATIAL >RANGE;T ( HYP]-
-PLEX FRAC;NS SEE @,S#68_4
;E_4 ,= SPATIAL >RANGE;T ( 3T9U$
FRAC;NS SEE @,S#69_4
,RULE ,,XIII--
,SUP]SCRIPTS & ,SUBSCRIPTS
,LEVEL ,9DICATORS
,BASE ,L9E "
,SUP]SCRIPT ^
,SUP]SCRIPT ) ,SUP]SCRIPT ^^
,NEME? ,MA?4 ,CODE1 #1972 A#HB
,SUP]SCRIPT ) ,SUBSCRIPT ^2
,SUP]SCRIPT ) ,SUP]SCRIPT ) ,SUP]-
SCRIPT ^^^
,SUP]SCRIPT ) ,SUP]SCRIPT ) ,SUB-
SCRIPT ^^;
,SUP]SCRIPT ) ,SUBSCRIPT ) ,SUP]-
SCRIPT ^;^
,SUP]SCRIPT ) ,SUBSCRIPT ) ,SUB-
SCRIPT ^;;
,SUBSCRIPT ;
,SUBSCRIPT ) ,SUP]SCRIPT ;^
,SUBSCRIPT ) ,SUBSCRIPT ;;
,SUBSCRIPT ) ,SUP]SCRIPT ) ,SUP]-
SCRIPT ;^^
,SUBSCRIPT ) ,SUP]SCRIPT ) ,SUB-
SCRIPT ;^;
,SUBSCRIPT ) ,SUBSCRIPT ) ,SUP]-
SCRIPT ;;^
,SUBSCRIPT ) ,SUBSCRIPT ) ,SUB-
SCRIPT ;;;
,3TRAC;N = ,-MA & ,OP;NAL ,SPACE AT
,SUP]SCRIPT OR ,SUBSCRIPT
,LEVEL [
#BAC
@,S#71_4 ..,NATURE ( ,SUP]- B#HB
SCRIPTS & .,SUBSCRIPTS3 ,X IS "*I/IC (
MA!MATICAL EXPRES.NS 6EMPLOY SIGNS1
USU,Y 9 SMALL] TYPE1 : >E ELEVAT$ OR
DEPRESS$ RELATIVE 6! .BASE .L9E4 ,A
SIGN : IS ELEVAT$ RELATIVE 6! BASE L9E
IS CALL$ A .SUP]SCRIPT2 "O : IS DE-
PRESS$ RELATIVE 6! BASE L9E IS CALL$ A
.SUBSCRIPT4 ,:5 AN 5TIRE EXPRES.N IS
AT ! SUP]SCRIPT OR SUBSCRIPT LEVEL1 X
%D 2 WRITT5 )\T AN 9DICATOR 9 BRL1 B
XS POSI;N M/ 2 EXPLA9$ 6! R1D] 0A
TRANSCRIB]'S NOTE4
-----------------------------------#HC
(1) ,9 X^2, #2 ,'ELEVAT$ 9 PR9T,'
IS A SUP]SCRIPT4
(2) ,9 X;N, ;N ,'DEPRESS$ 9 PR9T,'
IS ! SUBSCRIPT4
(3) ,! SIGN = .DEGREE IS .*_4
,'ELEVAT$ 9 PR9T,'
@,S#72_4 ..,HI]>*Y ( ,SUP]SCRIPTS &
.,SUBSCRIPTS3 ,SUP]SCRIPTS OR SUB-
SCRIPTS MAY C>RY SUP]SCRIPTS OR SUB-
SCRIPTS ( _! [N2 ! LATT] >E !N REF]R$
,NEME? ,MA?4 ,CODE1 #1972 A#HC
TO Z SUP]SCRIPTS OR SUBSCRIPTS ( .SEC-
OND .ORD]1 & >E ?US 4T+UI%$ F ! =M]1 :
>E CALL$ SUP]SCRIPTS OR SUBSCRIPTS (
.F/ .ORD]4 ,SECOND ORD] SUP]SCRIPTS OR
SUBSCRIPTS MAY1 9 TURN1 C>RY SUP]-
SCRIPTS OR SUBSCRIPTS ( _! [N1 : >E !N
CALL$ SUP]SCRIPTS OR SUBSCRIPTS (
.?IRD .ORD]4 ,:ILE X IS !ORETIC,Y POS-
SI# =A SUP]SCRIPT OR SUBSCRIPT 6BE (
ORD] HI<] ?AN ! ?IRD1 ? SITU,N R>ELY
>ISES 9 PRACTICE4
@,S#73_4 .,LEVEL .,9DICATORS3 ,A
LEVEL 9DICATOR O!R ?AN ! BASE-L9E 9DI-
CATOR ID5TIFIES ! SYMBOLS : FOLL[ X Z
REPRES5T+ A SUP]SCRIPT OR SUBSCRIPT4
,! .BASE-L9E 9DICATOR ID5TIFIES ! SYM-
BOLS : FOLL[ X Z REPRES5T+ SIGNS ON !
BASE L9E4 ,! DEGREE ( ELEV,N OR DE-
PRES.N SPECIFI$ 0A LEVEL 9DICATOR IS
..ALW ) RETPECT 6! BASE .L9E2 ! SYMBOL
: PREC$ES ! 9DICATOR1 IF X REPRES5TS A
SIGN AT "S O!R LEVEL1 PLAYS NO ROLE 9
? REG>D4
#BAE
@,S#74_4 ..,ORI5T,N 0,LEVEL B#HC
.,9DICATOR3
;A_4 ,! E6ECT (A LEVEL 9DICATOR )
"O -PON5T IS 6DIRECT ! R1D]'S ATT5;N
UPW>D OR D[NW>D F ! BASE L9E AC Z T
-PON5T IS ! SUP]SCRIPT OR ! SUBSCRIPT
9DICATOR4
(1) X^2
(;X SQU>$)
(2) Y^3
(;Y CUB$)
(3) X^@#
(;X C>RIES AN A/]ISK Z A SUP]SCRIPT)
(4) X^-2
(;X C>RIES M9US #2 Z A SUP]SCRIPT)
-----------------------------------#HD
(5) X;A
(;X SUB ;A)
(6) X;-2
(;X C>RIES M9US #2 Z A SUBSCRIPT)
;B_4 ,! E6ECT (A LEVEL 9DICATOR )
TWO -PON5TS MAY 2 ANALYZ$ Z FOLL[S3
;I_4 ,! F/ -PON5T DIRECTS ! R1D-
,NEME? ,MA?4 ,CODE1 #1972 A#HD
]'S ATT5;N UPW>D OR D[NW>D F ! BASE
L9E AC Z T -PON5T IS1 9 XF1 ! SUP]-
SCRIPT OR SUBSCRIPT 9DICATOR4
;II_4 ,! SECOND -PON5T !N DI-
RECTS ! R1D]'S ATT5;N UPW>D OR D[NW>D
F ? NEW POSI;N AC Z ! SECOND -PON5T
IS1 9 XF1 ! SUP]SCRIPT OR SUBSCRIPT
9DICATOR4
(1) N^X^^Y
(;N C>RIES A SUP]SCRIPT ;X : C>RIES
A SUP]SCRIPT ;Y)
(2) X^N^;A
(;X C>RIES A SUP]SCRIPT ;N : C>RIES
A SUBSCRIPT ;A)
(3) X;N;^A
(;X C>RIES A SUBSCRIPT ;N : C>RIES A
SUP]SCRIPT ;A)
(4) N;X;;Y
(;N C>RIES A SUBSCRIPT ;X : C>RIES A
SUBSCRIPT ;Y)
;C_4 ,! E6ECT (A LEVEL 9DICATOR )
?REE -PON5TS MAY 2 ANALYZ$ Z #BAG
FOLL[S3 B#HD
;I_4 ,! F/ TWO -PON5TS DIRECT !
R1D]'S ATT5;N F ! BASE L9E 6! POSI;N
DESCRIB$ 9 ;B ABV4
;II_4 ,! ?IRD -PON5T !N DIRECTS
! R1D]'S ATT5;N UPW>D OR D[NW>D F ?
NEW POSI;N1 AC Z T -PON5T IS1 9 XF1 !
SUP]SCRIPT OR ! SUBSCRIPT 9DICATOR4
(1) N^X^^Y^^^Z
(;N C>RIES A SUP]SCRIPT ;X : C>RIES
A SUP]SCRIPT ;Y : C>RIES A SUP]-
SCRIPT ;Z)
(2) N^X^^Y^^^Z^^^^'''
(THE ELLIPSIS 9DICATES ! PRES;E (
SUP]SCRIPTS ( 9CR1S+LY HI<] ORD]2 !
DOTS >E PR9T$ OBLIQUELY)
(3) X^Y^^Z^^;A
(;X C>RIES A SUP]SCRIPT ;Y : C>RIES
A SUP]SCRIPT ;Z : C>RIES A SUBSCRIPT
;A)
(4) X^Y^;A^;^N
(;X C>RIES A SUP]SCRIPT ;Y : C>RIES
A SUBSCRIPT ;A : C>RIES A SUP]SCRIPT
;N)
,NEME? ,MA?4 ,CODE1 #1972 #HE
(5) N^X^;A^;;J
(;N C>RIES A SUP]SCRIPT ;X : C>RIES
A SUBSCRIPT ;A : C>RIES A SUBSCRIPT
;J)
(6) X;A;^R;^^N
(;X C>RIES A SUBSCRIPT ;A : C>RIES A
SUP]SCRIPT ;R : C>RIES A SUP]SCRIPT
;N)
(7) X;A;^N;^;B
(;X C>RIES A SUBSCRIPT ;A : C>RIES A
SUP]SCRIPT ;N : C>RIES A SUBSCRIPT
;B)
(8) X;P;;A;;^M
(;X C>RIES A SUBSCRIPT ;P : C>RIES A
SUBSCRIPT ;A : C>RIES A SUP]SCRIPT
;M)
(9) N;X;;Y;;;Z
(;N C>RIES A SUBSCRIPT ;X : C>RIES A
SUBSCRIPT ;Y : C>RIES A SUBSCRIPT
;Z)
(10) N;X;;Y;;;Z;;;;'''
(THE ELLIPSIS 9DICATES ! PRES;E (
SUBSCRIPTS ( 9CR1S+LY HI<] ORD]2 !
DOTS >E PR9T$ OBLIQUELY) #BAI
;D_4 ,! E6ECT (A LEVEL A#HE
9DICATOR ) M ?AN ?REE -PON5TS MAY 2
ANALYZ$ 9 ! SAME MANN] SU7E/$ = LEVEL
9DICATORS ) TWO OR ?REE -PON5TS4
@,S#75_4 ..,LEFT ,SUP]SCRIPTS &
.,SUBSCRIPTS3 ,A SUP]SCRIPT OR SUB-
SCRIPT MAY O3UPY A POSI;N 6! LEFT1 Z
WELL Z 6! "R1 (! SIGN 6: X APPLIES4 ,!
^WS .LEFT OR ."R >E !N US$ )! ^WS
.SUP]SCRIPT OR .SUBSCRIPT 6MAKE ! DIS-
T9C;N 9 POSI;N4
,A "R OR LEFT SUP]SCRIPT OR SUB-
SCRIPT IS REPRES5T$ Z S* M]ELY 0PRE-
S]V+ ! RELATIVE HORIZONTAL POSI;NS (!
SUP]SCRIPT OR SUBSCRIPT SYMBOL &! SYM-
BOL 6: X APPLIES4 ,EA* M/ 2 PREC$$ 0XS
APPROPRIATE LEVEL 9DICATOR4
,LEFT SUP]SCRIPTS OR SUBSCRIPTS (!
?IRD OR HI<] ORD]1 AL? R>E1 >E TR1T$
9 ! MANN] SU7E/$ 0! EXAMPLES 2L4
(1) ^X"N
(;X IS A LEFT SUP]SCRIPT TO ;N)
(2) ^-"X
(THE M9US SIGN IS A LEFT SUP]SCRIPT
,NEME? ,MA?4 ,CODE1 #1972 B#HE
TO ;X)
(3) ;X"N
(;X IS A LEFT SUBSCRIPT TO ;N)
(4) ;X"N;Y
(;X IS A LEFT SUBSCRIPT TO ;N, ;Y IS
A "R SUBSCRIPT TO ;N)
-----------------------------------#HF
(5) #10^^-^4
(10 6! M9US #4 P[]2 ! M9US IS SUP]-
SCRIPT 6! #4)
(6) ^N^;A"X
(;N SUB ;A IS A LEFT SUP]SCRIPT TO
;X)
(7) ^;A^N"X
(;A IS A LEFT SUBSCRIPT TO ;N, !
-B9,N IS A LEFT SUP]SCRIPT TO ;X)
(8) ;N;^A"X
(;N 6! ;A P[] IS A LEFT SUBSCRIPT TO
;X)
(9) ;^A;N"X
(;A IS A LEFT SUP]SCRIPT TO ;N, !
-B9,N IS A LEFT SUBSCRIPT TO ;X)
(10) ;X;;Y"N
(;X SUB ;Y IS A LEFT SUB- #BBA
SCRIPT TO ;N) A#HF
(11) ;;Y;X"N
(;Y IS A LEFT SUBSCRIPT TO ;X, !
-B9,N IS A LEFT SUBSCRIPT TO ;N)
(12) P^B^C"X
(;P C>RIES A "R SUP]SCRIPT ;B & ;C
IS A LEFT SUP]SCRIPT TO ;X)
@,S#76_4 ..,DIRECT ,SUP]SCRIPTS &
.,SUBSCRIPTS3 ,A SUP]SCRIPT OR SUB-
SCRIPT : O3UPIES1 RESPECTIVELY1 A PO-
SI;N DIRECTLY OV] OR "U ! SIGN 6: X
APPLIES IS CALL$ A .MODIFI] (SEE ,RULE
,,XIV)_4
@,S#77_4 .,NUM]IC .,SUBSCRIPTS3 ,!
SUBSCRIPT 9DICATOR M/ N 2 US$ 69DICATE
A NUM]IC SUBSCRIPT PROVID$ T ALL (!
FOLL[+ 3DI;NS HOLD3
;I_4 ,! CORRESPOND+ NUM]IC SIGN
M/ 2 A ."R1 & N A LEFT1 SUBSCRIPT4
;II_4 ,! CORRESPOND+ NUM]IC SIGN
M/ 2 A SUBSCRIPT ( .F/ .ORD]1 & N (
HI<] ORD]4
;III_4 ,! SIGN ) : ! NUM]IC
SUBSCRIPT IS ASSOCIAT$ M/ 2 AN A2REVI-
,NEME? ,MA?4 ,CODE1 #1972 B#HF
AT$ FUNC;N "N OR A LR : HAS A SEP>ATE
ID5T;Y4 ,9 ! LATT] CASE1 ? LR M/ N 2
ANY LR : REPRES5TS A NUM]AL 9 A NON-
DECIMAL BASE4 ,O!RWISE1 ! LR MAY 2 F
ANY ALPHABET & 9 ANY TYPE =M1 & MAY 2
MODIFI$ 0"O OR M PRIMES1 A SUP]SCRIPT1
OR A DIRECT MODIFI]4 ,9 ! CASE (A TWO-
LR A2REVI,N =A *EMICAL -P.D1 ! A2REVI-
,N M/ 2 TR1T$ Z IF X 7 A LR4
;IV_4 ,! SUBSCRIPT M/ 3SI/ (
NUM]IC SYMBOLS ONLY1 & M/ C>RY NO
SUP]SCRIPTS OR SUBSCRIPTS ( XS [N4
(1) X1
(;X SUB #1_2 SUBSCRIPT 9DICATOR N
REQUIR$ 2C ALL 3DI;NS ;I-;IV HOLD)
-----------------------------------#HG
(2) X11
(;X SUB #1 #1_2 SUBSCRIPT 9DICATOR N
REQUIR$ 2C ALL 3DI;NS ;I-;IV HOLD)
(3) _,A1
(,G]MAN CAPITALIZ$ AH SUB #1_2
SUBSCRIPT 9DICATOR N REQUIR$ 2C ALL
3DI;NS ;I-;IV HOLD) #BBC
(4) X'1 A#HG
(;X PRIME SUB #1_2 SUBSCRIPT 9DICA-
TOR N REQUIR$ 2C ALL 3DI;NS ;I-;IV
HOLD)
(5) X''2
(;X D\# PRIME SUB #2_2 SUBSCRIPT
9DICATOR N REQUIR$ 2C ALL 3DI;NS
;I-;IV HOLD)
(6) ;3"X
(3 IS A LEFT SUBSCRIPT TO ;X_2 SUB-
SCRIPT 9DICATOR IS REQUIR$ 2C 3DI;N
;I DOES N HOLD)
(7) X;I;;1
(;X SUB ;I SUB #1_2 SUBSCRIPT 9DI-
CATOR IS REQUIR$ 2C 3DI;N ;II DOES N
HOLD)
(8) LOG2 X
(LOG 6! BASE #2 ( ;X_2 SUBSCRIPT
9DICATOR N REQUIR$ 2C ALL 3DI;NS
;I-;IV HOLD)
(9) #12;7
(12 SUB #7_2 SUBSCRIPT 9DICATOR IS
REQUIR$ 2C 3DI;N ;III DOES N HOLD)
(10) (,C,O3);2
(THE C>BONATE RADICAL TAK5 TWICE2
,NEME? ,MA?4 ,CODE1 #1972 B#HG
SUBSCRIPT 9DICATOR IS REQUIR$ 2F !
#2 2C 3DI;N ;III DOES N HOLD)
(11) ,NA2,C,O3
(SODIUM C>BONATE2 SUBSCRIPT 9DICATOR
N REQUIR$ 2C ALL 3DI;NS ;I-;IV HOLD)
(12) SEVEN;3
(SEV5 SUB #3_2 SUBSCRIPT 9DICATOR IS
REQUIR$ 2C 3DI;N ;III DOES N HOLD)
(13) X;1;;J
(;X SUB #1 SUB ;J_2 SUBSCRIPT 9DI-
CATOR IS REQUIR$ 2C 3DI;N ;IV DOES N
HOLD)
(14) X;2;^N
(;X C>RIES A SUBSCRIPT #2 : C>RIES A
SUP]SCRIPT ;N_2 SUBSCRIPT 9DICATOR
IS REQUIR$ 2C 3DI;N ;IV DOES N HOLD)
(15) X;2,
(;X SUB #2 PRIME2 SUBSCRIPT 9DICATOR
IS REQUIR$ 2C 3DI;N ;IV DOES N HOLD)
-----------------------------------#HH
(16) X;2+K
(;X C>RIES A SUBSCRIPT ( #2 PLUS
;K_2 SUBSCRIPT 9DICATOR IS REQUIR$
2C 3DI;N ;IV DOES N HOLD) #BBE
(17) X;1_/2 A#HH
(;X SUB "O-HALF2 SUBSCRIPT 9DICATOR
IS REQUIR$ 2C 3DI;N ;IV DOES N HOLD)
(18) ;3"X1
(3 IS A LEFT SUBSCRIPT TO ;X, #1 IS
A "R SUBSCRIPT TO ;X_2 SUBSCRIPT
9DICATOR IS REQUIR$ 2F ! #3 2C 3DI;N
;I DOES N HOLD)
(19) ,A;X1
(;,A SUB ;X #1_2 SUBSCRIPT 9DICATOR
IS REQUIR$ 2C 3DI;N ;IV DOES N HOLD)
(20) X10,000
(;X SUB #10,000_2 SUBSCRIPT 9DICATOR
N REQUIR$ 2C ALL 3DI;NS ;I-;IV HOLD)
(21) X1.2
(;X SUB #1.2_2 SUBSCRIPT 9DICATOR N
REQUIR$ 2C ALL 3DI;NS ;I-;IV HOLD)
(22) X.6
(;X SUB #.6_2 SUBSCRIPT 9DICATOR N
REQUIR$ 2C ALL 3DI;NS ;I-;IV HOLD)
(23) .,S0^N"A;K
(THE SUMM,N F Z]O TO ;N (A SUB ;K_2
SUBSCRIPT 9DICATOR IS N REQUIR$ 2C
ALL 3DI;NS ;I-;IV HOLD)
,NEME? ,MA?4 ,CODE1 #1972 B#HH
(24) .,P0^N"A;K
(THE PRODUCT F Z]O TO ;N ( ;A SUB
;K_2 SUBSCRIPT 9DICATOR IS N REQUIR$
2C ALL 3DI;NS ;I-;IV HOLD)
(25) #3AF;16
(;,A & ;,F REPRES5T A NUM]AL 9 BASE
#16_2 SUBSCRIPT 9DICATOR IS REQUIR$
2C 3DI;N ;III DOES N HOLD)
(26) !;0^>1-X^^2^]"F(X)DX
(THE 9TEGRAL F #0 6! SQU>E ROOT (
#1-X^2 ( ;F ( ;X DX_2 SUBSCRIPT
9DICATOR IS REQUIR$ 2C 3DI;N ;III
DOES N HOLD)
@,S#78_4 ..,-MA AT ,SUP]SCRIPT OR
,SUBSCRIPT .,LEVEL3 ,A -MONLY O3URR+
SUP]SCRIPT OR SUBSCRIPT NOT,N IS ! "O
9 : TWO 3SECUTIVE ITEMS >E SEP>AT$ 0A
-MA ) AN OP;NAL SPACE FOLL[+ ! -MA4 ,9
? 3FIGUR,N1 ! SYMBOL [ (DOTS #2-4-6)
M/ 2 US$ 6REPLACE ! -MA &! OP;NAL
SPACE US$ 9 ? WAY4 ,? 3TRACT$ =M M/ N
2 US$ 6REPLACE A -MA &! OP;NAL SPACE :
FOLL[S X 9 A 3FIGUR,N : IS ON ! #BBG
BASE L9E4 C#HH
(1) X;I[J[K
(EA* -MA IS FOLL[$ 0A SPACE 9 9K
PR9T)
-----------------------------------#HI
(2) X;(A[B)
(THE -MA IS N FOLL[$ 0A SPACE 9 9K
PR9T)
(3) X;1[2
(THE -MA IS FOLL[$ 0A SPACE 9 9K
PR9T)
(4) ,P;N;;X[Y
(THE -MA IS FOLL[$ 0A SPACE 9 9K
PR9T)
(5) X;N-1[N-1, X;N-1[N, X;N[N-1
(THE -MA & SPACE 2T ! ITEMS ON !
BASE L9E _C 2 3TRACT$)
(6) (X, Y)
(THE -MA & SPACE 2T ! ITEMS ON !
BASE L9E _C 2 3TRACT$)
@,S#79_4 ..,CIRCUM/.ES ,DET]M9+
,*ANGES ( .,LEVEL3 ,! SYMBOLS & SITUA-
;NS LI/$ 2L H ! FOLL[+ E6ECT 9 DET]M9+
,NEME? ,MA?4 ,CODE1 #1972 A#HI
*ANGES ( LEVEL4
;A_4 ,A LEVEL 9DICATOR T]M9ATES !
E6ECT (A PREVI\S LEVEL 9DICATOR &
9ITIATES ! LEVEL IMPLI$ 0! NEW 9DICA-
TOR4 ,9 ! CASE (! BASE-L9E LEVEL1 !
PREVI\S BASE-L9E 9DICATOR MAY ONLY 2
IMPLI$4
(1) X^2"+1
(THE SUP]SCRIPT 9DICATOR T]M9ATES !
PREVI\S IMPLI$ BASE-L9E LEVEL &
9ITIATES ! SUP]SCRIPT LEVEL1 ! BASE-
L9E 9DICATOR T]M9ATES ! PREVI\S
SUP]SCRIPT LEVEL & 9ITIATES ! BASE-
L9E LEVEL)
(2) X;A"+Y^2
(THE SUBSCRIPT 9DICATOR T]M9ATES !
PREVI\S IMPLI$ BASE-L9E LEVEL &
9ITIATES ! SUBSCRIPT LEVEL1 ! BASE-
L9E 9DICATOR T]M9ATES ! PREVI\S SUB-
SCRIPT LEVEL & 9ITIATES ! BASE-L9E
LEVEL1 ! SUP]SCRIPT 9DICATOR T]MI-
NATES ! PREVI\S BASE-L9E LEVEL &
9ITIATES ! SUP]SCRIPT LEVEL) #BBI
(3) ?E^X^^2"/2# B#HI
(THE SUP]SCRIPT 9DICATOR T]M9ATES !
PREVI\S IMPLI$ BASE-L9E LEVEL & 9I-
TIATES ! F/-ORD] SUP]SCRIPT LEVEL1 !
SECOND-ORD] SUP]SCRIPT 9DICATOR T]-
M9ATES ! PREVI\S F/-ORD] SUP]SCRIPT
LEVEL & 9ITIATES ! SECOND-ORD] SUP]-
SCRIPT LEVEL1 ! BASE-L9E 9DICATOR
T]M9ATES ! PREVI\S SECOND-ORD] SUP]-
SCRIPT LEVEL & 9ITIATES ! BASE-L9E
LEVEL)
(4) ,A^N+N+N ''' ^TO ^M ^N^_'S
.K A^MN
(THE SUP]SCRIPT 9DICATOR : FOLL[S !
;N PRES]VES ! E6ECT (! PREC$+ SUP]-
SCRIPT 9DICATOR2 O!RWISE1 ! PUNCTU,N
9DICATOR WD T]M9ATE ! E6ECT (! PRE-
VI\S LEVEL 9DICATOR & 9ITIATE !
BASE-L9E LEVEL)
-----------------------------------#IJ
;B_4 ,! PUNCTU,N 9DICATOR T]M9ATES
! E6ECT ( ANY PREVI\S LEVEL 9DICATOR &
9ITIATES ! BASE-L9E LEVEL4 ,9 A4I;N1 !
-MA1 PROVID$ X IS N A NUM]IC SYMBOL1
T]M9ATES ! E6ECT ( ANY PREVI\S 9DICA-
,NEME? ,MA?4 ,CODE1 #1972 A#IJ
TOR & 9ITIATES ! BASE-L9E LEVEL4 ,H[-
"E1 ! -MA1 :5 X IS A NUM]IC SYMBOL &!
3TRACT$ =M [ (DOTS #2-4-6), PRES]VES !
LEVEL T IS ALR 9 E6ECT4
(1) X^2_4
(THE P]IOD IS AT ! BASE-L9E LEVEL)
(2) X^2, X^3
(THE -MA IS AT ! BASE-L9E LEVEL)
(3) X^10,000
(THE -MA IS A NUM]IC SYMBOL & PRE-
S]VES ! SUP]SCRIPT LEVEL)
(4) X;I[J
(THE 3TRACT$ =M =A -MA & OP;NAL
SPACE PRES]VES ! SUBSCRIPT LEVEL T
IS ALR 9 E6ECT)
(5) ,P;N;;1;[N;;2;[ '''
(THE 3TRACT$ =M =A -MA & OP;NAL SPAC
PRES]VES ! 9DICAT$ SUBSCRIPT LEVEL)
;C_4 ,A SPACE OR ! TRANSI;N 6A NEW
BRL L9E : IS FOLL[$ 0LIT]>Y TEXT OR
UNRELAT$ MA!MATICAL TEXT T]M9ATES !
E6ECT ( ANY PREVI\S LEVEL 9DI- #BCA
CATOR & 9ITIATES ! BASE-L9E B#IJ
LEVEL4 ,H["E1 IF A SPACE O3URS 2T !
"PS ( AN A2REVI,N OR PHRESE1 ! APPRO-
PRIATE LEVEL 9DICATOR M/ 2 RE/AT$ 2F
EA* "P4
(1) #2P^2 IS ALW EV54
(THE SPACE 2F ! LIT]>Y TEXT T]M9ATES
! PREVI\S SUP]SCRIPT LEVEL & 9ITI-
ATES ! BASE-L9E LEVEL)
(2) #3@*10^4 ]GS
(THE SPACE 2F ! LIT]>Y TEXT T]M9ATES
! PREVI\S SUP]SCRIPT LEVEL & 9ITI-
ATES ! BASE-L9E LEVEL)
(3) #6.696@*10^8 MPH
(THE SPACE 2F ! LIT]>Y TEXT T]M9ATES
! PREVI\S SUP]SCRIPT LEVEL & 9ITI-
ATES ! BASE-L9E LEVEL)
(4) (X^2 Y^2")
(THESE ITEMS >E 5TRIES 9 A MATRIX &
H;E UNRELAT$2 ! SPACE T]M9ATES !
PREVI\S SUP]SCRIPT LEVEL & 9ITIATES
! BASE-L9E LEVEL)
(5) $T;REG4 ;POLYGON_4
(LEVEL 9DICATORS >E REQUIR$ 2T EA*
,NEME? ,MA?4 ,CODE1 #1972 C#IJ
"P ( ? A2REVI,N 6%[ T !Y >E BO? AT !
SUBSCRIPT LEVEL)
(6) $T;REGULAR ;POLYGON
(LEVEL 9DICATORS >E REQUIR$ 2T EA*
"P ( ? PHRASE 6%[ T !Y >E BO? AT !
SUBSCRIPT LEVEL)
-----------------------------------#IA
;D_4 ,! SPACE : IMMLY FOLL[S A
SYMBOL ( %APE1 AN A2REVIAT$ FUNC;N "N1
OR AN UNA2REVIAT$ FUNC;N "N1 PROVID$ !
LATT] IS 9 A MA!MATICAL 3TEXT1 PRE-
S]VES ! LEVEL T IS ALR 9 E6ECT4 ,IF ^!
ITEMS C>RY A SUP]SCRIPT OR SUBSCRIPT1
! SPACE : FOLL[S S* A SUP]SCRIPT OR
SUBSCRIPT RE9/ATES ! LEVEL T 0 9 E6ECT
2F4
(1) B;$T ,A,B,C
(THE SPACE PRES]VES ! SUBSCRIPT LEV-
EL AT : ! TRIANGLE APPE>S)
(2) SIN X
(THE SPACE PRES]VES ! BASE-L9E LEVEL
( SIN)
#BCC
(3) COS^2 X A#IA
(THE SPACE RE9/ATES ! BASE-L9E LEVEL
( COS)
(4) E^SIN X
(THE SPACE PRES]VES ! SUP]SCRIPT
LEVEL AT : SIN APPE>S)
(5) E^SIN X+ICOS X
(EA* SPACE PRES]VES ! SUP]SCRIPT
LEVEL AT : ! A2REVIAT$ FUNC;N "NS
APPE>)
(6) E^X+LN X
(THE SPACE PRES]VES ! SUP]SCRIPT
LEVEL AT : LN APPE>S)
(7) E^COS^^2 X
(THE SPACE RE9/ATES ! SUP]SCRIPT
LEVEL AT : COS APPE>S)
(8) E^SIN^^2 X+SIN^^2 Y
(EA* SPACE RE9/ATES ! SUP]SCRIPT
LEVEL AT : ! A2REVIAT$ FUNC;N "NS
APPE>)
(9) Q^LOG^;Q A
(THE SPACE RE9/ATES ! SUP]SCRIPT
LEVEL AT : LOG APPE>S)
(10) ,V;MAX (M[N)
(THE SPACE PRES]VES ! SUBSCRIPT LEV-
,NEME? ,MA?4 ,CODE1 #1972 B#IA
EL AT : MAX APPE>S)
;E_4 ,! SPACE : O3URS 9 A NUM]AL =
! PURPOSE ( DIVID+ X 96%ORT REGUL>
SEG;TS PRES]VES ! LEVEL ALR 9 E6ECT4
(1) E^3.14159 26535
;F_4 ,! SPACE : PREC$ES AN
ELLIPSIS OR L;G DA% PRES]VES ! E6ECT (
ANY PREVI\S LEVEL 9DICATOR4 ,! SPACE :
FOLL[S ! ELLIPSIS OR L;G DA% PRES]VES
! LEVEL T IS ALR 9 E6ECT4 ,H["E1 IF S*
A SPACE IS FOLL[$ 0LIT]>Y TEXT1 UNRE-
LAT$ MA!MATICAL TEXT1 OR A SIGN ( COM-
P>ISON1 ? SPACE 9ITIATES ! BASE-L9E
LEVEL4
-----------------------------------#IB
(1) X^1+1_/2+1_/3+ ''' +1_/N
(BO? SPACES PRES]VE ! SUP]SCRIPT
LEVEL)
(2) S1 ''' S;N
(BO? SPACES PRES]VE ! IMPLI$ BASE-
L9E LEVEL) #BCE
(3) #10^3+ ---- IS EQUAL A#IB
TO #10^5_4
(THE SPACE 2F ! DA% PRES]VES ! PRE-
VI\S SUP]SCRIPT LEVEL1 &! SPACE FOL-
L[+ ! DA% T]M9ATES ! E6ECT (! PRE-
VI\S SUP]SCRIPT LEVEL & 9ITIATES !
BASE-L9E LEVEL)
(4) #10^3+ ---- .K #10^5
(THE SPACE 2F ! DA% PRES]VES ! PRE-
VI\S SUP]SCRIPT LEVEL1 &! SPACE
FOLL[+ ! DA% T]M9ATES ! E6ECT (!
PREVI\S SUP]SCRIPT LEVEL & 9ITIATES
! BASE-L9E LEVEL)
;G_4 ,! SPACE OR TRANSI;N 6A NEW
BRL L9E : IS FOLL[$ 0A -P>ISON SYMBOL
T]M9ATES ! E6ECT (A LEVEL 9DICATOR ALR
9 E6ECT & 9ITIATES ! BASE-L9E LEVEL4
,! SPACE AF A -P>ISON SYMBOL PRES]VES
! LEVEL T IS ALR 9 E6ECT4
(1) X^2"+Y^2"+Z^2 .K R^2
(THE SPACE : IS FOLL[$ 0! EQUALS
SYMBOL T]M9ATES ! E6ECT (! PREC$+
SUP]SCRIPT LEVEL & 9ITIATES ! BASE-
,NEME? ,MA?4 ,CODE1 #1972 B#IB
L9E LEVEL1 ! SPACE AF ! EQUALS SYM-
BOL PRES]VES ! BASE-L9E LEVEL)
(2) #2^X "K #3^X
(THE SPACE : IS FOLL[$ 0! LESS ?AN
SYMBOL T]M9ATES ! E6ECT (! PREC$+
SUP]SCRIPT LEVEL & 9ITIATES ! BASE-
L9E LEVEL1 ! SPACE AF ! LESS ?AN
SYMBOL PRES]VES ! BASE-L9E LEVEL)
(3) Q^LOG^;Q A .K A
(THE SPACE : IS FOLL[$ 0! EQUALS
SYMBOL T]M9ATES ! E6ECT (! PREC$+
SUP]SCRIPT LEVEL & 9ITIATES ! BASE-
L9E LEVEL1 ! SPACE AF ! EQUALS SYM-
BOL PRES]VES ! BASE-L9E LEVEL)
(4) !;U ;.K A
(THE SUBSCRIPT 9DICATOR 2F ! EQUALS
SYMBOL KEEPS ? SYMBOL AT ! SUBSCRIPT
LEVEL2 ! SPACE AF ! EQUALS SYMBOL
PRES]VES ! LEVEL T IS ALR 9 E6ECT)
(5) (1-SIN^2 X)^2
.K COS^4 X
(THE TRANSI;N 6A NEW BRL L9E 2F !
EQUALS SYMBOL T]M9ATES ! PREVI\S
SUP]SCRIPT LEVEL & 9ITIATES #BCG
! BASE-L9E LEVEL) C#IB
;H_4 ,ANY SYMBOL OR SITU,N O!R ?AN
^? 9 ;A TO ;G PRES]VES ! LEVEL T IS
ALR 9 E6ECT4
-----------------------------------#IC
@,S#80_4 ..,USE ( ,LEVEL .,9DICA-
TORS3
;A_4 ,A LEVEL 9DICATOR M/ 2 US$ 2F
ANY BRL 9DICATOR OR GR\P+ SYMBOL :5"E
? BRL 9DICATOR OR GR\P+ SYMBOL APPLIES
6A LEVEL O!R ?AN ! "O CURR5TLY 9 EF-
FECT4
(1) >X^2"+Y^2"]
(THE T]M9,N 9DICATOR APPLIES 6! BASE
L9E1 "!=E ! BASE-L9E 9DICATOR IS RE-
QUIR$)
(2) E^>X^^2^+Y^^2^]
(THE T]M9,N 9DICATOR APPLIES 6! F/-
ORD] SUP]SCRIPT LEVEL1 "!=E ! SUP]-
SCRIPT 9DICATOR IS REQUIR$)
(3) ?1/X^2"#
(THE CLOS+ SIMPLE-FRAC;N 9DICATOR
APPLIES 6! BASE L9E1 "!=E ! BASE-L9E
,NEME? ,MA?4 ,CODE1 #1972 A#IC
9DICATOR IS REQUIR$)
(4) ,?D(?X/Y#),/1+(?X/Y#)^2",#
(THE CLOS+ -PLEX-FRAC;N 9DICATOR AP-
PLIES 6! BASE L9E1 "!=E ! BASE-L9E
9DICATOR IS REQUIR$)
(5) "X^2"<:]
(THE DIRECTLY-OV] 9DICATOR APPLIES
6! BASE L9E1 "!=E ! BASE-L9E 9DICA-
TOR IS REQUIR$)
(6) X^2"[Y^2"]
(THE OP5+ & CLOS+ C.ELL,N 9DICATORS
APPLY 6! BASE L9E1 "!=E ! BASE-L9E
9DICATOR IS REQUIR$)
(7) (X^2"+Y^2")
(THE CLOS+ P>5!SIS APPLIES 6! BASE
L9E1 "!=E ! BASE-L9E 9DICATOR IS RE-
QUIR$)
(8) X^(M^^N^)
(THE CLOS+ P>5!SIS APPLIES 6! F/-
ORD] SUP]SCRIPT LEVEL1 "!=E ! SUP]-
SCRIPT 9DICATOR IS REQUIR$)
;B_4 ,! SUP]SCRIPT 9DICATOR M/ 2
US$ 6RE/ATE ! SUP]SCRIPT LEVEL #BCI
:5 TWO SUP]SCRIPTS >E 3SECU- B#IC
TIVE B "O APPLIES 6! EXPRES.N : PRE-
C$ES X &! O!R APPLIES 6! EXPRES.N :
FOLL[S X4 ,SIMIL>LY1 ! SUBSCRIPT 9DI-
CATOR M/ 2 RE/AT$ :5 TWO SUBSCRIPTS >E
3SECUTIVE & "O APPLIES 6! EXPRES.N
PREC$+ X &! O!R APPLIES 6! EXPRES.N
FOLL[+ X4 ,A SUP]SCRIPT OR SUBSCRIPT
9DICATOR M/ 2 RE/AT$ 2F A MODIFI$ EX-
PRES.N : IS 9T]IOR 6! SUP]SCRIPT OR
SUBSCRIPT EXPRES.N1 PROVID$ T ! MULTI-
PURPOSE 9DICATOR IS AL US$4
(1) P^B^C"Q
(2) ,P;B;C",Q
(3) ,P1;2",Q
(4) ,A;"X<@:]+;"Y<@:]
(THE SUBSCRIPT 9DICATOR AF ! PLUS
SIGN M/ 2 RE/AT$ 2F ! MULTIPURPOSE
9DICATOR)
-----------------------------------#ID
;C_4 ,! APPROPRIATE LEVEL 9DICATOR
M/ 2 US$ 2F EA* "P ( AN A2REVI,N OR
PHRASE : IS AT A LEVEL O!R ?AN ! BASE
L9E4
,NEME? ,MA?4 ,CODE1 #1972 A#ID
(1) $T;REGULAR ;POLYGON
(2) A^N+N+N ''' ^TO ^M ^N^_'S
;D_4 ,:5"E SPACES >E LEFT =! PUR-
POSE ( A*IEV+ ALIGN;T1 LEVEL 9DICATORS
M/ 2 US$ Z ?\< S* SPACES 7 N PRES5T4
(1) 2X^3"- X^2"+ X+ 1
3X^3"+4X^2"-10X+ 7
5X^2" +12
-2X^3" - 6X
---------------------
3X^3"+8X^2"-15X+20
;E_4 ,! APPROPRIATE LEVEL 9DICATOR
M/ 2 US$ 2F ANY SYMBOL OR SITU,N 9 : A
*ANGE ( LEVEL IS REQUIR$ B ! *ANGE IS
N E6ECT$ 0ANY (! 3DI;NS ( @,S#79_4
(1) #360^.*"-9T]VAL
(THE BASE-L9E 9DICATOR PLACES ! HY-
PH5 AT ! BASE-L9E LEVEL)
#BDA
(2) !;U ;.K A B#ID
(THE SUBSCRIPT 9DICATOR 2F ! EQUALS
SYMBOL KEEPS ? SYMBOL AT ! SUBSCRIPT
LEVEL2 O!RWISE X WD 2 AT ! BASE-L9E
LEVEL)
(3) T@);T ;.K A^T ^.K B .K B-A
(THE SUBSCRIPT & SUP]SCRIPT 9DICA-
TORS 2F ! F/ TWO EQUALS SYMBOLS KEEP
^! AT ! SUBSCRIPT & SUP]SCRIPT LEV-
ELS RESPECTIVELY1 :ILE ! SPACE 2F !
LA/ EQUALS SYMBOL PLACES X AT !
BASE-L9E LEVEL)
(4) E^SIN X ^.K A .1 Y
(THE SUP]SCRIPT 9DICATOR 2F ! EQUALS
SYMBOL KEEPS ? SYMBOL AT ! SUP]-
SCRIPT LEVEL)
(5) ,P;S;;1;''' S;;N
(THE SUBSCRIPT 9DICATOR 2F !
ELLIPSIS PLACES ! ELLIPSIS AT ! F/-
ORD] SUBSCRIPT LEVEL)
-----------------------------------#IE
(6) ,P1^.A^;1"''' ,P;R^.A^;R
(THE BASE-L9E 9DICATOR PLACES !
ELLIPSIS AT ! BASE-L9E LEVEL)
,NEME? ,MA?4 ,CODE1 #1972 A#IE
(7) (1-SIN^2 X)^2
"COS^2 X
(IT IS ASSUM$ T ? EXPRES.N HAS _H TO
RUN OV] 6ANO!R BRL L9E AT ! PLACE
9DICAT$2 ! BASE-L9E 9DICATOR AT !
2G9N+ (! BRL L9E PLACES COS AT !
BASE-L9E LEVEL2 O!RWISE X WD H RE-
MA9$ AT ! SUP]SCRIPT LEVEL 9ITIAT$
ON ! PREVI\S BRL L9E)
@,S#81_4 ..,NON-,USE ( ,LEVEL .,9DI-
CATORS3
;A_4 ,! BASE-L9E 9DICATOR M/ N 2
US$ 6RETURN 6! BASE L9E F A NUM]IC
SUBSCRIPT IF ! SUBSCRIPT 9DICATOR HAS
N BE5 US$ 2F ! NUM]IC SUBSCRIPT4
(1) (X1+1)
(BASE-L9E 9DICATOR N REQUIR$ 2F !
PLUS SYMBOL)
(2) (X1Y1+X2Y2)
(BASE-L9E 9DICATOR N REQUIR$ AF ANY
(! NUM]IC SUBSCRPTS)
#BDC
;B_4 ,! BASE-L9E 9DICATOR B#IE
M/ N 2 US$ 2F A "R 5L>G$ GR\P+ SYMBOL
IF ? SYMBOL EI IS SEP>AT$ F XS PREC$+
MAT]IAL 0"O OR M SPACES1 OR IF ! MA-
T]IAL : PREC$ES ! "R GR\P+ SYMBOL IS N
! 5D ( AN EXPRES.N4
(1) .,(U .K X^2 .,)
.,(V .K X^2"+Y^2".,)
(2) ,\F(X, Y) X^2JK,\
,\ "+Y,\
,\G(X, Y) X^2JK,\
,\ "-Y,\
;C_4 ,A LEVEL 9DICATOR M/ N 2 US$ 2F
ANY CLOS+ GR\P+ SYMBOL : IS DRAWN IN4
,NEME? ,MA?4 ,CODE1 #1972 C#IE
(1) #1 #1 #1
X Y Z
X^2 Y^2 Z^2
-----------------------------------#IF
;D_4 ,A LEVEL 9DICATOR M/ N 2 US$
6*ANGE ! LEVEL IF ANY SYMBOL OR SITU,N
SPECIFI$ 9 @,S#79 HAS ALR E6ECT$ !
*ANGE 6! DESIR$ LEVEL4
@,S#82_4 ..,SIMULTANE\S & ,NON-
,SIMULTANE\S ,SUP]SCRIPTS & .,SUB-
SCRIPTS3
;A_4 ,:5 AN EXPRES.N SIMULTANE\SLY
C>RIES A SUP]SCRIPT & SUBSCRIPT1 !
SUBSCRIPT M/ 2 9DICAT$ F/1 EV5 IF !
SUBSCRIPT IS NUM]IC & DOES N REQUIRE !
SUBSCRIPT 9DICATOR4 ,H["E1 IF ? SIGN
C>RIES "O OR M PRIMES 9 A4I;N1 SEE
@,S#83_4
(1) X;A^N
(;X C>RIES SIMULTANE\SLY A SUBSCRIPT
( ;A &A SUP]SCRIPT ( ;N) #BDE
,NEME? ,MA?4 ,CODE1 #1972 A#IF
(2) ;A^N"X
(;X C>RIES SIMULTANE\SLY A LEFT SUB-
SCRIPT ( ;A &A LEFT SUP]SCRIPT ( ;N)
(3) X1^2
(;X C>RIES SIMULTANE\SLY A SUBSCRIPT
( #1 &A SUP]SCRIPT ( #2)
;B_4 ,:5 ! SAME EXPRES.N C>RIES A
SUP]SCRIPT &A SUBSCRIPT : >E N SIMUL-
TANE\S1 ! RELATIVE HORIZONTAL POSI;NS
(! SIGNS M/ 2 RETA9$ 9 ! TRANSCRIP;N1
B ! BASE-L9E 9DICATOR M/ 2 9S]T$ 2F
MAK+ ! TRANSI;N 6! O!R LEVEL4
(1) A^N";M
(THE SUP]SCRIPT IS CLOS] 6! ;A ?AN !
SUBSCRIPT)
(2) A;M"^N
(THE SUBSCRIPT IS CLOS] 6! ;A ?AN !
SUP]SCRIPT)
(3) ^A";B"X
(THE LEFT SUBSCRIPT IS CLOS] 6! ;X
?AN ! LEFT SUP]SCRIPT)
#BDG
(4) ;B"^A"X B#IF
(THE LEFT SUP]SCRIPT IS CLOS] 6! ;X
?AN ! LEFT SUBSCRIPT)
(5) X1"^2
(THE SUBSCRIPT IS CLOS] 6! ;X ?AN !
SUP]SCRIPT)
(6) X';A"^B
(THE SUBSCRIPT IS CLOS] TO ;X PRIME
?AN ! SUP]SCRIPT)
@,S#83_4 ..,PRIMES 9 ,A4I;N 6,SUP]-
SCRIPTS OR .,SUBSCRIPTS3
;A_4 ,! PRIME SYMBOL M/ N"E 2 PRE-
C$$ 0! SUP]SCRIPT 9DICATOR4
(1) X'
-----------------------------------#IG
;B_4 ,:5 AN EXPRES.N C>RIES "O OR
M PRIMES 9 A4I;N 6SUP]SCRIPTS OR SUB-
SCRIPTS1 ! PRIME SYMBOL OR SYMBOLS M/
2 9DICAT$ F/ UN.S S* SYMBOLS D N O3UR
AT ! 2G9N+ (! SUP]SCRIPT OR SUBSCRIPT1
9 : CASE !Y M/ RETA9 ! SAME POSI;N Z 9
9K PR9T4
,NEME? ,MA?4 ,CODE1 #1972 A#IG
(1) X';A
(2) X'^2
(3) X';A^B
(4) X''1^3
(5) X'^@#
(6) X^@#'
(7) ,A;UE^@#'
(8) ,A';UE^@#
;C_4 ,= PRIMES 9 O!R ROLES SEE
@,S#172_4
@,S#84_4 .,PLURALS .& .,POSSESSIVES3
,= PLURALS OR POSSESSIVES ( MA!MATICAL
EXPRES.NS : 5D )A SUP]SCRIPT OR SUB-
SCRIPT SEE @,S#39_4
,RULE ,,XIV--,MODIFI]S
,MODIFIC,N ,9DICATORS
,DIRECTLY ,OV]
,F/ ORD] <
,SECOND ORD] <<
#BDI
,NEME? ,MA?4 ,CODE1 #1972 B#IG
,DIRECTLY ,"U
,F/ ORD] %
,SECOND ORD] %%
,MULTIPURPOSE "
,SUP]POSI;N @
,T]M9,N ]
-----------------------------------#IH
,MODIFI]S
,>C
,3CAVE UPW>D $A
,3CAVE D[NW>D $'
,>R[
,B>B$ AT BO? 5DS $[33O
,B>B$ AT LEFT $[33
,B>B$ AT LEFT & DOTT$ AT "R
$[33*
,B>B$ AT "R
,3TRACT$ =M $O
#BEA
,NEME? ,MA?4 ,MODE1 #1972 A#IH
,UNCONTRACT$ =M $33O
,DOTT$ AT BO? 5DS $*33*
,DOTT$ AT LEFT (NO B>B) $*33
,DOTT$ AT LEFT & B>B$ AT "R
$*33O
,DOTT$ AT "R (NO B>B) $33*
,HOLL[ DOT AT BO? 5DS
$.*33.*
,HOLL[ DOT AT LEFT (NO B>B)
$.*33
,HOLL[ DOT AT LEFT & B>B$ AT "R
$.*33O
,HOLL[ DOT AT "R & B>B$ AT LEFT
$[33.*
#BEC
,NEME? ,MA?4 ,CODE1 #1972 B#IH
,HOLL[ DOT AT "R (NO B>B)
$33.*
,B>
,HORIZONTAL (MACRON) :
,V]TICAL \
-----------------------------------#II
,C>ET (CIRCUMFLEX) _<
,9V]T$ _%
,LEFT-PO9T+ ;<
,"R-PO9T+ ;%
,DOT *
,HOLL[ ,DOT .*
,"Q ,M>K _8
#BEE
,NEME? ,MA?4 ,CODE1 #1972 A#II
,TILDE
,EXT5D$ @,:
,SIMPLE @:
,TRIANGLE (EQUILAT]AL) $T
@,S#85_4 .,MODIFI]S3 ,A MODIFI] IS A
SUP]SCRIPT OR SUBSCRIPT : O3UPIES1 RE-
SPECTIVELY1 A POSI;N DIRECTLY OV] OR
DIRECTLY "U ! SIGN 6: X APPLIES4 ,!
MODIFI]S 9 ! LI/ AT ! 2G9N+ ( ? RULE
>E ^? MO/ -MONLY US$1 B O!R MODIFI]S
M/ 2 TR1T$ 9 ! SAME MANN]4
@,S#86_4 .,MODIFI$ .,EXPRES.NS3
;A_4 ..,! ,FIVE-,/EP ,RULE =
,TRANSCRIB+ ,MODIFI$ .,EXPRES.NS3 ,!
-PON5TS (A MODIFI$ EXPRES.N M/ APPE> 9
! FOLL[+ ORD]3
;I_4 ,MULTIPURPOSE 9DICATOR "
;II_4 ,EXPRES.N 2+ MODIFI$4
;III_4 ,DIRECTLY-OV] 9DICATOR <
OR DIRECTLY-"U 9DICATOR % #BEG
,NEME? ,MA?4 ,CODE1 #1972 B#II
;IV_4 ,MODIFI]4
;V_4 ,T]M9,N 9DICATOR ]
,^! FIVE -PON5TS MAY N"E 2 SEP>AT$ F
EA* O!R 0TRANSI;N 6ANO!R BRL L9E4 ,!
T]M9,N 9DICATOR T]M9ATES ONLY ! MODI-
FI$ EXPRES.N2 X DOES N A6ECT ! LEVEL
AT : ! MODIFI$ EXPRES.N O3URS4
(1) "X%:]
(;X ) SUBSCRIB$ B>)
----------------------------------#AJJ
(2) "X+Y<:]
(;X PLUS ;Y, ) SUP]SCRIB$ B>)
(3) "LIM%X $O #0] F(X)
(THE LIMIT1 Z ;X APPROA*ES #0, ( ;F
( ;X)
(4) "X^2"<:]
(;X SQU>$1 ) SUP]SCRIB[ B>)
(5) "X'<:]
(;X PRIM$1 ) SUP]SCRIB$ B>)
(6) "X1<:]
(;X SUB #1, ) SUP]SCRIB$ B>)
(7) "X;N"<:]
(;X SUB ;N, ) SUP]SCRIB$ B>) #BEI
;B_4 ,:5 ! EXPRES.N 2+ A#AJJ
MODIFI$ IS A S+LE DIGIT OR A LR1 L[]-
CASE OR CAPITALIZ$1 F ANY ALPHABET1 &
9 ANY TYPE =M1 & :5 ! MODIFI] IS !
HORIZONTAL B> DIRECTLY ABV S* A S+LE
DIGIT OR LR1 ! DIGIT OR LR1 FOLL[$ 0!
B>1 S]VES 6EXPRESS ! MODIFIC,N4 ,?
3/RUC;N %D 2 REG>D$ Z A 3TRACT$ =M (
EXPRES.N & M/ 2 US$ :5"E APPLICA#4 ,IF
! MODIFIC,N 9CLUDES A SUP]SCRIPT1 SUB-
SCRIPT1 OR PRIME1 ! FIVE-/EP RULE ( ;A
ABV M/ 2 FOLL[$4 ,! FIVE-/EP RULE MAY1
2 US$ 9 3JUNC;N )! 3TRACT$ =M )\T FE>
( 3FU.N4
(1) X:
(;X ) SUP]SCRIB$ B>)
(2) X:+Y:
(;X ) SUP]SCRIB$ B> PLUS ;Y ) SUP]-
SCRIB$ B>)
(3) X:Y
(;X ) SUP]SCRIB$ B>1 "TS ;Y)
(4) XY:Z
(;X, "TS ;Y ) SUP]SCRIB$ B>1 "TS
;Z)
,NEME? ,MA?4 ,CODE1 #1972 B#AJJ
(5) X:^2
(;X ) SUP]SCRIB$ B> SQU>$)
(6) X:'
(;X ) SUP]SCRIB$ B> PRIM$)
----------------------------------#AJA
(7) X:;1
(;X ) SUP]SCRIB$ B>1 SUB #1)
(8) X:;N
(;X ) SUP]SCRIB$ B>1 SUB ;N)
(9) _;,Z:
(BOLDFACE CAPITAL ;,Z ) SUP]SCRIB$
B>)
(10) #3.54:
(3.54, B> SUP]SCRIB$ 6! #4)
(11) "(A:_;,A+B:_;,B)<:]
(;A ) SUP]SCRIB$ B> "TS BOLDFACE
CAPITALIZ$ ;,A PLUS ;B ) SUP]SCRIB$
B> "TS BOLDFACE CAPITALIZ$ ;,B, !
:OLE EXPRES.N ) SUP]SCRIB$ B>)
(12) ,A;X:
(;,A )A "R SUBSCRIPT ( ;X ) SUP]-
SCRIB$ B>)
(13) ,A;X:+Y:
(;,A )A "R SUBSCRIPT ( ;X ) #BFA
SUP]SCRIB$ B> PLUS ;Y ) A#AJA
SUP]SCRIB$ B>)
(14) E^AX:
(;E )A "R SUP]SCRIPT ( ;A "TS ;X )
SUP]SCRIB$ B>)
(15) X:_'S
(THE PLURAL ( ;X ) SUP]SCRIB$ B>)
@,S#87_4 ..,MODIFI]S ( ,HI<] .,ORD]3
;A_4 ,A MODIFI] (! SECOND ORD] M/
2 PREC$$ 0! SECOND-ORD] DIRECTLY-OV]
OR DIRECTLY-"U 9DICATOR1 &A MODIFI] (!
?IRD ORD] M/ 2 PREC$$ 0! ?IRD-ORD]
DIRECTLY-OV] OR DIRECTLY-"U 9DICATOR4
,! T]M9,N 9DICATOR1 H["E1 M/ 2 US$
ONLY ONCE1 AF ! LA/ MODIFI] SYMBOL4
(1) "X+Y<:<<A .K #3]
(;X PLUS ;Y SUP]SCRIB$ 0A B>1 : 9
TURN IS SUP]SCRIB$ BY ;A EQUALS #3)
(2) "X+Y%:%%A .K #3%%%B .K #2]
(;X PLUS ;Y SUBSCRIB$ 0A B>1 : 9
TURN IS SUBSCRIB$ BY ;A EQUALS #3, :
9 TURN IS SUBSCRIB$ BY ;B EQUALS #2)
,NEME? ,MA?4 ,CODE1 #1972 B#AJA
;B_4 ,A MODIFI] ( ORD] HI<] ?AN !
?IRD M/ 2 TR1T$ 9 ! MANN] SU7E/$ 9 ;A
ABV4
----------------------------------#AJB
;C_4 ,A MODIFI]1 6BE ( ORD] HI<]
?AN ! F/1 M/ 2 ASSOCIAT$ )! SAME EX-
PRES.N Z A MODIFI] ( L[] ORD]4 ,9
@,S#86B (11) ABV1 ! L;G B> IS N A MOD-
IFI] ( SECOND ORD] 2C NO MODIFI] ( F/
ORD] IS ASSOCIAT$ )! SAME EXPRES.N Z !
L;G B>)
@,S#88_4 .,SIMULTANE\S .,MODIFI]S3
,:5 A MA!MATICAL EXPRES.N IS SIMUL-
TANE\SLY MODIFI$ ABV & 2L1 ! MODIFI]
2L M/ 2 9DICAT$ F/4 ,! T]M9,N 9DICA-
TOR1 H["E1 M/ 2 US$ ONLY ONCE1 AF !
LA/ MODIFI] SYMBOL4 ,IF ! MODIFI]S
9VOLV$ >E ( ORD] HI<] ?AN ! F/1 !Y >E
TR1T$ Z DESCRIB$ 9 @,S#87_4
(1) "X+Y%:<:]
(;X PLUS ;Y, ) SUBSCRIB$ & SUP]-
SCRIB$ B>S) #BFC
(2) A#AJB
"K,S%N .K #1<,=]?1/2^N"# .K #1
(THE ,GREEK CAPITALIZ$ SIGMA ) SUB-
SCRIB$ ;N EQUALS #1 & SUP]SCRIB$
9F9;Y SIGN)
(3) "X+Y%:%%A .K #3<:<<B .K #2]
(;X PLUS ;Y SUBSCRIB$ 0A B> : IS 9
TURN SUBSCRIB$ BY ;A EQUALS #3_2
SUP]SCRIB$ 0A B> : IS 9 TURN SUP]-
SCRIB$ BY ;B EQUALS #2)
@,S#89_4 .,P>ALLEL .,HORIZONTAL
.,B>S3 ,P>ALLEL HORIZONTAL B>S M/ N 2
REG>D$ Z ! EQUALS SIGN OR ! ID5T;Y
SIGN :5 !Y O3UR ABV OR 2L A MA!MATICAL
EXPRES.N O!R ?AN A -P>ISON SIGN4
,FUR!RMORE1 ! B> : IS M REMOTE F !
MA!MATICAL EXPRES.N 2+ MODIFI$ M/ N 2
REG>D$ Z A MODIFI] ( SECOND OR ?IRD
ORD]2 ! D\# OR TRIPLE B> M/ 2 REG>D$ Z
A S+LE MODIFI]4
(1) "X<::]
(;X SUP]SCRIB$ 0D\# HORIZONTAL B>S)
(2) "X%::<::]
,NEME? ,MA?4 ,CODE1 #1972 B#AJB
(;X SUBSCRIB$ & SUP]SCRIB$ 0D\#
HORIZONTAL B>S)
(3) "X%::<:]
(;X SUBSCRIB$ 0D\# HORIZONTAL B>S &
SUP]SCRIB$ 0S+LE HORIZONTAL B>)
(4) "X%:::]
(;X SUBSCRIB$ 0TRIPLE HORIZONTAL
B>S)
@,S#90_4 .,BINOMIAL .,COE6ICI5T3 ,!
TWO EXPRES.NS : 3/ITUTE A BINOMIAL CO-
E6ICI5T M/ 2 SEP>AT$ 0! DIRECTLY-"U
9DICATOR4 ,! EXPRES.N : FOLL[S ! OP5+
P>5!SIS & PREC$ES ! DIRECTLY-"U 9DI-
CATOR CORRESPONDS 6! UPP] SIGN
----------------------------------#AJC
9 ! BINOMIAL COE6ICI5T2 ! EXPRES.N :
FOLL[S ! DIRECTLY-"U 9DICATOR & PRE-
C$ES ! CLOS+ P>5!SIS CORRESPONDS 6!
L[] SIGN (! BINOMIAL COE6ICI5T4
(1) (N%K)
(THE BINOMIAL COE6ICI5T ) ;N Z !
UPP] SIGN & ;K Z ! L[] SIGN) #BFE
(2) (G;J"%A;J") A#AJC
(THE BINOMIAL COE6ICI5T ) ;G SUB ;J
Z ! UPP] SIGN & ;A SUB ;J Z ! L[]
SIGN)
,S#91_4 ..,MODIFI$ ,EXPRES.NS 9
,SUP]SCRIPTS & .,SUBSCRIPTS3 ,IF A
MODIFI$ EXPRES.N IS "P OR ALL (A "R
SUP]SCRIPT OR SUBSCRIPT1 ! MULTIPUR-
POSE 9DICATOR M/ 2 PREC$$ 0! APPROPRI-
ATE LEVEL 9DICATOR4 ,? W AUTOMATIC,Y 2
! CASE IF ! MODIFI$ EXPRES.N O3URS AT
! 2G9N+ (! SUP]SCRIPT OR SUBSCRIPT2 B
! APPROPRIATE LEVEL 9DICATOR M/ 2 RE-
/AT$ IF ! MODIFI$ EXPRES.N O3URS AT AN
9T]IOR POSI;N (! SUP]SCRIPT OR SUB-
SCRIPT4 ,IF ! 3TRACT$ =M =A MODIFI$
EXPRES.N IS US$ S T ! MULTIPURPOSE
9DICATOR DOES N APPE>1 ! APPROPRIATE
LEVEL 9DICATOR M/ N 2 RE/AT$4
(1) ,A;"X<@:]
(;,A C>RIES A SUBSCRIPT ( ;X )
SUP]SCRIB$ TILDE)
(2) ,A;"X<@:]+;"Y<@:]
,NEME? ,MA?4 ,CODE1 #1972 B#AJC
(;,A C>RIES A SUBSCRIPT ( ;X ) SUP]-
SCRIB$ TILDE PLUS ;Y ) SUP]SCRIB$
TILDE2 ! SUBSCRIPT LEVEL AF ! PLUS
SIGN M/ 2 RE/AT$ 2F ! MULTIPURPOSE
9DICATOR)
(3) ,A;X:+Y:
(;,A C>RIES A SUBSCRIPT ( ;X ) SUP]-
SCRIB$ B> PLUS ;Y ) SUP]SCRIB$ B>)
@,S#92_4 .,PLURAL .,MODIFI$ .,EX-
PRES.NS3 (,SEE @,S#39)_4
@,S#93_4 .,MODIFIC,N .BY .,SUP]POSI-
;N3 ,:5 "O SIGN MODIFIES ANO!R 0SUP]-
POSI;N1 9 DECID+ : IS ! BASIC SIGN & :
IS ! SUP]POS$ SIGN1 ! FOLL[+ HI]>*Y1
9 DESC5D+ ORD]1 %D 2 US$ Z A GUIDE3
;I_4 ,9TEGRAL SIGN
;II_4 ,OP],N SIGNS
;III_4 ,B>S--HORIZONTAL & V]TI-
CAL
;IV_4 ,%APE SIGNS
;V_4 ,-P>ISON SIGNS
;VI_4 ,SIGNS N COV]$ ABV
,A SIGN 2L;G+ 6A CATEGORY L[] #BFG
ON ! LI/ M/ 2 REG>D$ Z SUP]- C#AJC
POS$ ON A SIGN HI<] ON ! LI/1 &! SUP]-
POSI;N TRANSCRIB$ ACLY4 ,IF TWO SIGNS
2L;G 6! SAME CATEGORY1 X IS P]MISSI#
6REPRES5T ! SUP]POSI;N 9 EI
----------------------------------#AJD
ORD]1 PROVID$ T ! SAME ORD] IS US$
3SI/5TLY "?\T ! 5TIRE TRANSCRIP;N4 ,!
-PON5TS (A SIGN -P.D$ 0SUP]POSI;N M/ 2
JO9$ 0! SUP]POSI;N 9DICATOR & TRAN-
SCRIB$ UNSPAC$1 & )\T TRANSI;N 6ANO!R
BRL L9E4 ,! T]M9,N 9DICATOR M/ FOLL[ !
SECOND -PON5T4 (,FOR O!R EXAMPLES1 SEE
8,-P>ISON ,SIGNS ,-P.D$ 0,SUP]POSI;N0
PAGES #140-141, & #143_4)
(1) !@$R]
(INTEGRAL SIGN ) SUP]POS$ RECTANGLE)
(2) :@$4]
(HORIZONTAL B> ) SUP]POS$ SQU>E)
(3) *@.K]
(DOT 2T B>S ( EQUALS SIGN)
(4) .K@_"K]
(EQUALS SIGN ) SUP]POS$ 9CLU.N
SIGN)
,NEME? ,MA?4 ,CODE1 #1972 A#AJD
(5) _"K@.K]
(INCLUSION SIGN ) SUP]POS$ EQUALS
SIGN)
(6) $[@$A]
(ANGLE ) SUP]POS$ >C)
@,S#94_4 ..,9T]IOR ,MODIFI]S )
,SIGNS ( .,%APE3 ,SEE @,S#111_4
@,S#95_4 .,>C3
(1) ",A%$A]
(;,A ) SUBSCRIB$ >C 3CAVE UPW>D)
(2) ",A,B<$A]
(,A,B ) SUP]SCRIB$ >C 3CAVE UPW>D)
(3) ",A%$']
(;,A ) SUBSCRIB$ >C 3CAVE D[NW>D)
(4) ",A,B%$']
(,A,B ) SUBSCRIB$ >C 3CAVE D[NW>D)
@,S#96_4 .,>R[S3 ,>R[S M/ N 2 REG>D$
Z MODIFI]S :5 !Y O3UR DIRECTLY OV] OR
DIRECTLY "U A -P>ISON SIGN4 ,9 T EV5T1
!Y 2COME A -PON5T (A SIGN ( -P>ISON
-P.D$ V]TIC,Y4 #BFI
,:5 A "R-PO9T+ >R[ )A S+LE #AJE
%AFT ( ORD9>Y L5G? IS 9 REGUL> TYPE1
HAS A FULL B>B1 & IS N "P (A M -PLEX
3/RUC;N OR -P.D MODIFI]1 X M/ 2 TRAN-
SCRIB$ 9 XS 3TRACT$ =M4 ,IF S* AN >R[
IS 9 NON-REGUL> TYPE1 DOES N H A FULL
B>B OR %AFT ( ORD9>Y L5G?1 IS "P (A
-P.D MODIFI]1 OR IS XF MODIFI$1 X M/ 2
REPRES5T$ 9 XS UNCONTRACT$ =M4
(1) ",A,B<$O]
(,A,B ) SUP]SCRIB$ >R[ B>B$ AT "R)
(2) ",A,B<$[33]
(,A,B ) SUP]SCRIB$ >R[ B>B$ AT
LEFT)
(3) ",A,B<$[33O]
(,A,B ) SUP]SCRIB$ >R[ B>B$ AT BO?
5DS)
(4) ",A,B<$[33*]
(,A,B ) SUP]SCRIB$ >R[ B>B$ AT !
LEFT & DOTT$ AT ! "R)
(5) ",A,B<$*33*]
(,A,B ) SUP]SCRIB$ >R[ DOTT$ AT BO?
5DS)
(6) ",A,B<$.*33.*]
,NEME? ,MA?4 ,CODE1 #1972 A#AJE
(,A,B ) SUP]SCRIB$ >R[ ) HOLL[ DOTS
AT BO? 5DS)
(7) ",A,B<$*33]
(,A,B ) SUP]SCRIB$ >R[ DOTT$ AT
LEFT)
(8) ",A,B<$*33O]
(,A,B ) SUP]SCRIB$ >R[ DOTT$ AT LEFT
& B>B$ AT "R)
(9) ",A,B<$33*]
(,A,B ) SUP]SCRIB$ >R[ DOTT$ AT "R)
(10) ,X "$33O<F.*G] ,Y
(>R[ ) SUP]SCRIB$ ;F HOLL[ DOT ;G 2T
;,X & ;,Y)
@,S#97_4 .,HORIZONTAL .,B>3
;A_4 ,! HORIZONTAL B> M/ N 2 RE-
G>D$ Z A MODIFI] :5 X O3URS DIRECTLY
OV] OR DIRECTLY "U A -P>ISON SIGN4 ,9
T EV5T1 X 2COMES A -PON5T (A SIGN (
-P>ISON -P.D$ V]TIC,Y (SEE @,S#147)_4
,:5 ! HORIZONTAL B> IS XF MODIFI$ 0A
DOT "U X OR A C>ET DIRECTLY OV] OR "U
X1 ! -B9,N IS A MODIFI$ SIGN ( -P>ISON
(SEE @,S#146)_4 ,:5 ! HORIZON- #BGA
TAL B> IS XF MODIFI$ 0A DOT B#AJE
OV] X1 ! -B9,N IS A SIGN ( OP],N4
;B_4 ,! HORIZONTAL B> IS (T5 US$
69DICATE ! RECURR;E ( "O OR M DIGITS 9
A DECIMAL NUM]AL 0PLAC+ X OV] ! DIGITS
: RECUR4
----------------------------------#AJF
(1) #.3:
(DECIMAL PO9T #3, )A B> OV] ! #3)
(2) #."7128<:]
(DECIMAL PO9T #7128, )A B> OV] ! F\R
DIGITS)
(3) #3.57"29<:]
(3.5729, )A B> OV] ! #29)
;C_4 ,:5 ! HORIZONTAL B> O3URS OV]
OR "U ! 9TEGRAL SIGN1 OR OV] OR "U !
A2REVIAT$ OR UNA2REVIAT$ FUNC;N "N =
.LIMIT1 ! B> M/ N 2 TR1T$ Z A MODIFI]
(SEE @,S#171 & @,S#118, RESPECTIVE-
LY)_4
@,S#98_4 .,C>ET3
(1) "X<_<]
(;X ) SUP]SCRIB$ C>ET)
,NEME? ,MA?4 ,CODE1 #1972 A#AJF
(2) ".K<_<]
(EQUALS SIGN ) SUP]SCRIB$ C>ET)
(3) "X%_%]
(;X ) SUBSCRIB$ 9V]T$ C>ET)
(4) ".K<;<]
(EQUALS SIGN ) SUP]SCRIB$ LEFT-PO9T+
C>ET)
(5) ".K<;%]
(EQUALS SIGN ) SUP]SCRIB$ "R-PO9T+
C>ET)
@,S#99_4 .,DOT3
;A_4 ,! DOT IS FREQU5TLY US$ 69DI-
CATE ! RECURR;E ( "O OR M DIGITS 9 A
DECIMAL NUM]AL4 ,:5 US$ = ? PURPOSE1 A
DOT IS USU,Y PLAC$ 9 PR9T OV] EA* DIG-
IT (! RECURR+ SEQU;E4 ,9 BRL1 H["E1
ONLY A S+LE DOT M/ 2 US$ Z A MODIFI]4
(1) #."3<*]
(DECIMAL PO9T #3, )A DOT OV] ! #3)
----------------------------------#AJG
(2) #."135<*]
(DECIMAL PO9T #135, )A DOT #BGC
OV] EA* (! ?REE DIGITS) A#AJG
(3) #.13"5<*]
(DECIMAL PO9T #135, )A DOT OV] ! #5)
;B_4 ,AL? "! IS !ORETIC,Y NO LIMIT
6! NUMB] ( DOTS : MAY 2 PLAC$ OV] OR
"U A S+LE MA!MATICAL EXPRES.N1 9 PRAC-
TICE ! NUMB] R>ELY EXCE$S ?REE DOTS4
,H["E1 Z _M DOTS M/ 2 US$ 9 ! TRAN-
SCRIP;N Z >E PRES5T 9 ! PR9T$ TEXT1
EXCEPT 9 ! CASE ( RECURR+ DECIMALS Z 9
;A_4
(1) "X<**]
(;X ) TWO DOTS OV] IT)
(2) "X<***]
(;X ) ?REE DOTS OV] IT)
(3) "X%**]
(;X ) TWO DOTS "U IT)
@,S#100_4 .,HOLL[ .,DOT3
(1) ".K<.*]
(EQUALS SIGN ) SUP]SCRIB$ HOLL[
DOT)
,NEME? ,MA?4 ,CODE1 #1972 B#AJG
@,S#101_4 .,"Q .,M>K3
(1) ".K<_8]
(EQUALS SIGN ) SUP]SCRIB$ "Q M>K)
(2) ".K%_8]
(EQUALS SIGN ) SUBSCRIB$ "Q M>K)
@,S#102_4 .,TILDE3 ,! TILDE1 SIMPLE
OR EXT5D$1 M/ N 2 REG>D$ Z A MODIFI]
:5 X O3URS DIRECTLY OV] OR "U A COM-
P>ISON SIGN4 ,9 T EV5T1 X 2COMES A
-PON5T (A SIGN ( -P>ISON -P.D$ V]TIC,Y
(SEE @,S#147)_4 ,:5 ! TILDE1 SIMPLE OR
EXT5D$1 IS XF MODIFI$ 0A DOT OR A C>ET
DIRECTLY OV] OR "U X1 ! -B9,N IS A
MODIFI$ SIGN ( -P>ISON (SEE @,S#146)_4
----------------------------------#AJH
,RULE ,,XV--,RADICALS
,RADICAL (SQU>E ROOT) >
,RADICAL ,9DICATORS
,9DEX-(-,RADICAL < #BGE
,NEME? ,MA?4 ,CODE1 #1972 A#AJH
,ORD]-(-,RADICAL
,F/ 9N] RADICAL .
,SECOND 9N] RADICAL ..
,?IRD 9N] RADICAL ...
,T]M9,N ]
@,S#103_4 .,SIMPLE .,RADICALS3 ,!
MO/ -MONLY O3URR+ RADICAL IS ! SQU>E
ROOT4
;A_4 ,:5 ! SQU>E ROOT SIGN HAS
A V9CULUM (HORIZONTAL B>) : SPECIFIES
! EXT5T 6: ! RADICAL SIGN IS E6ECTIVE1
! TRANSCRIP;N ( S* A RADICAL IS A3OM-
PLI%$ 0! FOLL[+ ?REE /EPS3
;I_4 ,! RADICAL SYMBOL >
;II_4 ,! EXPRES.N 6: X APPLIES
(RADICAND)_4
;III_4 ,! T]M9,N 9DICATOR ]
(1) >2]
(2) >X+Y]
(3) >X^2"+1]
(4) >X^2"+Y^2"]
(5) >?X/Y#] #BGG
(6) #3>A] B#AJH
(7) >X]^3
----------------------------------#AJI
;B_4 ,:5 ! SQU>E ROOT SIGN O3URS
)\T A RADIC&1 Z :5 ATT5;N IS 2+ CALL$
6A SIGN 9 9K PR9T1 OR :5 ! EXT5T 6: !
RADICAL IS E6ECTIVE IS N 9DICAT$ 9 9K
PR9T 0! V9CULUM1 ! T]M9,N 9DICATOR M/
2 OMITT$4
(1) ,! > M1NS 8SQU>E ROOT40
(2) >(X+Y)
(NO V9CULUM 9 9K PR9T)
@,S#104_4 .,9DEX .( .,RADICAL3
,RADICALS ( 9DEX O!R ?AN #2 REQUIRE A
SPECIFIC 9DEX4 ,! TRANSCRIP;N ( S* A
RADICAL IS A3OMPLI%$ 0! FOLL[+ ?REE
/EPS3
;I_4 ,! 9DEX-(-RADICAL 9DICATOR
<
;II_4 ,! 9DEX (! RADICAL4
;III_4 ,!N PROCE$ AC 6! ?REE
/EPS 9 @,S#103A_4
,NEME? ,MA?4 ,CODE1 #1972 A#AJI
(1) <3>2]
(2) #3<3>X+Y]
(3) <N>A]
(4) <M+N>P+Q]
@,S#105_4 .,NE/$ .,RADICALS3 ,OC-
CA.N,Y1 RADICALS >E NE/$ "O )9 ! O!R4
,! F/ 9N] RADICAL IS !N REG>D$ Z HAV+
A DEP? ( ORD] #1, ! SECOND 9N] RADICAL
Z HAV+ A DEP? ( ORD] #2, & S ON4 ,9 S*
CASES1 ! ORD]-(-RADICAL 9DICATOR .
(DOTS #4-6) M/ 2 REP1T$ 2F BO? ! RADI-
CAL SYMBOL & XS ASSOCIAT$ T]M9,N 9DI-
CATOR Z _M "TS Z X IS NEC 69DICATE !
DEP? ( T RADICAL4 ,IF "O (! 9N] RADI-
CALS IS ASSOCIAT$ ) AN 9DEX1 ! PROP]
NUMB] ( ORD]-(-RADICAL 9DICATORS M/ 2
PLAC$ 2F ! 9DEX-(-RADICAL 9DICATOR R
?AN 2F ! RADICAL SYMBOL XF4 ,! ORD]-(-
RADICAL 9DICATORS >E PROVID$ =! PUR-
POSE ( 5ABL+ ! R1D] 6KEEP TRACK (!
DEP? (! RADICAL 6: X APPLIES4
#BGI
(1) >X+.>X+Y.]+Z] B#AJI
(THE SQU>E ROOT (! SUM ( ?REE T]MS2
! F/ T]M IS ;X, ! SECOND T]M IS !
SQU>E ROOT ( ;X PLUS ;Y_2 ! ?IRD T]M
IS ;Z)
(2) <3>X^2"+.<3>X^2"+Y^2".]+Y^2"]
(THE CUBE ROOT (! SUM ( ;X SQU>$1 !
CUBE ROOT ( ;X SQU>$ PLUS ;Y SQU>$1
& ;Y SQU>$)
----------------------------------#AAJ
(3) >.<3>X.]] .K <3>.>X.]]
(THE SQU>E ROOT (! CUBE ROOT ( ;X
EQUALS ! CUBE ROOT (! SQU>E ROOT (
;X)
(4) >X+.>Y+..>Z..].]]
(A NE/ ( ?REE RADICALS2 ! \T] RADI-
CAL 3TA9S ;X PLUS ! 9N] RADICALS1 !
F/ 9N] RADICAL 3TA9S ;Y PLUS ! SEC-
OND 9N] RADICAL1 &! SECOND 9N] RADI-
CAL 3TA9S ;Z)
,,5D ,,( ,,VOLUME ,,II