,,US+ ,,! ,,CRANM] ,,ABACUS ,,= ,,! ,,BL ,0,,FR$ ,L4 ,,GISSONI ,EMBOSS$ 9 ,"O ,VOLUME ,PAGES ;I-;VII & #A-ACC ,EMBOSS$ & ,PR9T$ 0,P]MIS.N (! ,AU?OR ,AM]ICAN ,PR9T+ ,H\SE =! ,BL ,L\ISVILLE1 ,K5TUCKY #DJBJF-JJHE #AIHF ,? ,BOOK IS ,D$ICAT$ TO ,MR4 ,TAKA%I ,KOJIMA ( ,TOKYO1 ,JAPAN ,US+ ! ,CRANM] ,ABACUS ;III ,,TA# ,,( ,,3T5TS ,PAGE ,D$IC,N """""""""""""""""""""""" ;II ,TA# ( ,3T5TS """"""""""""""""" ;III ,APPRECI,N """""""""""""""""""" ;VII ,BACKGR.D """"""""""""""""""""""" #A ,DEVELOP;T (! ,CRANM] ,ABACUS """ #D ,! ,PROMISE (! ,ABACUS """""""""" #E ,H[ 6,/UDY """""""""""""""""""""" #F ,MEET ! ,ABACUS """"""""""""""""" #H ,! ,LANGUAGE (! ,ABACUS """"""""" #I ,SETT+ ,NUMB]S """"""""""""""""" #AJ ,A4+ ,"O-DIGIT ,NUMB]S """"""""" #AC ,! ,SECRETS ( ,A4I;N """"""""""" #AG ,PRACTICE ,EX]CISES """""""""""" #BB ,! ,A4I;N ( ,TWO-DIGIT ,NUMB]S #BC ,PRACTICE ,EX]CISES """""""""""" #BF ,! ,A4I;N ( ,NUMB]S ,HAV+ ,?REE OR ,M ,DIGITS """""""""""""""" #BG ,A4+ ,SUMS ( ,M"OY """"""""""""" #BH ,PAGE ,PRACTICE ,EX]CISES """""""""""" #BI ,MULTIPLIC,N """"""""""""""""""" #CJ ,! ,POSI;N ,3CEPT """""""""""""" #CE ,DET]M9+ ! ,L5G? 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ABACUS FL\RI%$1 RCVG WIDE- SPR1D ATT5;N F ,*9ESE & ,JAPANESE MA!MATICIANS & O!R S*OL>S4 ,TD1 XS POPUL>;Y 9 ,JAPAN AL"O IS 5 6/A7] ! IMAG9,N4 ,6CITE J "O EXAMPLE ( XS POPUL>;Y1 N9ETY 7#IJ7 P] C5T ( ALL BUSI;S >I?METIC 9 ,JAPAN IS D"O ON ! ABACUS1 OR SOROBAN Z X IS CALL$ 9 ,JAPANESE4 ,"! >E _M R1SONS =! POPU- L>;Y (! ABACUS4 .,SPE$3 ,! SPE$ ( ABACUS CALCUL,N IS NO?+ %ORT ( BR1?- TAK+ 6"S"O WIT;S+ AN ABACUS DEMON- /R,N =! F/ "T4 ,9 3TE/ AF 3TE/1 SKILL$ SIE 9 %ORT SUPPLY1 AN- O!R SAV+ IS R1LIZ$ 9 T ONLY ! F9AL ANSW] (A DI6ICULT SET ( CALCUL,NS NE$ 2 RECORD$1 ?US DO+ AWAY )! NE$ ( PAGES & PAGES ( S-CALL$ 8SCRAT* PA- P]04 .,PORTABIL;Y3 ,! ABACUS IS SMALL1 LIRY4 ,UN- LIKE MOD]N CALCULAT+ EQUIP;T1 : RE- ,US+ ! ,CRANM] ,ABACUS #C QUIRES A S\RCE ( ELECTRICAL P[] OR AT L1/ REPLACE;T ( PAP] TAPE1 ! ABA- CUS IS SELF-3TA9$ &1 IF Y H ! DE- VICE1 Y H "EY?+ NE$$ 6D ! MO/ -PLI- CAT$ ( CALCUL,NS4 ..,FREEDOM F ,M5- TAL .,EF=T3 ,ANO!R -PELL+ R1SON =! POPUL>;Y (! ABACUS IS ! FACT T1 ONCE HAV+ MA/]$ ! PROCESSES ( ABACUS CAL- CUL,NS1 ! OP]ATOR IS -PLETELY FREE F ! NE$ 6D M5TAL >I?METIC1 = ABACUS CALCUL,N IS -PLETELY AUTOMATIC & ME*ANICAL 9 NATURE4 ,UN=TUNATELY1 ! ABACUS1 Z US$ BY SIE FREE 6TRAVEL BACK & =? ON _! COLUMNS ) VIRTU,Y NO RE- SI/.E4 ,! M]E/ T\* (A F+] W MOVE A C.T]4 ,? 3DI;N1 : MAKES ! ABACUS S USE;L 6SIN+ 6OP]ATE ! DEVICE4 ,ATTA9;T ( ? GOAL WD H BE5 IMPOSSI# 7 X N =! MAN 6:OM ? BOOK IS D$ICAT$1 ,MR4 ,TAKA%I ,KOJIMA1 ( ,TOKYO1 ,JA- PAN4 ,"? ! PAGES (! BOOK1 ,,! ,,JAP- ANESE ,,ABACUS1 ,,XS ,,USE ,,& ,,!O- RY1 PUBLI%$ 0! ,*>LES ,E4 ,TUTTLE ,-PANY ( ,RUTL&1 ,V]MONT1 ,MR4 ,KO- JIMA 2CAME \R T1*] & FR4 ,"? 8 BOOK1 HE HAS GIV5 U ! MO/ PRICE.S GIFT "O MAN C 2/[ ^U ANO!R1 ! GIFT ( K4 ,X IS = ? R1SON T ? "W IS D$ICAT$ 6HM4 ,,US+ ! ,CRANM] ,ABACUS #E ,! ,PROMISE (! ,ABACUS ,SOON1 ,CRANM] & ^? ( U ASSOCIAT$ ) HM 9 ? "W1 2GAN EXPLOR+ ! MY/]IES ( ABACUS CALCUL,NS4 ,! DEEP] WE DELV$1 ! M WE LE>N$4 ,! M WE LE>N$1 ! M FAS- C9AT$ WE 2CAME4 ,=! 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OP],NS : WE W LE>N & USE 9 A4I;N >E AL V IMPORTANT 6MULTIPLIC,N1 SUBTRAC;N1 & ,US+ ! ,CRANM] ,ABACUS #AC DIVI.N4 ,"!=E1 X IS .MO/ IMPORTANT T WE LE>N H[ 6ADD & H[ 6ADD WELL 2F WE LE>N ANY?+ ELSE4 ,A4+ ,"O ,DIGIT ,NUMB]S ,WE 2G9 LE>N+ 6ADD 0A4+ ! NUMB] #A 6XF A NUMB] ( "TS4 ,WE W SET ! NUMB] #A & 3T9UE 6ADD X 6! "R-MO/ COLUMN ( ! ABACUS4 ,9 ORD] T WE W ALL 2 TALK+ ! SAME LANGUAGE1 WE W LABEL ? COLUMN ,COLUMN ;C4 ,! "O IMMLY 6XS LEFT W 2 COLUMN ;B &! "O IMMLY 6! LEFT ( COL- UMN ;B W 2 COLUMN ;A4 ,ALL ! REMA9+ COLUMNS ON ! ABACUS W 2 UNUS$ 9 \R "W =! PRES5T "T4 ,? LABEL+ ( COLUMNS ) LRS W ONLY S]VE 6ID5TIFY COLUMNS 9 S F> Z !Y PLAY A "P 9 A GIV5 PRO#M4 ,! LABEL+ IS >BITR>Y1 & ":1 9 "O CASE1 COLUMN ;A MAY 2 NE> ! "R 5D ( YR ABACUS1 9 ANO!R X MAY 2 AT ! MID- DLE OR LEFT4 ,ON COLUMN ;C SET ! NUMB] #A 0MOV+ "O (! L[] B1DS UP 6! B>4 ,NEXT1 ADD #A 6? #A4 ,AG1 ADD ANO!R #A4 ,N[1 ADD / "O M #A4 ,COL- UMN ;C %D N[ 2 %[+ ! 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RULE WE M/ RE- MOVE ! #I F COLUMN ;C & SET #A ON COLUMN ;B4 ,COLUMN ;B N[ %[S #A1 COLUMN ;C #J1 &! NUMB] #AJ1 T IS ! DIGITS #A1 #J APPE> ON COLUMNS ;B & ;C4 ,IF WE WANT 6ADD / M "OS !N WE M/ SET !M ON COLUMN ;C4 ,6! #AJ ON COLUMNS ;BC ADD #A 0SETT+ #A ON COL- UMN ;C4 ,WE N[ H ! DIGITS #AA ON COLUMNS ;B & ;C4 ,? REPRES5TS ! NUM- B] #AA4 ,N[1 3T9UE A4+ "OS Z 2F4 ,:5 Y GET 6#AD Y W AG H 6USE ! F/ RULE = ! 9DIRECT A4I;N ( #A1 T IS1 SET #E1 CLE> #D )! D[NW>D SWEEP (! 9DEX F+]4 ,? W GIVE Y A TOTAL ( #AE4 ,AG1 :5 Y H R1*$ #AI Y W H 6APPLY ! SECOND RULE OR SECRET =! 9DIRECT A4I;N ( #A : SAYS3 6ADD #A1 CLE> #I1 SET #A LEFT4 ,? W GIVE Y #B ON ;B1 #J ON ;C1 &! NUMB] #BJ ON COLUMNS ;BC4 ,N[ ON COLUMN ;C ADD #A4 ,? GIVES #BA4 ,3T9UE UNTIL Y R1* #DD4 ,AG1 SET #E1 CLE> #D )! D[NW>D SWEEP (! 9DEX F+] ON COLUMN ;C1 & ? W GIVE #DE4 ,3T9UE A4+ #A 6! #DE UNTIL Y R1* ! NUMB] #DI4 ,ONCE AG X IS NEC 6CLE> #I & SET #A LEFT1 T IS1 CLE> ! #I F ;C & SET #A ON COLUMN ;B4 ,9 ORD] 6SET #A ON COLUMN ;B Y W H 6D X 9DIRECTLY4 ,T IS1 Y W H 6SET #E & CLE> #D ON COLUMN ;B ) ANO!R D[NW>D SWEEP (! 9DEX F+]2 ONLY 9 ? CASE1 9/1D ( SETT+ #E1 CLE>+ #D1 Y W 2 SETT+ #EJ1 & CLE>+ #DJ4 ,N[ 3T9UE TO ADD #A ON UP "? ! #EJ'S1 #FJ'S1 #GJ'S1 #HJ'S1 & #IJ'S4 ,EV5TU,Y1 Y W R1* #IH4 ,ADD #A & Y W H #II1 & X AG 2COMES NEC 6ADD #A 9DIRECTLY4 ,6D ?1 CLE> #I1 SET #A LEFT4 ,CLE> ! #I F COLUMN ;C4 ,!N1 NOTICE T X IS IMPOS- SI# 6SET #A ON COLUMN ;B 2C ALL (! C.T]S ON ;B >E ACTIVE1 T IS1 >E AG/ ! B>4 ,S1 Y M/ CLE> ! #I F ;B & SET #A ON COLUMN ;A4 ,? GIVES Y #A ON ;A & #J'S ON ;B & ;C =! ?REE-DIGIT NUM- B] #AJJ4 ,US+ ! ,CRANM] ,ABACUS #AG ,! ,SECRETS ( ,A4I;N ,WE H ALR TOLD Y T ! ,CRANM] ,ABA- CUS IS BAS$ ^U ! ,JAPANESE ,ABACUS4 ,9 US+ ! ABACUS1 :5 X IS IMPOSSI# TO ADD ! NUMB] DIRECTLY1 S* Z WE H ALR SE5 9 ! CASE ( #D PLUS #A1 #I PLUS #A1 ! ,JAPANESE MAKE USE ( 9DIRECT M1NS4 ,EXCEPT =! NUMB] #E1 ! NUMB]S F #A 6#I C 2 A4$ 9 ?REE DI6]5T WAYS2 EI 0DIRECT M1NS OR 0A *OICE ( TWO 9DIRECT WAYS4 ,! NUMB] #E C ONLY 2 A4$ 9 TWO WAYS1 "O DIRECT & "O 9DI- RECT4 ,! ,JAPANESE H CALL$ ! 9DIRECT ME?ODS ( A4+ NUMB]S .SECRETS1 & ,JAPANESE S*OOL *N1 LE>N+ 6USE ! SOROBAN1 MEMORIZE ^! SECRETS UNTIL _! USE 2COMES AUTOMATIC4 ,SOON WE W LI/ ALL ! SECRETS = AD- DI;N 9 CASE Y WD L 6MEMORIZE !M4 ,B F/1 = ^? :O WD R LE>N ! BASIC PR9AI- PLE ^U : ! ABACUS OP]ATES1 WE W TRY 6EXPLA9 ! BASIS = USE (! SECRETS4 ,WE M/ MAKE USE ( "O (! SECRETS = A4I;N :5 X IS N POSSI# = U 6ADD A NUMB] DIRECTLY4 ,= EXAMPLE1 #D PLUS #A EQUALS #E4 ,SET ! NUMB] #D ON A COLUMN4 ,Y W NOTE T ) ALL #D "O-UNIT C.T]S MOV$ UP 6! B>1 X IS N POSSI# 6ADD #A DIRECTLY4 ,F\R PLUS #A IS ! SAME Z #D PLUS 7#E M9US #D74 ,AT F/ ? MAY SEEM Q -PLICAT$1 B ?9K ( X ? WAY4 ,SUPPOSE 9 YR POCKET Y H #D P5NIES4 ,SUPPOSE1 TOO1 T ,I [E Y A P5NY & AM WILL+ 6PAY Y4 ,UN- =TUNATELY1 ,I D N H A S+LE P5NY1 B ,I D H A FIVE-C5T PIECE4 ,IF ,I GIVE Y ! FIVE-C5T PIECE & Y1 9 TURN1 GIVE ME F\R C5TS *ANGE1 :AT ,I H ACTU,Y D"O IS 6ADD "O C5T 6! F\R C5TS : Y ALR _H1 GIV+ Y A GR& TOTAL ( FIVE C5TS4 ,6PUT X 96! LANGUAGE (! RULE OR SECRET1 96YR POCKET ,I H SET #E1 CLE>$ #D C5TS4 ,! SAME ?+ IS TRUE :5 WE 3SID] ! PRO#M #I PLUS #A EQUALS #AJ4 ,N9E PLUS #A IS ! SAME Z #I M9US 7#I PLUS #AJ74 ,:AT 0 TRUE (! FIVE-C5T PIECE ,US+ ! ,CRANM] ,ABACUS #AI IS AL TRUE (! T5-C5T PIECE4 ,IF Y H N9E C5TS & ,I WANT 6GIVE Y A P5NY B H NO "O-C5T PIECE1 ,I C GIVE Y A DIME1 & Y 9 TURN C GIVE ME YR N9E C5TS4 ,? RESULTS 9 ! A4I;N ( #A C5T 6#I1 =A TOTAL ( #AJ C5TS4 ,! SECRETS MAKE USE ( :AT >E CALL$ .-PLE;T>Y .NUMB]S4 ,^! MIE NUMB]S T REPRES5T ! DI6];E 2T ! NUMB] Y WANT 6ADD & EI #E OR #AJ4 ,! -PLEM5- T>Y NUMB]S = #E1 >E #A & #D1 OR #B & #C4 ,! -PLE;T>Y NUMB]S = #AJ1 >E #A & #I1 #B & #H1 #C & #G1 #D & #F1 & #E & #E4 ,IF Y WD GA9 9SIY NUMB]S X %D HELP Y TO KEEP 9 M9D ! PR9CIPLES ( MAK+ *ANGE 9 A TRANSAC;N 9VOLV+ ! EX*ANGE ( M"OY4 ,6GIVE Y ANO!R EXAMPLE (! AP- PLIC,N (A SECRET1 3SID] #H PLUS #G EQUALS #AE4 ,WE H #H SET ON A COL- UMN1 & 6? #H WE WANT 6ADD #G4 ,! DI6];E OR -PLE;T>Y NUMB] = #G1 9 ORD] 6>RIVE AT #AJ1 IS #C4 ,EIE Z FOLL[S3 .,TO .,ADD .,SECRET #A """""" ,SET #E1 CLE> #D #A """""" ,CLE> #I1 SET #A LEFT #B """""" ,SET #E1 CLE> #C ,US+ ! ,CRANM] ,ABACUS #BA .,TO .,ADD .,SECRET 7,3T9U$7 #B """""" ,CLE> #H1 SET 0A LEFT #C """""" ,SET #E1 CLE> #B #C """""" ,CLE> #G1 SET #A LEFT #D """""" ,SET #E1 CLE> #A #D """""" ,CLE> #F1 SET #A LEFT #E """""" ,CLE> #E1 SET #A LEFT #F """""" ,SET #A1 CLE> #E1 SET #A LEFT #F """""" ,CLE> #D1 SET #A LEFT #G """""" ,SET #B1 CLE> #E1 SET #A LEFT #G """""" ,CLE> #C1 SET #A LEFT #H """""" ,SET #C1 CLE> #E1 SET #A LEFT #H """""" ,CLE> #B1 SET #A LEFT #I """""" ,SET #D1 CLE> #E1 SET #A LEFT #I """""" ,CLE> #A1 SET #A LEFT ,ONCE AG1 6SET #A LEFT M1NS 6SET ! DIGIT #A ON ! COLUMN IMMLY 6! LEFT ( ! "O OO : WE >E "W+4 ,IF : COLUMN 3TA9S A #I Y ( C\RSE _C SET #A ON X DIRECTLY4 ,S1 CLE> ! #I1 & SET #A ON ! NEXT COLUMN OV] 6! LEFT4 ,PRACTICE ,EX]CISES ,9 ORD] = Y 6GA9 SKILL 9 ! USE (! ABACUS X W 2 NEC T Y PRACTICE1 PRAC- TICE1 & PRACTICE AG4 ,AN EX]CISE : W HELP Y 6GA9 SKILL IS 62G9 )! NUMB] #A2 6?1 ADD #B PLUS #C1 PLUS #D1 PLUS #E1 PLUS #F1 PLUS #G1 PLUS #H1 PLUS #I4 ,IF Y H D"O ALL (! /EPS CORRECTLY YR ANSW] %D 2 #DE4 ,IF Y 3T9UE ! PROCESS & 6#DE1 ADD #A PLUS #B1 PLUS #C1 PLUS #D1 & S ON UP 6#I1 YR NEXT ANSW] %D 2 #IJ4 ,EA* "T Y -PLETE A R.D F #A 6#I YR ANSW] W IN- CR1SE 0#DE4 ,:5 Y H D"O X #BJ "TS YR ANSW] W 2 #IJJ4 ,= V>IETY1 Y MII?METIC D _! A4I;N F "R TO LEFT4 ,= EXAMPLE1 9 "W+ ! PRO#M #AH PLUS #BG PLUS #CF EQUALS #HA1 Y WD ADD F/ ! #H ( #AH1 ! #G ( #BG1 &! #F ( #CF1 =A TOTAL ( #BA4 ,Y WD SET D[N ! #A ( #BA & C>RY ! #B 6! NEXT COL- UMN4 ,A4+ ! #B ( #BA1 ! #A ( #AH1 ! #B ( #BG1 &! #C ( #CF1 =A TOTAL ( #H4 ,? #H WD 2 SET IMMLY 6! LEFT (! #A OBTA9$ AT ! 5D (! F/ COLUMN ( FIG- URES1 & ? WD GIVE ! F9AL TOTAL ( #HA4 ,US+ ! ABACUS1 A4I;N IS D"O F LEFT 6"R1 & "! IS NO NE$ 6C>RY ANY NUMB]S AT ALL4 ,LET'S D ? SAME PRO#M ON ! ABACUS4 ,EI #C1 SET #A LEFT4 ,S1 REMOVE #C F COLUMN ;B1 & SET #A ON COLUMN ;A4 ,? N[ GIVES #D1 SET ON ;AB4 ,F9,Y1 6#AH PLUS #BG1 WE WANT 6ADD #CF4 ,SET #C ON ;A & #F ON ;B4 ,9 ORD] 6ADD #C 6;A WE M/ USE ! F/ SECRET = A4+ #C4 ,? IS 6ADD #C1 SET #E1 CLE> #B4 ,? GIVES U #G ON ;A & #GE ON ;AB4 ,9 ORD] 6ADD #F 6;B WE M/ USE ! F/ SECRET =! A4I;N (! NUMB] #F4 ,6ADD #F1 SET #A1 CLE> #E1 SET #A LEFT4 ,S1 !N1 ON ;B MAKE "O UPW>D MO;N )! 9DEX F+] : SETS #A & CLE>S #E4 ,!N %IFT YR ATT5;N 6COLUMN ;A1 & SET #A4 ,? GIVES Y #H ON ;A & #HA ON ,US+ ! ,CRANM] ,ABACUS #BE ;AB1 : IS \R F9AL TOTAL4 ,N[1 ALL ( ? MAY SEEM 6BE UNDULY -PLICAT$1 B REMEMB] WE >E GO+ /EP-BY-/EP "? A PROCESS : IS ACTU,Y EASI] 6D ?AN TO TALK AB4 ,:5 Y H -MITT$ ! SECRETS TO MEMORY1 OR1 H S -PLETELY "U/OOD ! !ORY ( _! OP],N Z 6MAKE YR F+] MOVE- ;TS AUTOMATIC1 Y W 2 A# 65T] & *ANGE NUMB]S )\T 3SCI\S ?"\ OR ANY M5TAL EF=T AT CALCUL,N4 ,2C (! NATURE ( XS OP],N1 ! ABACUS IS N A RECORD+ DEVICE1 T IS1 X IS N POSSI# 6RECORD ! /EP-BY-/EP PROCESS WE G "? 9 ORD] 6>RIVE AT A F9AL AN- SW]4 ,ALL WE H ON ! ABACUS IS A F9AL RESULT4 ,? M1NS "! IS NO WAY = U TO *ECK \R "W EXCEPT 6D X AG4 ,H["E1 ! SPE$ ) : ! ABACUS ALL[S U 6"W PRO#MS 9 >I?METIC M ?AN MAKES UP = ? APP>5T 4ADVANTAGE4 ,F ! EXAMPLE : WE J GAVE1 T IS1 #AH PLUS #BG1 PLUS #CF1 Y %D NOTE T1 Z WE 5T]$ NUMB]S ON ! ABACUS1 WE AU- TOMATIC,Y CALCULAT$ !M4 ,EIY ! OP],N 0CLE>+ YR ABACUS & TRY+ X AG1 ? "T />T+ ) #II PLUS #HH1 PLUS #GG1 PLUS #FF1 & ON D[N 6PLUS #AA--AT : PO9T Y %D AG H A TOTAL ( #DIE4 ,B1 ! ME?OD ( SETT+ NUMB]S W 2 DI6]5T4 ,REMEMB] .PRACTICE .MAKES .P]FECT1 & 2F GO+ ON )! /UDY (! ABACUS1 2 SURE T Y "U/& & C D P]FECTLY "EY?+ WE H COV]$ S F>4 ,! ,A4I;N ( ,NUMB]S ,HAV+ ,?REE OR ,M ,DIGITS ,NUMB]S HAV+ ?REE OR M DIGITS1 RE- G>D.S ( H[ _M1 >E A4$ 9 EXACTLY ! SAME WAY Z TWO DIGIT NUMB]S1 T IS1 ! A4I;N IS D"O F LEFT 6"R1 &! ONLY SPECIAL C>E T IS NE$$ IS 6BE SURE T EA* DIGIT IS SET ON A COLUMN ": X 2L;GS4 ,LET U "W AT ! "R 5D (! ABA- CUS1 & IF WE >E A4+ TWO ?REE-DIGIT NUMB]S1 WE W USE ! ?REE "R-MO/ COL- UMNS ON ! ABACUS4 ,SET ! DIGITS1 & X MAY WELL 2 T Z WE SET \R NUMB]S \R "W W SPILL OV] ONTO ! COLUMN IMMLY 6! LEFT1 9 O!R ^WS1 MAK+ USE (! F\R "R-MO/ COLUMNS ON ! ABACUS4 ,IF WE H A SEV5-DIGIT NUMB]1 & 6X WE WANT TO ADD A ?REE-DIGIT NUMB]1 WE SET ! ?REE-DIGIT NUMB] 9 XS PROP] PLACE1 ON ! ?REE COLUMNS AT ! "R 5D (! ABA- CUS1 &! ANSW] W =M AUTOMATIC,Y4 ,A4+ ,SUMS ( ,M"OY ,:5 A4+ SUMS ( M"OY1 : 3TA9 BO? DOLL>S & C5TS1 Y %D USE YR UNIT M>K4 ,IMAG9E ! UNIT M>K Z ! DECIMAL PO9T SEP>AT+ ! C5TS F ! DOLL>S4 ,! AM.T 4#CDG.AH WD 2 WRITT5 9 ? WAY3 ,US+ ! ,CRANM] ,ABACUS #BI #C1 #D1 #G1 )! #G ON A COLUMN IMMLY 6! LEFT (! UNIT M>K2 #A1 #H1 )! #A ( #AH ON ! COLUMN IMMLY 6! "R (! UNIT M>K4 ,IF1 6? AM.T1 WE WANT 6ADD 4#G.EJ1 WE USE ?REE COLUMNS1 SETT+ ! #G ( 4#G.EJ ON ! COLUMN IMMLY 6! LEFT (! UNIT M>K1 & IF 6? AM.T WE WANT 6ADD 4#I.JJ1 WE SET ! #I ON ? SAME COLUMN1 T IS1 ! COLUMN IMMLY 6! LEFT (! UNIT M>K1 2C WE >E ACTU,Y A4+ #I.JJ4 ,PRACTICE ,EX]CISES ,ON YR ABACUS SET #GHI4 ,6?1 ADD #GHI PLUS #GHI1 PLUS #GHI1 PLUS #GHI1 UNTIL Y H SET ? NUMB] #GHI =A TOTAL ( N9E "TS4 ,! CORRECT ANSW] 6? PRO#M ( A4I;N IS #G1AJA4 ,= ANO!R EX]CISE1 ADD #AAA1 PLUS #BBB1 PLUS #CCC1 PLUS #DDD1 PLUS #EEE1 ETC41 UNTIL Y H #III4 ,YR ANSW] %D 2 #D1IIE4 ,AG1 ON YR ABACUS SET #E1JDI4 ,? IS A F\R- DIGIT NUMB]4 ,6? NUMB] ADD #AAA PLUS #BBB1 PLUS #CCC1 ETC41 UNTIL Y H A4$ ALL (! NUMB]S UP TO & 9CLUD+ #III4 ,^! >E ?REE-DIGIT NUMB]S & %D 2 5T]$ CORRECTLY ON ! ?REE "R-MO/ COLUMNS4 ,IF Y D ALL ! /EPS CORRECTLY1 YR FI- NAL ANSW] W 2 #AJ1JDD4 ,ANO!R 9T]E/+ EX]CISE IS 6TR1T Z "O CYCLE ! NUMB]S #ABC PLUS #DEF1 PLUS #GHI4 ,REP1T ? CYCLE N9E "TS & Y W OBTA9 A F9AL RE- SULT ( #AB1CAB4 ,2F Y G ON 9 ? "W1 2 SURE Y "U/& :AT HAS BE5 COV]$ S F>4 ,2 SURE1 TOO1 T Y >E A# 6D ALL ! OP],NS : WE H \TL9$ S T Y W 2 9 A BETT] POSI;N 6"U/& & 6D ! /EPS : W FOLL[4 ,MULTIPLIC,N ,IF Y H PRACTIC$ 5 S T Y >E N[ A# 6D A4I;N ON ! ABACUS -=TABLY1 & >E A# 6"U/& :AT Y >E DO+1 Y >E N[ R1EY 62G9 ! /UDY ( MULTIPLIC,N4 ,MULTIPLIC,N IS RE,Y A RAPID =M ( A4I;N4 ,= EXAMPLE1 #I MULTIPLI$ 0#F IS #ED4 ,? IS ! SAME Z A4+ #I PLUS ,US+ ! ,CRANM] ,ABACUS #CA #I1 PLUS #I1 PLUS #I1 PLUS #I1 PLUS #I4 ,?US1 6MULTIPLY A NUMB] 0#C IS 6ADD T NUMB] ?REE "TS2 6MULTIPLY X 0#GD IS 6ADD ! SAME NUMB] SEV5TY-F\R "TS1 ETC4 ,EA* "P (A PRO#M 9 MULTIPLIC,N HAS A "N1 & 9 ORD] 64CUSS MULTIPLIC,N 9TELLIG5TLY1 WE %D LE>N ^! "NS S T WE W ALL 2 SP1K+ ! SAME LANGUAGE4 ,! NUMB] : IS MULTIPLI$ IS ! .MULTIPLI- C&4 ,! NUMB] 0: ! MULTIPLY+ IS D"O IS ! .MULTIPLI]1 &! ANSW] IS CALL$ ! .PRODUCT4 ,9 ! EXAMPLE #F MULTIPLI$ 0#C1 #F IS ! MULTIPLIC&1 MULTIPLI$ 0#C1 ! MULTIPLI]1 & X GIVES U A PRODUCT ( #AH4 ,9 "W+ \ A PRO#M 9 MULTIPLIC,N ON ! ABACUS ! PROC$URE IS Q SIMPLE4 ,! MULTIPLIC& IS MULTIPLI$ 0! MULTIPLI] &! PRODUCT IS SET 9 ! CORRECT POSI;N 6! "R (! MULTIPLIC&4 ,IF ! MULTIPLI- C& IS A NUMB] HAV+ M ?AN "O DIGIT1 EA* DIGIT (! MULTIPLIC& IS MULTIPLI$ 0ALL ! DIGITS (! MULTIPLI]1 &! PRODUCT 7 ALL ^! /EPS ( MULTIPLIC,N IS SET ON ! ABACUS 6! "R (! MULTI- PLIC& 9 A3ORD.E ) RULES ( POSI;N+ : W 2 5CUSS$ LAT]4 ,:5 A DIGIT (! MUL- TIPLIC& HAS BE5 MULTIPLI$ 0ALL ! DIGITS (! MULTIPLI]1 X IS CLE>$ & \R ATT5;N IS C5T]$ ON ! NEXT DIGIT 6! LEFT4 ,:5 WE H -PLET$ MULTIPLY+ ALL ! DIGITS (! MULTIPLIC& 0ALL DIGITS ( ! MULTIPLI]1 ! MULTIPLIC& W H BE5 CLE>$1 & \R TOTAL ANSW] OR GR& PROD- UCT W H =M$ AUTOMATIC,Y ^U ! ABACUS BO>D4 ,= \R F/ EXAMPLE ( MULTIPLIC,N1 LET'S MULTIPLY A "O-DIGIT NUMB] BY ANO!R "O-DIGIT NUMB]4 ,SIX MULTIPLI$ 0#G IS #DB4 ,ON ! F/ COLUMN AT ! EX- TREME LEFT-H& 5D (! ABACUS SET ! MULTIPLI] #G4 ,NEXT1 L1VE TWO COL- UMNS UNUS$ & SET ! MULTIPLIC& #F4 ,? M1NS T #G %D 2 SET ON COLUMN ;A1 COLUMNS ;B & ;C W 2 UNUS$1 &! MUL- TIPLIC&1 #F1 W 2 SET ON ;D4 ,9 ALL ! EXAMPLES : WE % GIVE1 WE W ASSUME T ! MULTIPLIC& IS SET UP 9 S* A WA# T ,US+ ! ,CRANM] ,ABACUS #CC XS LA/ DIGIT W O3UR ON A UNIT COL- UMN1 T IS1 ON A COLUMN IMMLY 6! LEFT (A UNIT M>K4 ,S1 :5"E WE SAY T ! MULTIPLIC& IS SET ON COLUMN ;D1 OR ON COLUMNS ;DEF1 WE W ASSUME T ! LA/ DIGIT (! MULTIPLIC& IS SET ^U A UNIT COLUMN4 ,BE> ? 9 M9D & FOLL[ ? PRAC- TICE4 ,N[1 9 MULTIPLY+ #F 0#G WE H ALR LE>N$ ! MULTIPLIC,N TA#1 S WE AUTOMATIC,Y SET ! PRODUCT #DB IMMLY 6! "R (! MULTIPLIC&1 OR ON COLUMNS ;EF4 ,IF Y H GD MEMORY X IS N NEC TO SET ! MULTIPLI] ( E PRO#M1 B Y %D "K H[ 6SET D[N AN 5TIRE PRO#M S T Y W 2 A# 6D X :5 "W+ ) EXTREMELY -PLEX PRO#MS4 ,N[1 LET'S D A PRO#M HAV+ A "O-DIGIT MULTIPLIC& &A TWO-DIGIT MULTIPLI]4 ,! PRO#M3 #G MULTIPLI$ BY #DB IS #BID4 ,F/1 SET ! MULTIPLI] #DB ON COLUMNS ;AB4 ,L1VE ;C & ;D UNUS$4 ,!N1 SET ! MULTIPLIC& #G ON COLUMN ;E : M/ 2 A UNIT COLUMN4 ,N[1 WE >E R1DY 62G9 MULTIPLY+4 ,! F/ /EP IS 6MULTIPLY ! #G 0! #D (! #DB4 ,SEV5 0#D IS #BH4 ,S1 SET #BH ON COLUMNS ;FG4 ,NEXT1 MULTIPLY ! MUL- TIPLIC& #G 0! #B ( #DB4 ,SEV5 0#B IS #AD4 ,SET ! #A (! #AD ON COLUMN ;G & ! #D ON COLUMN ;H4 ,NEXT1 CLE> ! MULTIPLIC& #G F COLUMN ;E &! JOB W 2 D"O1 L1V+ ! PRODUCT #BID1 ON COLUMNS ;FGH4 ,4CUS.N ,NOTICE T ! F/ "T WE MULTIPLI$ ! MULTIPLIC& #G1 WE MULTIPLI$ X 0! #D (! #DB1 GIV+ U ! PRODUCT ( #BH4 ,:5 ? 0 D"O1 WE !N MULTIPLI$ X 0! #B ( #DB1 GIV+ U ! PRODUCT ( #AD4 ,0MUL- TIPLY+ 9 ? WAY1 T IS1 0APPLY+ ! DIG- ITS (! MULTIPLI] 9 ! ORD] ( _! O3UR- R;E1 WE D AWAY )! NE$ 6C>RY NUMB]S1 & \R PRODUCT IS =M$ AUTOMATIC,Y4 ,NOTICE1 TOO1 ! WAY 9 : ! DIGITS (! PRODUCT 7 5T]$ ONTO ! ABACUS4 ,F/1 WE MULTIPLI$ ! MULTIPLIC& #G 0#D ( #DB4 ,SEV5 0#D IS #BH1 & ? PRODUCT 0 ,US+ ! ,CRANM] ,ABACUS #CE SET D[N IMMLY 6! "R (! MULTIPLIC&1 OR ON COLUMNS ;FG4 ,!N1 WE MULTIPLI$ ? #G 0! #B ( #DB4 ,SEV5 0#B IS #AD4 ,! #A (! #AD 0 SET ON COLUMN ;G &! #D ON COLUMN ;H4 ,9 O!R ^WS1 ! #H ( #BH &! #A ( #AD OV]LAPP$4 ,? IS AN EXTREMELY IMPORTANT 3CEPT = Y 6"U/&4 ,! F/ "T WE MULTIPLY ! MULTIPLIC& 0A DIGIT (! MULTIPLI]1 ! RESULT+ PRODUCT IS 5T]$ ^U ! F/ POSI;N4 ,!N1 ! PRODUCT RESULT+ F ! SECOND MULTI- PLIC,N (! MULTIPLIC& IS 5T]$ ^U ! SECOND POSI;N4 ,SEV5 0#D IS #BH1 F/ MULTIPLIC,N1 F/ POSI;N4 ,SEV5 0#B IS #AD1 SECOND MULTIPLIC,N1 SECOND PO- SI;N4 ,! ,POSI;N ,3CEPT ,A POSI;N 3SI/S ( TWO COLUMNS4 ,EXACTLY : COLUMNS W DEP5D ^U ! POSI- ;N'S RANK1 T IS1 :E!R X IS ! F/ PO- SI;N1 ! SECOND POSI;N1 ?IRD POSI;N1 & S ON4 ,! F/ POSI;N IS MADE UP ( COLUMNS #A & #B IMMLY 6! "R ( T DIG- IT (! MULTIPLIC& 6: ! POSI;N 2L;GS2 ! SECOND POSI;N IS MADE UP ( COLUMNS #B & #C2 ! ?IRD POSI;N1 COLUMNS #C & #D2 ! F\R? POSI;N1 COLUMNS #D & #E2 & S ON4 ,F ?1 Y C SEE T1 IF X 7 NEC 6MULTIPLY A DIGIT 9 ! MULTIPLIC& #BG "TS1 ! TW5TY-SEV5? POSI;N WD 2 COL- UMNS #BG & #BH 6! "R ( T DIGIT4 ,! F/ "T T A DIGIT (! MULTIPLIC& IS MULTIPLI$1 ! RESULT+ PRODUCT IS 5T]$ 9 ! F/ POSI;N4 ,PRODUCT (! SECOND MULTIPLIC,N IS 5T]$ 96! SECOND PO- SI;N2 ?IRD MULTIPLIC,N1 ?IRD POSI;N2 F\R? MULTIPLIC,N1 F\R? POSI;N4 ,Y H N YET HE>D ! 5TIRE /ORY ( POSI;NS1 B ? IS Z F> Z WE %D G =! PRES5T4 ,LET'S FIRM UP \R "U/&+ ) ANO!R EX- AMPLE4 ,FIVE MULTIPLI$ 0#CG IS #AHE4 ,ON COLUMNS ;AB1 SET ! MULTIPLI] #CG4 ,L1VE ;C & ;D UNUS$1 & SET ! MULTI- PLIC& #E ON COLUMN ;E1 : M/ 2 A UNIT COLUMN4 ,N[1 MULTIPLY ! MULTIPLIC& #E 0! #C ( #CG4 ,FIVE 0#C IS #AE4 ,US+ ! ,CRANM] ,ABACUS #CG ,S1 SET #AE 9 ! F/ POSI;N1 ON COL- UMNS ;FG4 ,NEXT1 MULTIPLY ! #E 9 ! MULTIPLIC& 0! #G ( #CG4 ,FIVE 0#G IS #CE4 ,SET #CE ON COLUMNS ;GH4 ,C.EL ! MULTIPLIC& #E4 ,! JOB IS D"O1 &! CORRECT PRODUCT #AHE W 2 F.D ON COL- UMNS ;FGH4 ,4CUS.N ,! F/ MULTIPLIC,N ( #E RESULT$ 9 A PRODUCT ( #AE4 ,? 0 5T]$ 96! F/ PO- SI;N4 ,! PRODUCT (! SECOND MULTIPLI- C,N ( #E1 ? "T 0! #G ( #CG 0 #CE4 ,! SECOND MULTIPLIC,N1 "!=E1 5T] 96! SECOND POSI;N4 ,E>LI]1 WE SD T A ?OR\< "W+ K (! SECRETS ( A4I;N 0 ES- S5TIAL 2F GO+ ON )! /UDY (! ABACUS4 ,LET'S %[ Y :AT WE M1N 0ANO!R EXAM- PLE4 ,? "T1 MULTIPLY #F 0#CG4 ,5T] ! MULTIPLI] #CG1 ON COLUMNS ;AB4 ,L1VE ;C & ;D UNUS$1 & SET ! MULTIPLIC& #F ON COLUMN ;E1 AG A UNIT COLUMN4 ,SIX 0#C IS #AH4 ,S1 SET ! PRODUCT1 #AH ON COLUMNS ;FG4 ,NEXT1 #F 0#G IS #DB4 ,S SET ! PRODUCT #DB ON COLUMNS ;GH4 ,F9,Y1 CLE> COLUMN ;E (! MUL- TIPLIC& #F1 &! JOB IS D"O1 L1V+ ! CORRECT PRODUCT ( #BBB ON COLUMNS ;FGH4 ,4CUS.N ,SIX 0#C 0 ! F/ MULTIPLIC,N1 & XS PRODUCT #AH 0 5T]$ 96! F/ POSI;N4 ,SIX 0#G 0 ! SECOND MULTIPLIC,N (! MULTIPLIC&1 &! PRODUCT #DB1 0 5T]$ 96! SECOND POSI;N4 ,B1 9 ORD] 65T] ! #D ( #DB ON COLUMN ;G1 ! COLUMN CON- TA9+ ! #H ( #AH1 X 0 NEC 6MAKE USE ( "O (! SECRETS =! A4I;N ( #D4 ,2C1 BY 5T]+ #D ^U A COLUMN ALR 3TA9+ ! NUM- B] #H1 WE 7 A4+ #D 6#H4 ,! ONLY SE- CRET : WE CD POSSIBLY H APPLI$ 0 6ADD #D1 CLE> #F1 SET #A LEFT4 ,:5 WE LEFT \R 4CUS.N (! POSI;N 3CEPT1 WE SD T WE _H N TOLD ! 5TIRE /ORY4 ,Y W NOTE T 9 ALL ! EXAMPLES WE H US$ S F>1 E "T WE MULTIPLI$ ! ,US+ ! ,CRANM] ,ABACUS #CI MULTIPLIC& 0A DIGIT (! MULTIPLI]1 WE OBTA9$ A TWO-DIGIT SUB-PRODUCT4 ,WE H SD T A POSI;N 3SI/S ( TWO COLUMNS4 ,S1 IF WE H TWO COLUMNS & TWO DIG- ITS1 "EY?+ %D -E \ EV54 ,B :AT HAP- P5S :5 WE H A SITU,N 9 : MULTIPLIC,N RESULTS 9 A "O-DIGIT PRODUCT8 ,LET'S D AN EXAMPLE & F9D \4 ,? "T1 #D MUL- TIPLI$ 0#DB EQUALS #AFH4 ,ON COLUMNS ;AB1 SET ! MULTIPLI] #DB4 ,L1VE ;C & ;D UNUS$1 & SET ! MULTIPLIC& #D ON ! UNIT COLUMN ;E4 ,F\R 0#D IS #AF4 ,S1 SET ! PRODUCT #AF ON COLUMNS ;FG4 ,F\R 0#B IS #H4 ,S1 SET #H ON COLUMN ;H4 ,!N CLE> COLUMN ;E ( XS MULTI- PLIC& #D1 &! JOB IS D"O1 L1V+ #AFH ON COLUMNS ;FGH4 ,4CUS.N ,WE H SD T :5 A DIGIT (! MULTIPLI- C& IS MULTIPLI$ 0A DIGIT (! MULTI- PLI]1 ! RESULT+ PRODUCT M/ 2 5T]$ 9 XS APPROPRIATE POSI;N4 ,:5 ? PRODUCT IS A TWO-DIGIT NUMB] WE H NO PRO#M S9CE WE USE BO? COLUMNS (! POSI;N4 ,:5 ! PRODUCT IS A S+LE-DIGIT NUMB]1 WE W H NO PRO#M EI1 IF WE W REMEMB] T .S+LE-DIGIT .PRODUCTS >E 5T]$ ^U ! .SECOND .COLUMN (! POSI;N4 ,9 ! EXAMPLE WE J DID1 #D 0#DB1 #D 0#D 0 #AF--F/ MULTIPLIC,N1 F/ PO- SI;N1 RESULT+ 9 #AF 2+ 5T]$ ON COL- UMNS ;FG4 ,!N1 #D 0#B 0 #H--! SECOND MULTIPLIC,N1 "!=E1 SECOND POSI;N4 ,B1 #H IS A S+LE-DIGIT NUMB]4 ,"!=E1 X M/ 2 5T]$ ^U ! .SECOND .COLUMN ( XS POSI;N4 ,"! IS A SIMLE TE*NIQUE = LOCAT+ N ONLY ! CORRECT POSI;N B ! CORRECT COLUMN )9 A POSI;N =! 5TRY (A GIV5 DIGIT4 ,SET UP YR ABACUS ) #DB ON ;AB1 L1VE ;C & ;D UNUS$1 & SET #D ON COLUMN ;E4 ,! DIGIT (! MULTIPLIC& ) : Y >E "W+ AT A GIV5 "T IS CALL$ ! .M>K] .COLUMN1 & POSI;NS >E M>K$ (F 6! "R ( ? COLUMN4 ,LOCATE COLUMN ;E1 ! M>K] COLUMN1 )! 9DEX F+] ( YR LEFT H&4 ,N[1 PLACE ! 9DEX F+] ( YR "R H& ,US+ ! ,CRANM] ,ABACUS #DA ON COLUMN ;F &! MI4LE F+] ( YR "R H& ON COLUMN ;G4 ,! 9DEX & MI4LE F+?S ( YR "R H& >E N[ RE/+ ^U ! F/ POSI;N4 ,MOVE ? PAIR ( F+]S "O PLACE 6! "R4 ,! 9DEX F+] %D 2 ON COLUMN ;G1 ! MI4LE F+] ON COLUMN ;H4 ,OBS]VE T ? IS ! SECOND POSI;N1 & T ! MI4LE F+] ( YR "R H& IS RE/+ ^U ! .SECOND .COLUMN ( ? POSI;N4 ,9 ACTUAL PRAC- TICE1 !N1 :5 ! PRODUCT 6BE 5T]$ ON A GIV5 POSI;N IS A TWO-DIGIT NUMB]1 ! F/ DIGIT W 2 5T]$ ^U ! COLUMN ON : YR 9DEX F+] RE/S1 &! SECOND DIGIT W 2 5T]$ ^U ! COLUMN T\*$ 0YR MI4LE F+]4 ,9 ! SAME WAY1 IF ! PRODUCT IS A S+LE DIGIT NUMB] XS 5TRY W 2 MADE ^U A COLUMN ON : YR MI4LE F+] RE/S 2C YR MI4LE F+] W ALW RE/ ^U ! .SEC- OND .COLUMN (A POSI;N4 ,S F> WE H "W$ ONLY ) "O-DIGIT MULTIPLIC&S4 ,N[1 LET'S TRY AN EXAM- PLE )A ?REE-DIGIT MULTIPLIC& B A "O- DIGIT MULTIPLI]4 ,EI ;E (! #A ( #HAI1 & MUL- TIPLY ! #H ( #HAI 0#I4 ,EI ;D ( XS #H1 &! JOB IS D"O1 L1V+ ! CORRECT PRODUCT #G1CGA ON COLUMNS ;EFGH4 ,4CUS.N ,:5 WE MULTIPLI$ ! #I ( #HAI 0#I WE _H A PRODUCT ( #HA4 ,? 0 ! F/1 & ONLY1 MULTIPLIC,N ( #I4 ,S1 X 0 5T]$ 96! F/ POSI;N1 OR ^U COLUMNS ;GH4 ,!N1 WE C.ELL$ ! #I 9 ! MULTIPLIC&4 ,NEXT1 WE MULTIPLI$ ! #A ( #HAI 0\R MULTIPLI] #I4 ,"O 0#I IS #I4 ,? 0 ! ,US+ ! ,CRANM] ,ABACUS #DC F/ MULTIPLIC,N ( ? DIGIT (! MULTI- PLIC&1 & X RESULT$ 9 A "O DIGIT PRODUCT1 : IS 6BE 5T]$ 96! F/ POSI- ;N4 ,REMEMB]1 ?\<1 T "O DIGIT PROD- UCTS >E 5T]$ ^U ! SECOND COLUMN ( _! POSI;NS4 ,! SECOND COLUMN (! F/ PO- SI;N =! MULTIPLIC& DIGIT #A IS COL- UMN ;G4 ,S1 WE 5T] ! #I ON COLUMN ;G4 ,6D ?1 Y M/ USE A SECRET ( A4I- ;N1 : SAYS1 6ADD #I1 CLE> #A1 SET #A LEFT4 ,S1 REMOVE #A F COLUMN ;G & SET #A ON COLUMN ;F1 L1V+ #AGA ON COLUMNS ;FGH4 ,NEXT1 CLE> ;E ( XS #A4 ,N[1 WE >E R1DY =! F9AL /EP1 #H ( #HAI MULTIPLI$ 0\R MULTIPLI] #I4 ,EID MANN]4 ,CLE> ;D ( XS #H1 L1V+ #G1CGA ON ;EFGH4 ,N[ LET'S D AN EXAMPLE HAV+ A TWO- DIGIT MULTIPLI] &A ?REE-DIGIT MUL- TIPLIC&4 ,! PRO#M #HAI MULTIPLI$ BY #BC EQUALS #AH1HCG4 ,ON COLUMNS ;AB1 SET ! MULTIPLI] #BC4 ,L1VE ;C & ;D UNUS$1 & SET ! MULTIPLIC& #HAI ON COLUMNS ;EFG1 ) ;G Z A UNIT COLUMN4 ,MULTIPLY ! #I ( #HAI 0! #B ( #BC4 ,N9E 0#B IS #AH4 ,SET ? PRODUCT ON COLUMNS ;HI4 ,NEXT1 MULTIPLY ! #I ( #HAI 0! #C ( #BC4 ,N9E 0#C IS #BG1 & ? PRODUCT IS 6BE SET ON COLUMNS ;IJ4 ,NEXT1 CLE> ;G ( XS #I1 & %IFT YR ATT5;N 6COLUMN ;F4 ,MULTIPLY ! #A ( #HAI 0! #B ( #BC4 ,"O 0#B IS #B1 & ? PRODUCT %D 2 SET ON COLUMN ;H4 ,NEXT1 MULTIPLY ! #A 0#C4 ,"O 0#C IS #C & ? PRODUCT 2L;GS 6COLUMN ;I4 ,!N1 CLE> #A F COLUMN ;F & TURN YR ATT5;N 6COLUMN ;E : 3TA9S ! #H ( #HAI4 ,EI ;E ( XS #H1 &! JOB IS D"O1 L1V+ ! PRODUCT #AH1HCG ON COLUMNS ;FGHIJ4 ,4CUS.N ,\R F/ /EP 0 6MULTIPLY ! #I ( #HAI 0! #B ( #BC4 ,N9E 0#B RESULT$ 9 A ,US+ ! ,CRANM] ,ABACUS #DE PRODUCT ( #AH4 ,? 0 ! F/ MULTIPLIC,N ( #I1 & #AH 0 5T]$ 9 ! F/ POSI;N4 ,WE NEXT MULTIPLI$ #I 0! #C ( #BC1 GIV+ U ! PRODUCT ( #BG4 ,? 0 ! SEC- OND MULTIPLIC,N ( #I1 S1 5TRY ( ? PRODUCT 0 9 ! SECOND POSI;N4 ,65T] ! #B ( #BG ^U ! COLUMN 3TA9+ ! #H ( #AH1 X 0 NEC 6APPLY T SECRET =! A4I- ;N ( #B : SAYS 6ADD #B1 CLE> #H1 SET #A LEFT4 ,AT ? PO9T \R #I _H BE5 MULTIPLI$ 0ALL ! DIGITS (! MULTI- PLI]4 ,S1 #I IS R1DY 6BE CLE>$1 & WE >E R1DY 6TURN \R ATT5;N 6! DIGIT #A ( #HAI4 ,"O 0#B IS #B4 ,? IS ! F/ MULTIPLIC,N (! DIGIT #A1 &! RESULT+ PRODUCT IS 6BE 5T]$ ^U ! F/ POSI;N4 ,NEXT1 #A 0#C IS #C4 ,? IS ! SECOND MULTIPLIC,N ( #A1 "!=E1 5TRY IS ^U ! SECOND POSI;N1 & AG1 S9CE #C IS A "O-DIGIT NUMB]1 X IS 5T]$ ^U ! SEC- OND COLUMN ( XS POSI;N4 ,"O1 HAV+ BE5 MULTIPLI$ 0ALL DIGITS (! MULTI- PLI]1 IS R1DY 6BE CLE>$4 ,NEXT1 WE MULTIPLY ! #H ( #HAI 0#B4 ,EI #A4 ,N[ T ! #H HAS BE5 MULTIPLI$ 0ALL DIGITS (! MULTIPLI]1 X MAY 2 CLE>$1 L1V+ \R PRODUCT #AH1HCG ON ;FGHIJ4 ,REMEMB] T1 REG>D.S (! NUMB] ( DIGITS 9 ! MULTIPLI] OR ! MULTIPLIC&1 IF 5TRY ( PRODUCTS IS MADE ^U ! PROP] POSI;NS & IF ! CORRECT SECRETS =! A4I;N ( NUMB]S >E APPLI$1 YR MULTIPLIC,N W 2 D"O AUTOMATIC,Y1 &) ABSOLUTELY NO M5TAL EF=T ON YR "P4 ,( C\RSE1 Y M/ H A ?OR\< K (! MULTIPLIC,N TA#S UP TO & 9CLUD+ #I 0#I4 ,N[1 LET'S TRY "O M EXAMPLE4 ,WE'LL USE ! SAME MUL- TIPLIC&1 #HAI1 B ? "T \R MULTIPLI] W 2 A ?REE-DIGIT NUMB]1 #ABC1 & \R AN- SW]1 OR PRODUCT1 W 2 #AJJ1GCG4 ,ON COLUMNS ;ABC1 SET ! MULTIPLI] #ABC4 ,L1VE ;D & ;E UNUS$1 & SET #HAI1 ! MULTIPLIC&1 ON ;FGH1 ) ;H Z A UNIT ,US+ ! ,CRANM] ,ABACUS #DG COLUMN4 ,N[1 MULTIPLY ! #I ( #HAI BY #A4 ,N9E 0#A IS #I4 ,S1 SET #I ON COLUMN ;J4 ,NEXT1 MULTIPLY ! #I 0! #B ( #ABC4 ,N9E 0#B IS #AH4 ,SET #AH ON ;JK4 ,F9,Y1 MULTIPLY ! #I 0! #C ( #ABC4 ,N9E 0#C IS #BG1 & ? PRODUCT IS 5T]$ ^U ;KL4 ,N[1 CLE> #I F COL- UMN ;H & TURN YR ATT5;N 6COLUMN ;G4 ,MULTIPLY ! #A ( #HAI 0! #A ( #ABC4 ,"O 0#A IS #A4 ,S1 SET ? PRODUCT ON COLUMN ;I4 ,NEXT1 MULTIPLY #A 0#B4 ,"O 0#B IS #B4 ,? IS 6BE SET ON COL- UMN ;J1 & F9,Y1 #A 0#C IS #C1 & ? PRODUCT IS 6BE SET ON COLUMN ;K4 ,N[1 CLE> COLUMN ;G ( XS #A & TURN YR ATT5;N 6COLUMN ;F4 ,MULTIPLY ! #H ( #HAI 0! #A ( #ABC4 ,EI ;F ( XS #H &! JOB IS D"O1 L1V+ #AJJ1GCG Z ! PRODUCT ON ;GHIJKL4 ,4CUS.N ,! F/ "T WE MULTIPLI$ ! #I ( #HAI X 0 MULTIPLI$ 0#A1 GIV+ A PRODUCT ( #I4 ,? 0 6BE 5T]$ ^U ! F/ POSI;N4 ,N9E IS A "O-DIGIT NUMB]2 "!=E1 5TRY %D 2 MADE ON ! SECOND COLUMN (! F/ POSI;N4 ,NEXT1 #I 0#B GIVES U A PRODUCT ( #AH--SECOND MULTIPLIC,N1 "!=E1 SECOND POSI;N4 ,65T] #A ( #AH ON ! COLUMN ALR 3TA9+ ! #I ( \R PRE- VI\S MULTIPLIC,N1 X IS NEC 6USE T SECRET =! A4I;N ( #A1 : SAYS1 6ADD #A1 CLE> #I1 SET #A LEFT4 ,! #H ( #AH C !N 2 5T]$ 9 A /RAID WAY ON ! SECOND COLUMN ( XS POSI;N4 ,NEXT1 #I 0#C IS #BG4 ,65T] #B1 WE M/ USE ! SECRET = A4I;N : SAYS1 6ADD #B1 CLE> #H1 SET #A LEFT4 ,SEV5 C 2 5T]$ DIRECTLY4 ,? 0 ! ?IRD MULTIPLI- C,N1 "!=E1 5TRY 0 ^U ! ?IRD POSI;N1 & WE CLE>$ \R #I F ! MULTIPLIC&4 ,NEXT1 #A 0#A GAVE A PRODUCT ( #A-- F/ MULTIPLIC,N1 F/ POSI;N4 ,B1 S9CE #A IS A "O-DIGIT NUMB]1 X 0 5T]$ ^U ,US+ ! ,CRANM] ,ABACUS #DI ! SECOND COLUMN (! F/ POSI;N4 ,NEXT1 #A 0#B IS #B--SECOND MULTIPLIC,N1 "!=E1 5TRY IS MADE ^U ! SECOND POSI- ;N1 2C #B IS A "O-DIGIT NUMB]4 ,NEXT1 #A 0#C IS #C--?IRD MULTIPLI- C,N1 ?IRD POSI;N1 B ! SECOND COLUMN ( ? POSI;N 2C #C IS A "O DIGIT PROD- UCT4 ,N[1 WE CLE> #A F ! MULTIPLIC& & TURN \R ATT5;N 6! #H ( #HAI4 ,EI #H1 SET #A LEFT4 ,S1 WE CLE> ! #H1 B LOOK+ AT ! COLUMN IMMLY 6XS LEFT WE SEE T X 3TA9S A #I4 ,S1 WE CLE> ? #I1 & SET #A 6XS LEFT1 OR ON COLUMN ;G4 ,F9,Y1 65T] ! #D ( #BD1 WE M/ USE ! SECRET =! A4I;N ( #D : SAYS1 6ADD #D1 SET #E1 CLE> #A4 ,? IS D"O ON COLUMN ;J4 ,WE !N CLE> #H F ! MULTIPLIC&1 & \R PRODUCT1 #AJJ1GCG IS LEFT ON COLUMNS ;GHIJKL4 ,9 ALL ! PRO#MS : WE H D"O1 WE H SET BO? ! MULTIPLI] &! MULTIPLIC&4 ,WE H SD T "S P MIKS4 ,DO+ ? W SAVE Y BO? "T & EF=T :5 TRY+ 6LOCATE ! LA/ DIGIT ( YR PROD- UCT 9 CASE ! PRODUCT 3TA9S A NUMB] ( Z]OS4 ,DET]M9+ ! ,L5G? (! ,PRODUCT 6%[ :AT WE M1N 0?1 3SID] ! PRO#M #ABE MULTIPLIF 0#H EQUALS #A1JJJ4 ,! NUMB] #A1JJJ IS MADE UP (! DIG- ,US+ ! ,CRANM] ,ABACUS #EA ITS #A1 #J1 #J1 #J4 ,B1 ON YR ABACUS A Z]O MAY EASILY 2 3FUS$ = AN UNUS$ COLUMN4 ,H["E1 ? 3FU.N IS N POSSI# IF Y PLACE ! LA/ DIGIT (! MULTIPLIC& ^U A UNIT COLUMN & OBS]VE ! FOLL[+ RULE4 ,! LA/ DIGIT (! PRODUCT FALLS 6! "R (! LA/ DIGIT (! MULTIPLIC& 0Z _M PLACES Z "! >E :OLE-NUMB] DIGITS 9 ! MULTIPLI] PLUS "O A4I;NAL PLACE 2C ? IS A PRO#M 9 MULTIPLIC,N4 ,6SEE ? RULE 9 OP],N1 SET YR MULTIPLIC& #ABE 9 S* A WAY T ! #E ( #ABE FALLS ON A UNIT COLUMN1 T IS1 A COLUMN IMMLY 6! LEFT (A UNIT M>K4 ,LET'S CALL ? COL- UMN1 COLUMN ;E4 ,N[1 MULTIPLY #ABE BY #H1 FOLL[+ ! PROC$URES Y H ALR LE>N$4 ,FIVE 0#H IS #DJ--F/ MULTIPLIC,N1 F/ POSI;N4 ,TWO 0#H IS #AF--F/ MULTI- PLIC,N1 F/ POSI;N4 ,65T] #F1 USE ! SECRET 6ADD #F1 CLE> #D1 SET "O LEFT4 ,"O 0#H IS #H--F/ MULTIPLIC,N1 F/ POSI;N4 ,65T] #H1 USE ! SECRET : SAYS1 6ADD #H1 CLE> #B1 SET #A LEFT4 ,( C\RSE1 X GOES )\T SAY+ T AF EA* MULTIPLIC,N Y H C.ELL$ T DIGIT (! MULTIPLIC& : Y H J MULTIPLI$4 ,N[1 ! ABACUS %D 2 P]FECTLY CLE> EXCEPT =! DIGIT #A1 O3URR+ ON A C]TA9 COLUMN2 X DOESN'T MATT] : "O AT ! PRES5T "T4 ,WE "K T #ABE MULTIPLI$ 0#H IS N #A1 B J H[ _M #J'S FOLL[ ? #A8 ,RUN YR F+] AL;G ! FACE (! L[] $GE (! FRAME UNTIL Y F9D ! UNIT M>K 6! LEFT ( : Y H J SET ! #E ( #ABE4 ,! DOT IMMLY 6! LEFT ( ? UNIT M>K W !N M>K ! COLUMN ON : ? #E _H BE5 SET4 ,PLACE YR F+] ^U ? COLUMN & !N MOVE X 6! "R F/ "O PLACE 2C "! IS J "O :OLE NUMB] DIGIT 9 ! MULTIPLI]1 & !N "O A4I;NAL PLACE 2C ? IS A PRO#M 9 MULTIPLIC,N4 ,YR F+] %D N[ 2 RE/+ ON COLUMN ;G4 ,? IS N AN UNUS$ COLUMN !N1 B A COLUMN : 3TA9S ! DIGIT #J4 ,S1 ;G IS "O1 MOV+ 6! LEFT2 ;F IS TWO1 ;E IS ?REE #J'S1 & COLUMN ;D 3TA9S A #A4 ,S1 ! ANSW] IS "O FOLL[$ 0?REE #J'S1 OR #A1JJJ4 ,? %D GIVE Y ! IDEA (! IMPORT.E ( PLACE;T (! LA/ DIGIT (! MULTIPLIC& ^U A UNIT COLUMN4 ,B1 ! SUBJECT W RCV M ?OR\< 4CUS.N :5 WE TALK AB ! ,US+ ! ,CRANM] ,ABACUS #EC MULTIPLIC,N ( DECIMALS4 ,! ,TR1T;T ( ,Z]OS ,SUPPOSE Y H A MULTIPLI] S* Z #C1JJF1 &! MULTIPLIC& IS #G4 ,! F/ MULTIPLIC,N GIVES A PRODUCT ( #BA2 #G 0#C IS #BA4 ,SEV5 0#J IS #J1 #G 0#J IS #J1 #G 0#F IS #DB4 ,? PRODUCT1 #DB1 M/ 2 5T]$ 96! F\R? POSI;N4 ,9 O!R ^WS1 :5 Z]OS O3UR 9 ! MI4LE (A MULTIPLI]1 !Y M/ 2 TR1T$ Z ANY O!R DIGIT S F> Z ! POSI;N 3CEPT IS 3C]N$4 ,2F GO+ ON 9 ? "W1 2 SURE Y "U/& ?OR\N$ S F> 9 ! /UDY ( A4I;N & MULTIPLIC,N4 ,PRACTICE ,EX]CISES ,= PRACTICE1 MULTIPLY ! NUMB] #ABC1DEF1GHI1 A N9E DIGIT NUMB]1 F/ 0#AH1 !N 0#BG1 #CF1 #DE1 #ED1 #FC1 #GB1 #HA1 #II4 ,!N1 PRACTICE AG4 ,? "T MULTIPLY #IHG1FED1CBA BY #AH1 #BG1 #CF1 #DE1 #ED1 #FC1 #GB1 #HA1 #II4 ,WE C AL MAKE USE ( "S ( \R PRO#MS 9 A4I;N4 ,SEV5 HUNDR$ EIE TWO NUMB]S =! PURPOSE ( LE>N+ ! DI6];E 2T !M4 ,IF WE WANT TO "K ! DI6];E 2T #CGE & #IGE1 WE SUB- TRACT #CGE F #IGE1 & WE LE>N T ! DI6];E IS #FJJ4 ,! V>I\S "PS (A PRO#M 9 SUBTRAC;N >E "ND &1 9 ORD] 6AVOID 3FU.N1 WE %D LE>N ^! "QS 2F GO+ ANY FUR!R4 ,! NUMB] F : A NUMB] IS SUBTRACT$ IS CALL$ ! .M9U5D4 ,! NUMB] : IS SUBTRACT$ IS CALL$ ! .SUBTRAH5D1 &! DI6];E OR ANSW] : RE- SULTS IS CALL$ ! .REMA9D]4 ,9 \R #CGE & #IGE EXAMPLE1 #IGE1 ! NUMB] F : ! SUBTRAC;N IS D"O1 IS ! M9U5D2 #CGE1 ! NUMB] : IS SUBTRACT$1 IS ! SUBTRAH5D2 &! DI6];E1 #FJJ1 IS ! ,US+ ! ,CRANM] ,ABACUS #EE REMA9D]4 ,9 WRITT5 >I?METIC1 SUBTRACT;N IS D"O F "R 6LEFT4 ,= EXAMPLE1 9 WRITT5 >I?METIC ! #E ( #CGE WD 2 SUBTRACT$ F ! #E ( #IGE4 ,!N1 ! #G ( #CGE WD 2 SUBTRACT$ F ! #G ( #IGE4 ,!N1 & FI- N,Y1 ! #C ( #CGE IS SUBTRACT$ F ! #I ( #IGE1 L1V+ ! REMA9D] ( #FJJ4 ,:5 WE USE ! ABACUS1 H["E1 WE "W F LEFT 6"R1 J Z WE D 9 ALL ! PROCESSES ( >I?METIC4 ,S1 WE WD SET #IGE ON ! ABCUS1 & F ! #I WE WD SUBTRACT #C1 F ! #G WE WD SUBTRACT #G1 & F ! #E WE WD SUBTRACJ #E1 & AUTOMATIC,Y =M #FJJ Z \R REMA9D]4 ,Z 9 ! CASE ( A4I;N1 :5 A NUMB] IS 6BE SUBTRACT$ DIRECTLY1 ! NEWCOM] 6! USE (! ABACUS IS PRES5T$ ) NO S]I\S PRO#M4 ,= EXAMPLE1 X'S NO TRICK AT ALL 6SUBTRACT #B F #G4 ,ALL Y NE$ D IS MOVE ! TWO "O-UNIT C.T]S AWAY F ! B>4 ,? L1VES A REMA9D] ( #E4 ,B1 SUPPOSE Y WANT$ 6SUBTRACT #C F #G4 ,? 2COMES ANO!R /ORY4 ,X IS N POSSI# 6MOVE ?REE "O-UNIT C.T]S AWAY F ! B>1 & S WE M/ RESORT 6AN 9DIRECT ME?OD ( SUBTRAC;N4 ,Z 9 ! CASE ( AD- DI;N1 WE M/ MAKE USE (! SY/EM ( COM- PLE;T>Y NUMB]S OR1 IF Y PREF]1 Y MAY MEMORIZE ! SECRETS ( SUBTRAC;N 9/1D4 ,9 ORD] 6SUBTRACT #C F #G1 X WD 2 NEC 6SET #B & CLE> #E4 ,SUPPOSE ,I H #G C5TS1 #A FIVE-C5T PIECE & #B "O- C5T PIECES 9 MY H&1 & ,I WANT 6SUB- TRACT #C C5TS F ? AM.T4 ,IF Y GAVE ME #B C5TS1 & ,I1 9 TURN GAVE Y MY FIVE-C5T PIECE1 WE WD H SUBTRACT$ #C C5TS F MY #G C5TS1 L1V+ ME )! RE- MA9D] ( #D C5TS4 ,IF WE LE>N ! SE- CRETS ( SUBTRAC;N1 OR1 IF WE LE>N TO APPLY A PR9CIPLE (! USE ( -PLE;T>Y NUMB]S S QKLY T X 2COMES AUTOMATIC1 WE C SUBTRACT ) ABSOLUTELY NO M5TAL EF=T4 ,Z 9 ! CASE (! O!R PROCESSES1 SUBTRAC;N WD 2 A PURELY ME*ANICAL OP],N4 ,"! >E SEV5TE5 SECRETS = SUB- TRAC;N1 J Z "! >E SEV5TE5 SECRETS = A4I;N4 ,!Y >E Z FOLL[S3 ,US+ ! ,CRANM] ,ABACUS #EG .,TO .,SUB- .,SECRET TRACT #A """"""" ,CLE> #A LEFT1 SET #I #A """"""" ,SET #D1 CLE> #E #B """"""" ,CLE> #A LEFT1 SET #H #B """"""" ,SET #C1 CLE> #E #C """"""" ,CLE> #A LEFT1 SET #G #C """"""" ,SET #B1 CLE> #E #D """"""" ,CLE> #A LEFT1 SET #F #D """"""" ,SET #A1 CLE> #E #E """"""" ,CLE> #A LEFT1 SET #E #F """"""" ,CLE> #A LEFT1 SET #D #F """"""" ,CLE> #A LEFT1 SET #E1 CLE> #A #G """"""" ,CLE> #A LEFT1 SET #C #G """"""" ,CLE> #A LEFT1 SET #E1 CLE> #B #H """"""" ,CLE> #A LEFT1 SET #B #H """"""" ,CLE> #A LEFT1 SET #E1 CLE> #C #I """"""" ,CLE> #A LEFT1 SET #A #I """"""" ,CLE> #A LEFT1 SET #E1 CLE> #D ,9 US+ ^! SECRETS :5 WE SP1K ( CLE>+ #A LEFT1 WE M1N REMOVE #A "O- UNIT C.T] F ! F/ COLUMN IMMLY 6! LEFT (! COLUMN ON : \R SUBTRAC;N IS 2+ D"O4 ,IF ? COLUMN HAPP5S 63TA9 A #J1 X W1 ( C\RSE1 2 IMPOSSI# 6REMOVE #A F ? COLUMN4 ,S1 Y %D MOVE OV] 6! LEFT "O M COLUMN4 ,REMOVE #A F ? COLUMN1 & *ANGE ! #J 6A #I4 ,9 ! CASE ( TWO OR ?REE OR M #J'S1 3T9UE MOV+ 6! LEFT UNTIL Y F9D A COLUMN F : X IS POSSI# 6REMOVE #A4 ,REMOVE ! #A & *ANGE ALL (! 9T]V5+ #J'S TO #I'S4 ,IF Y LE>N 6D A4I;N & MULTI- PLIC,N E6ICI5TLY1 SUBTRAC;N %D -E 6Y Q EASILY4 ,X W J 2 A MATT] ( LE>N+ 6REV]SE ! SECRETS ( A4I;N4 ,0N[1 Y %D H 5 SKILL 9 US+ ! ABACUS 6MAKE ! USE ( NUM]\S EXAMPLES UNNEC4 ,H["E1 WE W %[ "O PRO#M 9 SUBTRAC;N /EP-BY- /EP1 & LET Y1 ! R1D]1 TAKE X F "!4 ,! PRO#M3 #A1BCD M9US #EFG4 ,USE ! F\R "R-MO/ COLUMNS ON ! ABACUS1 & WE W CALL !M ;ABCD4 ,ON ;ABCD1 SET #A1BCD4 ,F ! #B ( #A1BCD1 SUBTRACT ! ,US+ ! ,CRANM] ,ABACUS #EI #E ( #EFG4 ,FIVE _C 2 SUBTRACT$ DI- RECTLY F ! #B1 S1 WE M/ USE ! SE- CRET = SUBTRACT+ #E1 : IS1 6SUBTRACT #E1 CLE> #A LEFT1 SET #E4 ,S1 F COL- UMN ;A1 CLE> #A & SET #E ON ;B4 ,ON ;BCD WE N[ H #GCD4 ,F ! #C ( #A1BCD1 SUBTRACT ! #F ( #EFG4 ,AG1 #F _C 2 SUBTRACT$ F #C DIRECTLY4 ,S1 WE M/ USE ! SECOND SECRET = SUBTRACT+ #F1 : IS1 6SUBTRACT #F1 CLE> #A LEFT1 SET #E1 CLE> #A4 ,S1 CLE> #A F ;B & SET #E1 CLE> #A ON COLUMN ;C4 ,? *ANGES ! #G ON ;B 6#F & GIVES U #FGD ON ;BCD4 ,N[1 F ! #D ( #A1BCD1 SUB- TRACT #G ( #EFG4 ,9 ORD] 6D ?1 Y M/ USE T SECRET =! SUBTRAC;N ( #G : SAYS1 6SUBTRACT #G1 CLE> #A LEFT1 SET #E1 CLE> #B4 ,CLE> #A F ;C1 & ON ;D SET #E1 CLE> #B4 ,? GIVES Y ! FI- NAL ANSW]1 OR REMA9D]1 ( #FFG ON ;BCD4 ,6V]IFY ?1 6! #FFG ON ;BCD1 ADD #EFG4 ,Y W SEE T YR ANSW] IS #A1BCD4 ,PRACTICE ,EX]CISES ,= PRACTICE 9 SUBTRAC;N1 WE SU7E/ T Y REV]SE ^? EX]CISES GIV5 Z PRAC- TICE = A4I;N4 ,= EXAMPLE1 ON ! ABA- CUS SET #G1AJA1 & F ? SUBTRACT #GHI1 M9US #GHI1 M9US #GHI1 ETC4 ,:5 Y H REP1T$ ? SUBTRAC;N N9E "TS1 YR ANSW] %D 2 #J4 ,OR1 SET #DE1 & F X SUB- TRACT #A1 M9US #B1 M9US #C1 M9US #D1 M9US #E1 M9US #F1 M9US #G1 M9US #H1 M9US #I4 ,YR ANSW] %D AG 2 #J4 ,IF Y >E SUBTRACT+ A ?REE-DIGIT NUMB] F A F\R- OR FIVE-DIGIT NUMB]1 X IS IM- PORTANT T Y D YR SUBTRACT+ F ! AP- PROPRIATE COLUMN4 ,9 SUBTRACT+ #GHI F #G1AJA1 ! #G ( #GHI %D 2 SUBTRACT$ F ! COLUMN : HOLDS ! F/ #A ( #G1AJA4 ,9 SUBTRACT+ #A F #DE1 #A IS SUB- TRACT$ F ! COLUMN 3TA9+ ! #E ( #DE4 ,WE >E N[ R1DY 62G9 ! /UDY (! PROCESS ( DIVI.N1 B1 2F Y G ON 6LE>N ANY?+ NEW1 PL1SE REVIEW & 2 SURE Y "U/& ?OR\N$ S F>4 ,US+ ! ,CRANM] ,ABACUS #FA ,DIVI.N ,J Z MULTIPLIC,N IS A RAPID =M ( A4I;N1 S DIVI.N IS ! RAPID =M ( SUB- TRAC;N1 T IS1 0DIVI.N1 WE >E A# TO TELL H[ _M "TS "O NUMB] C 2 SUB- TRACT$ F ANO!R4 ,= EXAMPLE1 #GD DI- VID$ 96#A1DHJ GIVES AN ANSW] ( #BJ4 ,? M1NS T ! NUMB] #GD C 2 SUBTRACT$ F ! NUMB] #A1DHJ A TOTAL ( TW5TY "TS4 ,SIMIL>LY1 #GD 96#A1DHE GOES TW5TY "TS )A REMA9D] ( #E4 ,? IS ! SAME Z SAY+ #GD C 2 SUBTRACT$ F #A1DHE TW5TY "TS L1V+ A REMA9D] ( #E4 ,? M1NS1 ( C\RSE1 T1 ": X C 2 US$1 DIVI.N IS M* FA/]1 M* M A3URATE ?AN REP1T$ SUBTRAC;N1 J Z MULTIPLI- C,N IS M* QK] & EASI] ?AN REP1T$ AD- DI;N4 ,J Z 9 ! CASE ( MULTIPLIC,N & SUB- TRAC;N1 V>I\S "PS (A PRO#M 9 DIVI.N H "NS1 Z WE %D LE>N 2F ACTU,Y /UDY+ ! PROCESS ( DIVI.N XF4 ,! NUMB] : DOES ! DIVID+ IS ! .DIVISOR4 ,! NUM- B] 96: X IS DIVID$ IS ! .DIVID5D1 &! ANSW] IS ! .QUOTI5T4 ,?US1 9 ! PRO#M #A1DHJ DIVID$ 0#GD EQUALS #BJ1 #GD IS ! DIVISOR1 #A1DHJ IS ! DIVID5D1 & #BJ IS ! QUOTI5T4 ,:5 SETT+ UP A PRO#M 9 DIVI.N ON YR ABACUS1 ! DIVISOR IS SET AT ! EX- TREME LEFT 5D (! BO>D4 ,6! "R (! DI- VISOR F\R COLUMNS >E LEFT UNUS$1 & !N ! DIVID5D IS SET4 ,J Z 9 ! CASE ( MULTIPLIC,N1 ONCE Y H GA9$ SKILL 9 ! USE (! ABACUS1 Y MAY PREF] N 6SET D[N ! DIVISOR AT ALL1 B 6KEEP X 9 YR MEMORY4 ,9 S* A CASE1 L1VE ! TWO LEFT-MO/ COLUMNS ON YR ABACUS UNUS$1 & 2G9 SETT+ ! DIVID5D ON ! ?IRD COL- UMN1 T IS1 L1VE COLUMNS ;A & ;B UN- US$1 & SET ! F/ DIGIT (! DIVID5D ON COLUMN 2C4 ,AT .L1/ TWO UNUS$ COL- UMNS M/ 2 PROVID$ AT ! LEFT 5D (! ABACUS1 B 9 ANY CASE 5T] ! DIVID5D 9 S* A WAY T XS LA/ DIGIT FALLS ^U A UNIT COLUMN1 T IS1 A COLUMN IMMLY 6! LEFT ( "O (! LL UNIT M>KS F.D ON ! ,US+ ! ,CRANM] ,ABACUS #FC FACE (! BOTTOM MEMB] (! ABACUS FRAME4 ,9 DIVI.N1 ! QUOTI5T FIGURES W AP- PE> 6! LEFT (! DIVID5D1 & ? IS :Y AT L1/ TWO UNUS$ COLUMNS M/ 2 PROVID$4 ,9 ! EXAMPLES T WE % D1 WE % SET BO? ! DIVISOR &! DIVID5D J 6%[ Y H[ X IS D"O4 ,ASSUME ALW T ! LA/ NUMB] ( \R DIVID5D IS SET ^U A UNIT COLUMN4 ,%ORT ,DIVI.N ,%ORT DIVI.N IS A PROCESS 0: DIVID5D1 MADE UP ( ANY NUMB] ( DIG- ITS1 IS DIVID$ 0A DIVISOR HAV+ ONLY "O DIGIT4 ,! PROCESS IS "W$ 0-P>+ ! DIVISOR ) "O OR M DIGITS (! DIVID5D4 ,? RESULTS 9 A QUOTI5T FIGURE : IS 5T]$ ON ! ABACUS4 ,? QUOTI5T FIGURE IS !N MULTIPLI$ 0! DIVISOR1 &! PROD- UCT RESULT+ F ? MULTIPLIC,N IS SUB- TRACT$ F ! DIVID5D4 ,! SUBTRAC;N IS D"O FOLL[+ RULES ( POSI;N+ SIMIL> TO ^? US$ 9 MULTIPLIC,N4 ,^! W 2 4CUSS$ LAT]4 ,2F GO+ ANY FUR!R1 LET U D ! EXAM- PLE #GCF DIVID$ 0#D4 ,6D ?1 WE W USE COLUMNS ;A "? ;H4 ,SET ! DIVISOR #D ON COLUMN ;A1 L1VE ;BCDE UNUS$1 & SET ! DIVID5D1 #GCF ON COLUMNS ;FGH4 ,WE ASSUME ;H 6BE A UNIT COLUMN4 ,N[ -P>E ! #D : IS \R DIVISOR )! #G ( #GCF4 ,F\R W G 96#G J #A "T4 ,S1 \R F/ QUOTI5T FIGURE IS #A4 ,SET ? QUO- TI5T FIGURE #A ON ! SECOND COLUMN 6! LEFT (! #G ( #GCF1 9 O!R ^WS ON COL- UMN ;D4 ,NEXT1 MULTIPLY ! QUOTI5T FIGURE #A 0! DIVISOR #D4 ,"O 0#D IS #D4 ,S1 SUBTRACT ? #D F COLUMN ;F1 OR F ! #G ( #GCF4 ,? L1VES #C ON COLUMN ;F1 & #CCF ON ;FGH4 ,NEXT1 -P>E ! DIVISOR #D )! #C ON COLUMN ;F4 ,F\R W N G 96#C4 ,S1 Y M/ -P>E ? #D )! F/ TWO DIGITS (! DIVID5D1 T IS1 )! #CC ON COLUMNS ;FG4 ,F\R W G 96#CC EIG] ?AN #A1 WE M/ -P>E \R DIVISOR #D )! #AF REMA9+ ON ;GH4 ,F\R 96#AF GOES F\R "TS4 ,S1 SET ! NOXT QUOTI5T FIG- URE #D ON COLUMN ;F1 IMMLY 6! LEFT ( ! DIGIT #A ON COLUMN ;G4 ,MULTIPLY ? QUOTI5T FIGURE #D 0! DIVISOR #D4 ,F\R 0#D IS #AF4 ,SUBTRACT ? PRODUCT #AF F ! #AF ON ;GH4 ,? CLE>S AWAY ! DIVID5D1 & L1VES ! F9AL QUOTI5T ( #AHD1 ON COLUMNS ;DEF4 ,6V]IFY YR RESULTS1 MULTIPLY #AHD 0#D & Y W OB- TA9 A PRODUCT ( #GCF4 ,4CUS.N ,Y W REMEMB] T :5 WE -P>$ \R DI- VISOR #D )! #G ( #GCF1 WE SET \R QUOTI5T FIGURE #A TWO COLUMNS 6! LEFT (! DIVID5D4 ,B1 LAT] :5 WE -P>$ #D ) #CC1 & AG ) #AF1 WE SET ! RESULT+ QUOTI5T FIGURES IMMLY 6! LEFT (! DIVID5D4 ,? 0 NO A3ID5T4 ,! RULE =! PLACE;T (! QUOTI5T FIGURE 9 %ORT DIVI.N IS Z FOLL[S3 ,:5 ! DI- VISOR IS A NUMB] : IS SMALL] ?AN1 OR EQUAL TO1 ! F/ DIGIT (! DIVID5D ) : X IS 2+ -P>$1 ! RESULT+ QUOTI5T FIG- URE M/ 2 PLAC$ ON ! SECOND COLUMN 6! LEFT (! DIVID5D4 ,9 O!R ^WS1 ! QUO- TI5T FIGURE & DIVID5D M/ 2 SEP>AT$ 0"O UNUS$ COLUMN4 ,:5 ! DIVISOR IS L>G] ?AN ! F/ DIGIT (! DIVID5D 6: X IS 2+ -P>$1 ! RESULT+ QUOTI5T FIGURE IS SET IMMLY 6! LEFT (! DIVID5D4 ,9 ? CASE1 "! IS NO UNUS$ COLUMN 2T QUOTI5T FIGURE & DIVID5D4 ,REVIEW ! EXAMPLE & Y W SEE T1 9 ! F/ -P>ISON1 #D 0 SMALL] ?AN ! #G ( #GCF4 ,S1 ! QUOTI5T FIGURE #A 0 SEP- >AT$ F ! DIVID5D #G 0"O UNUS$ COL- UMN4 ,9 ! SECOND -P>ISON1 ! #D 0 L>G] ?AY ! #C ( #CC1 S ! QUOTI5T FIGURE #H 0 SET IMMLY 6! LEFT ( ? F/ #C4 ,F9,Y1 #D IS L>G] ?AN ! #A ( ,US+ ! ,CRANM] ,ABACUS #FG #AF4 ,S1 AG1 ! QUOTI5T FIGURE #D M/ 2 SET IMMLY 6! LEFT (! #A ( #AF4 ,WE H "W$ "O EXAMPLE & 4CUSS$ X 9 GRT DETAIL4 ,WE W N[ D ANO!R )\T B5EFIT ( FOLL[-UP 4CUS.N4 ,\R PRO#M3 #GCH DIVID$ 0#F4 ,SET ! DIVISOR #F ON COLUMN ;A4 ,L1VE ;BCDE UNUS$1 & ON ;FGH SET ! DIVID5D #GCH4 ,AG1 WE ASSUME ;H 6BE A UNIT COLUMN4 ,-P>E ! DIVISOR #F )! #G ( #GCH4 ,SIX W G 96#G ONCE4 ,S1 SET YR QUOTI5T FIG- URE #A ON COLUMN ;D4 ,NEXT MULTIPLY ! QUOTI5T FIGURE 0! DIVISOR4 ,"O BY #F IS #F4 ,SUBTRACT ! PRODUCT #F F COLUMN ;F4 ,NEXT1 -P>E ! #C (! DI- VISOR )! #A ON COLUMN ;F4 ,SIX W N G 96#A4 ,S1 Y M/ -P>E ! #F )! #AC ON ;FG4 ,SIX W G 96#AC TWICE4 ,SET YR QUOTI5T FIGURE #B ON COLUMN ;E4 ,MULTIPLY ! QUOTI5T FIGURE 0! DI- VISOR4 ,TWO 0#F IS #AB4 ,SUBTRACT ! PRODUCT #AB F COLUMNS ;FG4 ,NEXT1 -P>E ! DIVISOR #F )! #AH ON COLUMNS ;GH4 ,SIX W G 96#AH ?REE "TS4 ,SET YR QUOTI5T FIGURE #C ON COLUMN ;F4 ,MULTIPLY ! QUOTI5T FIGURE 0! DI- VISOR4 ,?REE 0#F IS #AH4 ,SUBTRACT ! PRODUCT #AH F ! #AH ON COLUMNS ;GH4 ,? CLE>S ! ABACUS1 & L1VES ! QUOTI5T #ABC ON COLUMNS ;DEF4 ,2F LE>N+ ANY?+ NEW1 REVIEW T : Y H LE>N$ S F>1 & MAKE SURE T Y >E -PLETELY -=TA# ) %ORT DIVI.N1 SUB- TRAC;N1 MULTIPLIC,N1 & A4I;N4 ,PRAC- TICE PRO#MS 9 %ORT DIVI.N & V]IFY YR RESULTS 0MULTIPLIC,N4 ,? W GIVE Y SKILL 9 %ORT DIVI.N1 & %>P5 SKILLS ALR GA9$ 9 MULTIPLIC,N4 ,DIVIDE YR TELEPH"O NUMB] 0! F/ DIGIT ( YR H\SE NUMB]4 ,OR1 DIVIDE YR TELEPH"O NUMB] 0#B1 #C1 #D1 #E1 #F1 #G1 #H1 #I4 ,MULTIPLY YR QUOTI5T 0! DIVISOR EA* "T 6*ECK YR A3URACY4 ,MULTIPLY YR AGE 9 YE>S AT YR LA/ BIR?"D 0#CFE4 ,& !N1 ADD A "D = EA* L1P YE> S9CE YR BIR?4 ,? W GIVE Y YR AGE 9 "DS AT YR LA/ BIR?"D4 ,DIVIDE ? FIGURE 0#G1 & ? W GIVE Y YR AGE 9 WEEKS4 ,? W AL GIVE Y LOTS ( PRAC- ,US+ ! ,CRANM] ,ABACUS #FI TICE4 ,AN 9T]E/+ %ORT DIVI.N PRO#M IS ? "O3 ,DIVIDE ! T5-DIGIT NUMB] #A1AAA1AAA1AJA 0#I4 ,!N LOOK AT YR QUOTI5T4 ,9T]E/+1 ISN'T X8 ,ONLY Y C 2 ! JUDGE ( :5 Y >E R1DY 6G ON )! NEXT /EP4 ,HAV+ DECID$ T Y C ADD1 MULTIPLY1 SUBTRACT1 & D %ORT DIVI.N -=TABLY1 Y >E N[ R DY 6TAKE UP ! /UDY ( L;G DIVI.N4 ,L;G ,DIVI.N ,? IS ! PROCESS 0: A DIVID5D MADE UP ( ANY NUMB] ( DIGITS IS DIVID$ 0A DIVISOR MADE UP ( TWO OR M DIG- ITS4 ,9 SETT+ A L;G DIVI.N PRO#M ON YR ABACUS1 Y PROCE$ J Z 9 ! CASE ( %ORT DIVI.N4 ,MAKE SURE T ! LA/ DIG- IT (! DIVID5D IS SET ^U A UNIT COL- UMN4 ,! ACTUAL PROCESS = DIVI.N IS ESS5TI,Y ! SAME1 AL? X REQUIRES A FEW A4I;NAL /EPS4 ,USU,Y1 ! F/ DIGIT (! DIVISOR IS -P>$ )! F/ DIGIT (! DIVID5D4 ,! RESULT+ QUOTI5T FIGURE IS 5T]$ ^U ! APPROPRIATE COLUMN1 & ? QUOTI5T FIGURE IS !N MULTIPLI$ 0EA* DIGIT (! DIVISOR1 & EA* PRODUCT RE- SULT+ F ? MULTIPLIC,N IS SUBTRACT$ F APPROPRIATE COLUMNS (! DIVID5D 9 AC- CORD.E )! RULE ( POSI;N+4 ,= \R F/ EXAMPLE1 LET'S DIVIDE #EGF 0#BD4 ,SET ! DIVISOR #BD ON COLUMNS ;AB4 ,L1VE ;CDEF UNUS$1 & SET #EGF ON ;GHI4 ,WE ASSUME ;I 6BE A UNIT COLUMN4 ,NEXT1 -P>E ! #B ( #BD )! #E ( #EGF4 ,TWO GOES 96#E TWO "TS1 GIV+ Y A QUOTI5T FIGURE ( #B4 ,SET ? QUOTI5T FIGURE #B ON COLUMN ;E4 ,N[ MULTIPLY ! QUOTI5T FIGURE 0! F/ DIGIT (! DIVISOR4 ,TWO 0#B IS #D4 ,S1 SUBTRACT ! #D F COLUMN ;G4 ,? L1VES #A ON ;G & #AGF ON ;GHI4 ,NEXT1 MULTIPLY ! QUOTI5T FIGURE 0! SECOND DIGIT (! DIVISOR4 ,TWO 0#D IS #H4 ,S1 SUBTRACT ! #H F COLUMN ;H1 L1V+ #I ON ;H1 & #IF ON COLUMNS ;HI4 ,NEXT1 -P>E ! #B ( #BD )! #I ( #IF4 ,TWO 96#I GOES #D "TS4 ,SET YR QUO- TI5T FIGURE #D ON ;F4 ,NEXT1 MULTI- ,US+ ! ,CRANM] ,ABACUS #GA PLY ! QUOTI5T FIGURE 0! F/ DIGIT (! DIVISOR4 ,F\R 0#B IS #H4 ,SUBTRACT ? #H F ;H & MULTIPLY ! QUOTI5T FIGURE 0! SECOND DIGIT (! DIVISOR4 ,F\R 0#D IS #AF4 ,? IS SUBTRACT$ F ! #AF ON ;HI1 CLE>+ ! ABACUS ( XS DIVID5D1 & L1V+ ! CORRECT QUOTI5T #BD ON COL- UMNS ;EF4 ,4CUS.N ,IF WE BE> 9 M9D ! RULE =! PLACE;T (! QUOTI5T FIGURE : WE LE>N$ 9 \R /UDY ( %ORT DIVI.N1 X IS EASY 6SEE H[ ? PRO#M 0 "W$ "? 6A SU3ESS;L SO- LU;N4 ,9 \R F/ -P>ISON )! #B ( #BD & ! #E ( #EGF1 #B IS SMALL] ?AN #E1 S T ! RESULT+ QUOTI5T FIGURE 0 SEP>AT$ F ! DIVID5D 0"O UNUS$ COLUMN4 ,AG1 9 ! SECOND -P>ISON1 ! #B ( #BD 0 SMALL] ?AN ! #I ( #IF1 RESULT+ 9 A SEP>,N ( QUOTI5T FIGURE & DIVID5D BY "O UNUS$ COLUMN4 ,! RULE = QUOTI5T FIGURE PLACE;T 9 %ORT DIVI.N SAYS T :5 ! DIVISOR IS SMALL] ?AN1 OR EQUAL TO1 ! F/ DIGIT (! DIVID5D1 ! RESULT+ QUOTI5T FIGURE IS SEP>AT$ F ! DIVI- D5D 0"O UNUS$ COLUMN4 ,:5 ! DIVISOR IS GRT] ?AN ! F/ DIGIT (! DIVID5D1 "! IS NO UNUS$ COLUMN SEP>AT+ ! QUO- TI5T FIGURE F ! DIVID5D4 ,? RULE IS ALL "R Z F> Z X GOES1 B = L;G DIVI- .N1 X DOES N G Q F> 5\<4 ,9 MO/ CASES1 X C 2 TAK5 AT FACE VALUE1 B 9 "S 9/.ES X NE$S 6BE C>RI$ FUR!R4 ,? IS 2C ! DIVISOR 9 A L;G DIVI.N PRO#M 3SI/S ( TWO OR M DIGITS4 ,? M1NS T "S"TS ! F/ DIGIT (! DIVID5D &! F/ DIGIT (! DIVISOR MAY 2 ! SAME4 ,:5 ? HAPP5S1 ! SECOND DIGITS ( BO? ! DI- VISOR & DIVID5D M/ 2 -P>$4 ,IF ^! DIGITS >E EQUAL1 !N1 ! ?IRD DIGITS M/ 2 -P>$1 ETC41 UNTIL A PO9T ( IN- EQUAL;Y IS R1*$4 ,X IS AT ? "T T ! RULE =! PLACE;T ( QUOTI5T FIGURE IS APPLI$4 ,? RULE HAS BE5 /AT$ PREVI- \SLY1 T IS1 IF ! DIGIT 9 ! DIVISOR IS SMALL] ?AN ! DIGIT 9 ! DIVID5D 6: X IS -P>$1 ! QUOTI5T FIGURE IS ,US+ ! ,CRANM] ,ABACUS #GC SEP>AT$ F ! DIVID5D 0"O UNUS$ COL- UMN4 ,B IF ! DIGIT 9 ! DIVISOR IS L>G] ?AN T 9 ! DIVID5D 6: X IS 2+ -P>$1 !N ! QUOTI5T FIGURE IS PLAC$ IMMLY 6! LEFT (! DIVID5D ) NO SEP>,N 0AN UNUS$ COLUMN4 ,3SID] ! PRO#M #BBE DIVID$ 0#BE4 ,SET #BE ON ;AB4 ,L1VE ;CDEF UNUS$1 & ON ;GHI SET #BBE4 ,AG1 ;I IS A UNIT COLUMN4 ,N[1 -P>E ! #B ( #BE )! #B ( #BBE4 ,!Y >E EQUAL1 &1 ON ! FACE ( X1 Y MIAT$ F ! DIVID5D 0"O UNUS$ COLUMN4 ,2F MAK+ A F9AL DECI.N1 LET'S -P>E ! SECOND DIGITS (! DIVISOR & DIVID5D4 ,! #E ( #BE IS L>G] ?AN ! SECOND #B ( #BBE4 ,S1 WE M/ OBEY ! RULE : SAYS T1 :5 ! DIGIT 9 ! DIVISOR IS GRT] ?AN ! DIG- IT 9 ! DIVID5D 6: X IS 2+ -P>$1 ! RE- SULT+ QUOTI5T FIGURE IS SET IMMLY 6! LEFT (! F/ DIGIT (! DIVID5D1 & N SEP>AT$ 0AN UNUS$ COLUMN4 ,S1 9 ? CASE1 WE WD SET \R QUOTI5T FIGURE ON COLUMN ;F4 ,\R QUOTI5T FIGURE 9 ? CASE IS #I4 ,S1 ) ? #I SET ON ;F LET'S MULTIPLY X 0! F/ DIGIT (! DI- VISOR4 ,N9E 0#B IS #AH4 ,S1 SUBTRACT #AH F ;GH4 ,? RESULTS 9 #DE 2+ LEFT ON COLUMNS ;HI4 ,N[ WE MULTIPLY ! QUOTI5T FIGURE #I 0! SECOND DIGIT (! DIVISOR #E4 ,N9E 0#E IS #DE4 ,WE SUBTRACT #DE F ! #DE ON ;HI1 CLE>+ ! ABACUS1 & L1V+ #I Z \R QUOTI5T4 ,4CUS.N ,! ?"\;L R1D] W WOND] :Y WE DID N EXPLA9 H[ WE >RIV$ AT \R QUOTI5T FIGURE #I :ILE "W+ ! PREVI\S PRO#M4 ,? OMIS.N 0 DELIB]ATE 9 ORD] 6AVOID 3FU.N &! EXPLAN,N IS GIV5 "H 9/1D4 ,:5"E Y 5C.T] A PRO#M ( ? TYPE1 T IS1 "O 9 : ! F/ DIGITS (! DIVISOR &! DIVID5D >E ! SAME B ! SECOND DIGIT ( ! DIVISOR IS L>G] ?AN ! SECOND DIGIT (! DIVID5D1 A QUOTI5T FIGURE ( #I /&S A BETT] ?AN EV5 *.E ( 2+ COR- RECT4 ,? IS N AN EXPLAN,N1 ( C\RSE1 ,US+ ! ,CRANM] ,ABACUS #GE B IS A /ATE;T (A G5]ALIZ$ RULE ( ?UMB4 ,N[1 LET'S SET UP ! PRO#M AG & LOOK AT X SCI5TIFIC,Y4 ,ON ;AB1 SET #BE1 ON ;GHI1 #BBE4 ,IF Y -P>E ! #B ( #BE )! #B ( #BBE Y WD CORRECTLY AS- SUME ! RESULT+ QUOTI5T FIGURE 6BE #A4 ,B1 IF Y 7 6TRY #A1 X WD N 2 L;G 2F YR MISTAKE WD 2COME APP>5T4 ,S1 -P>E ! #B ( #BE )! #BB ( #BBE4 ,TWO GOES 96#BB ELEV5 "TS4 ,B1 #AA IS A TWO-DIGIT NUMB]4 ,S1 X M/ 2 REJECT$ Z A QUOTI5T FIGURE S9CE WE DEVELOP \R QUOTI5T FIGURES "O DIGIT AT A "T1 & V]IFY !M "? MULTIPLIC,N 0! DIVISOR & APPROPRIATE SUBTRAC;N F ! DIVID5D4 ,N[1 =A M9UTE1 LET'S ?9K \ \R DI- VISOR #BE N Z ! :OLE NUMB] #BE1 B R Z ! DECIMAL FRAC;N #B.E1 OR1 #B-A/B1 OR Z A NUMB] LY+ "S": 2T #B & #C4 ,WE H SD T -P>+ ! #B ( #BE ) #BB ( #BBE GIVES A QUOTI5T FIGURE ( #AA4 ,B1 S9CE WE >E N[ ?9K+ ( \R DIVISOR Z A NUMB] : LIES "S": 2T ! :OLE NUM- B]S #B & #C1 LET'S -P>E #C )! #BB ( #BBE4 ,?REE GOES 96#BB SEV5 "TS4 ,IF WE *OSE #G Z \R QUOTI5T FIGURE1 NO H>M WD 2 D"O1 S9CE1 DUR+ ! PROCESS ( SOLU;N1 ? QUOTI5T FIGURE WD 2COME CORRECT$ 6#I AUTOMATIC,Y4 ,J H[ ? WD -E AB W 2 DESCRIB$ LAT]4 ,S1 0-P>+ #B ) #BB WE H #AA1 A QUOTI5T FIGURE : IS TOO L>GE2 & 0-P>+ #C ) #BB WE H #G1 A QUOTI5T FIGURE : IS TOO SMALL4 ,9 ! CASE ( #BBE DIVID$ 0#BE1 ?9K ( \R DIVISOR Z A NUMB] HALF-WAY 2T #B & #C1 &! QUOTI5T FIGURE %D 2 A NUMB] HALF-WAY 2T #AA & #G1 OR 9 ? CASE #I1 : TURNS \ 6BE CORRECT4 ,X IS N 9T5D$ T Y %D G "? M5TAL GYMNA/ICS E "T ? K9D ( SITU,N O3URS4 ,? EXPLAN,N IS GIV5 ONLY 6SATISFY ^? :O >E 9T]E/$ 9 !ORETICAL 3SID],NS4 ,=! SAKE ( AUTOMATIC DIVI.N1 FREE F M5TAL EF=T1 WE SU7E/ T Y APPLY ! G5- ]ALIZ$ RULE ( ?UMB1 /AT$ AT ! 2G9N+ ( ? EXPLAN,N4 ,! PROCESS ( DIVI.N C 2 J Z AUTOMATIC1 J Z FREE F M5TAL EF=T Z >E A4I;N1 MULTIPLIC,N1 & SUB- TRAC;N4 ,IF AUTOMATIC DIVI.N IS YR ,US+ ! ,CRANM] ,ABACUS #GG GOAL1 Z X %D BE1 Y M/ LE>N 6"U/& & MA/] TWO SECRETS ( SU3ESS4 ,F/1 IS ! PLACE;T (! QUOTI5T FIGURE ^U ! CORRECT COLUMN4 ,WE H ALR COV]$ ? PO9T4 ,ONCE ! QUOTI5T FIGURE HAS BE5 SET ON ! ABACUS1 X IS MULTIPLI$ 0E DIGIT (! DIVISOR1 &! PRODUCTS ( ? MULTIPLIC,N >E SUBTRACT$ 9 AN ORD]LY WAY F ! CORRECT COLUMNS (! DIVID5D4 ,? SUBTRAC;N IS D"O 9 A3ORD.E )! RULES ( POSI;N+ : WE H ALR LE>N$ 9 MULTIPLIC,N4 , ,:5 WE SET \R QUOTI5T FIGURE ON ! ABACUS & !N PROCE$ 6MULTIPLY X 0! V>I\S DIGITS (! DIVISOR1 WE >E RE- DUC+ \R DIVID5D "? A -B9,N (! PROC- ESSES ( MULTIPLIC,N & SUBTRAC;N4 ,9 S* A CASE1 ! QUOTI5T FIGURE IS ! MULTIPLIC&1 &! DIVISOR IS ! MULTI- PLI]4 ,9 MULTIPLIC,N1 REMEMB] T EA* "T Y MULTIPLY A DIGIT (! MULTIPLIC& 0! DIGITS (! MULTIPLI] YR PRODUCT IS SET 9 ! APPROPRIATE POSI;N4 ,SUPPOSE Y H A "O-DIGIT MULTIPLIC& &A FIVE- DIGIT MULTIPLI]4 ,! PRODUCT : RE- SULTS F ! F/ MULTIPLIC,N ( ? MUKTI- PLIC& IS SET 9 ! F/ POSI;N2 ! PROD- UCT (! SECOND MULTIPLIC,N1 9 ! SEC- OND POSI;N1 & S ON1 UNTIL Y MULTIPLY ! MULTIPLIC& 0! FIF? DIGIT (! MULTI- PLI]4 ,? IS ! FIF? "T ! MULTIPLIC& IS MULTIPLI$1 &! PRODUCT : RESULTS IS SET 9 ! FIF? POSI;N4 ,! F/ POSI- ;N1 ( C\RSE1 IS MADE UP (! F/ TWO COLUMNS IMMLY 6! "R (! MULTIPLIC&4 ,! SECOND POSI;N IS ! SECOND & ?IRD2 ! ?IRD POSI;N1 ! ?IRD & F\R?2 &! F\R? POSI;N1 F\R? & FIF? COLUMNS IMMLY 6! "R (! MULTIPLIC&4 ,?US1 ! DIGIT (! MULTIPLIC& : IS CURR5TLY 2+ MULTIPLI$ S]VES Z ! .M>K] .DIGIT1 & POSI;NS MAY 2 DET]M9$ 0C.T+ 6! "R F ? M>K] COLUMN1 J Z 9 ! CASE ( MULTI- PLIC,N3 F/ POSI;N1 COLUMNS "O & TWO2 SECOND POSI;N1 COLUMNS TWO & ?REE2 ?IRD POSI;N1 COLUMNS ?REE & F\R1 ETC4 ,N[1 9 ! CASE ( DIVI.N1 ! QUO- TI5T FIGURE REPRES5TS ! MULTIPLIC&1 ! DIVISOR1 ! MULTIPLI]4 ,S1 HAV+ ,US+ ! ,CRANM] ,ABACUS #GI OBEY$ ! RULES =! PROP] PLACE;T (! QUOTI5T FIGURE1 ! QUOTI5T FIGURE IS !N MULTIPLI$ 0EA* DIGIT (! DIVISOR1 &! RESULT+ PRODUCTS >E SUBTRACT$ F ! DIVID5D AC 6^! PR9CIPLES ( POSI;N+4 ,F/ MULTIPLIC,N IS SUBTRACT$ F ! F/ POSI;N2 SECOND MULTIPLIC,N IS SUB- TRACT$ F ! SECOND POSI;N1 ETC4 ,IF WE H A ?REE-DIGIT DIVISOR1 ! ?IRD PRODUCT W 2 SUBTRACT$ F ! ?IRD PO- SI;N1 &! ?IRD POSI;N W 2 ! ?IRD & F\R? COLUMNS 6! "R (! QUOTI5T FIG- URE1 : 9 ? CASE S]VES Z \R M>K] COL- UMN4 ,LET'S "W "? AN EXAMPLE & FOLL[ X UP )A 4CUS.N4 ,= \R PRO#M WE W DI- VIDE #EFG 0#BA4 ,SET ! DIVISOR #BA ON ;AB4 ,L1VE ;CDEF UNUS$1 & ON ;GHI SET #EFG1 \R DIVID5D2 ;I AG1 IS A UNIT COLUMN4 ,-P>E ! #B ( #BA )! #E ( #EFG4 ,? RESULTS 9 A QUOTI5T FIG- URE ( #B1 S9CE #B W G 96#E TWO "TS4 ,SET #B ! QUOTI5T FIGURE1 ON ;E4 ,NEXT1 MULTIPLY X 0! DIVISOR #BA4 ,TWO 0#B IS #D4 ,S1 SUBTRACT #D ^ COLUMN ;G4 ,NEXT1 MULTIPLY ! QUOTI5T FIGURE #B 0! #A ( #BA4 ,TWO 0#A IS #B4 ,S1 SUBTRACT ? #B F ;H4 ,N[1 -P>E ! #B ( #BA )! #AD ON ;GH4 ,TWO W G 96#AD SEV5 "TS4 ,S1 SET YR QUO- TI5T FIGURE #G ON COLUMN ;F4 ,NEXT1 MULTIPLY ? #G 0! #B ( #BA4 ,SEV5 0#B IS #AD4 ,SUBTRACT X F ! #AD ON ;GH4 ,NEXT1 MULTIPLY #G 0! #A ( #BA4 ,SEV5 0#A IS #G4 ,SUBTRACT ? F ! #G ON ;I4 ,? CLE>S ! ABACUS (! DIVID5D1 & L1VES ! QUOTI5T #BG ON ;EF4 ,4CUS.N ,:5 WE -P>$ #B ( #BA )! #E ( #EFG ! #B 0 SMALL] ?AN ! #E1 S T ! RE- SULT+ QUOTI5T FIGURE #B 0 SEP>AT$ F ! #E 0"O UNUS$ COLUMN4 ,? M1NS T COLUMN ;E1 ! COLUMN ON : ! QUOTI5T FIGURE #B 0 SET1 2CAME ! M>K] COL- UMN4 ,WE !N MULTIPLI$ ? #B ON COLUMN ;E 0! #B 9 ! DIVISOR4 ,TWO 0#B IS #D1 & X REPRES5TS ! F/ MULTIPLIC,N ( ? QUOTI5T FIGURE4 ,? F/ MULTIPLIC,N ,US+ ! ,CRANM] ,ABACUS #HA M/ H XS PRODUCT SUBTRACT$ F ! F/ PO- SI;N4 ,F\R IS A "O-DIGIT NUMB]4 ,S X M/ 2 SUBTRACT$ F ! .SECOND .COLUMN (! POSI;N4 ,) COLUMN ;E Z \R M>K] COLUMN1 WE C.T 6! "R TWO COLUMNS4 ;,F IS ! F/1 ;G IS ! SECOND4 ,S WE SUBTRACT \R #D F ;G4 ,NEXT1 WE MUL- TIPLY \R QUOTI5T FIGURE #B ON COLUMN ;E 0! #A ( #BA4 ,? RESULTS 9 A PROD- UCT ( #B4 ,? IS ! SECOND MULTIPLIC,N ( \R QUOTI5T FIGURE1 & X M/ 2 SUB- TRACT$ F ! SECOND POSI;N4 ,! SECOND POSI;N IS MADE UP (! SECOND & ?IRD COLUMNS 6! "R ( \R M>K] COLUMN1 & S9CE #B1 ! PRODUCT 6BE SUBTRACT$1 IS A "O-DIGIT NUMB]1 X M/ 2 SUBTRACT$ F ! SECOND COLUMN ( ? POSI;N4 ,S POSI- ;N TWO 3SI/S ( COLUMNS TWO & ?REE 6! "R (! M>K] COLUMN4 ,"!=E1 ! S+LE DIGIT #B M/ 2 SUBTRACT$ F ! ?IRD COLUMN4 ,C.T+ 6! "R1 WE F9D T ;F IS ! F/1 ;G IS ! SECOND1 & ;H IS ! ?IRD4 ,S1 WE SUBTRACT #B F COLUMN ;H4 ,9 \R NEXT -P>ISON1 ! #B ( #BA IS -P>$ ) #AD ON ;GH4 ,S9CE ! #B 9 \R DIVISOR IS GRT] ?AN ! #A 9 ! DIVI- D5D1 WE SET ! RESULT+ QUOTI5T FIG- URE #G ON ;F1 T IS1 IMMLY 6! LEFT (! DIVID5D1 ) NO SEP>,N 0AN UNUS$ COL- UMN4 ,! F/ MULTIPLIC,N ( ? QUOTI5T FIGURE #G IS 0! DIGIT #B (! DIVISOR4 ,SEV5 0#B IS #AD1 & ? SUBTRAC;N IS F ! F/ POSI;N4 ,COLUMN ;F IS N[ \R M>K] COLUMN1 S9CE ? IS ! COLUMN ON : \R MULTIPLIC& IS SET4 ,S SUBTRACT #AD F ! F/ POSI;N1 T IS1 F ! TWO COLUMNS IMMLY 6! "R ( ? M>K] COLUMN4 ,! TWO COLUMNS 6! "R ( ;F >E ;G & ;H4 ,SUBTRACT #AD F ;GH4 ,NEXT1 WE MULTIPLY ! #G 0#A & OBTA9 ! PRODUCT ( #G4 ,? IS ! SECOND MULTIPLIC,N ( \R QUOTI5T FIGURE #G1 & X M/ 2 SUB- TRACT$ F ! SECOND POSI;N4 ,B #G IS A "O-DIGIT NUMB]4 ,S X M/ 2 SUB- TRACT$ F ! SECOND COLUMN ( ? POSI;N4 ,! SECOND POSI;N IS COLUMNS TWO & ?REE 6! "R (! M>K] COLUMN1 OR1 9 O!R ^WS1 COLUMNS ;H & ;I; ,S9CE ! PROD- ,US+ ! ,CRANM] ,ABACUS #HC UCT IS A "O-DIGIT NUMB]1 WE M/ SUB- TRACT X F COLUMN ;I4 ,SEV5 SUBTRACT$ F #G CLE>S \R ABACUS & L1VES ! QUO- TI5T #BG ON ;EF4 ,MULTIPLIC,N ( #BG 0#BA W GIVE U #EFG & W V]IFY ! COR- RECT;S ( \R "W4 ,AN EASY WAY 6F9D YR POSI;N IS TO PLACE ! 9DEX F+] ( YR LEFT H& ON ! M>K] COLUMN1 ! COLUMN ON : ! QUOTI5T FIGURE "U 3SID],N IS SET4 ,!N1 PLACE ! 9DEX F+] & MI4LE F+] (! "R H& ON ! TWO COLUMNS 6! "R ( ? M>K] COLUMN4 ,? IS POSI;N "O4 ,MOV+ YR F+]S OV] "O PLACE 6! "R W PLACE Y ON POSI;N TWO1 & S ON4 ,! MI4LE F+] ( YR "R H& W AUTOMATIC,Y RE/ ON ! SECOND COLUMN ( EA* POSI;N1 & X IS F ? COLUMN T ! S+LE-DIGIT PRODUCT M/ 2 SUBTRACT$4 ,TWO-DIGIT PRODUCTS M/ 2 SUBTRACT$ F ! F/ & SECOND COLUMNS (! POSI;N1 OR1 F ! COLUMNS ON : BO? ! 9DEX F+] & MI4LE F+] ( YR "R H& >E RE/+4 ,CORRECT+ ! ,QUOTI5T ,FIGURE ,9 L;G DIVI.N1 WE >RIVE AT \R QUO- TI5T FIGURE 0-P>+ ! F/ DIGIT (! DI- VISOR )! F/ "O OR TWO DIGITS (! DIV- ID5D4 ,S9CE ! DIVISOR 3SI/S ( M ?AN "O NUMB]1 ! QUOTI5T FIGURE OBTA9$ F ONLY F/-DIGIT -P>ISONS IS SUBJECT TO ]ROR4 ,:5 S* AN ]ROR IS MADE1 ! QUO- TI5T FIGURE M/ 2 CORRECT$4 ,IF TOO SMALL A QUOTI5T FIGURE IS *OS51 X M/ 2 RAIS$ 6! CORRECT FIGURE 0.UPW>D .CORREC;N4 ,IF TOO L>GE A QUOTI5T FIGURE IS *OS51 X M/ 2 REDUC$ 0M1NS (A .D[NW>D .CORREC;N4 ,! ,UPW>D ,CORREC;N ,:5 TOO SMALL A QUOTI5T FIGURE HAS BE5 *OS51 ! PROCESS 0: X IS CORRECT$ UPW>D IS .SELI-CORRECTIVE 9 NATURE4 ,6F9D \ J H[ X "WS1 LET'S D AG ! PRO#M #BBE DIVID$ 0#BE4 ,ON ;AB SET #BE4 ,L1VE ;CDEF UNUS$1 & SET #BBE ON ;GHI4 ,Y W REMEMB] T 9 ! EXPLAN,N ,US+ ! ,CRANM] ,ABACUS #HE FOLL[+ ? PRO#M1 WE SD T IF WE ?"\ (! #BE Z #B.E1 X WD 2 A NUMB] : WD O3UR "S": 2T ! :OLE NUMB]S #B & #C1 & T IF WE -P>$ #B ) #BB WE WD OBTA9 A QUOTI5T FIGURE ( #AA1 : WD N 2 USEA# 2C X IS A TWO-DIGIT NUMB]4 ,ON ! O!R H&1 IF WE *OSE 6-P>E #C ) #BB1 WE WD OBTA9 A QUOTI5T FIGURE ( #G1 : WD 2 TOO SMALL4 ,B1 J =! SAKE (! EXAMPLE1 LET U SET #G Z \R F/ QUOTI5T FIGURE4 ,SET #G ON COLUMN ;F4 ,MULTIPLY #G 0! #B ( #BE4 ,SEV5 0#B IS #AD4 ,S SUBTRACT #AD F ;GH4 ,NEXT1 MULTIPLY #G 0#E1 & SUBTRACT ! RESULT+ PROD- UCT #CE F ;HI4 ,N[1 -P>E ! #B ( \R DIVISOR #BE )! #E ( \R REMA9+ DIVI- D5D #EJ ON COLUMNS ;HI4 ,TWO GOES 96#E TWICE4 ,SET #B ON COLUMN ;F4 ,? *ANGES ! #G ALR EXI/+ ON ;F 6A #I4 ,? NEW QUOTI5T FIGURE ( #B IS MULTIPLI$ 0! #B ( #BE &! RESULT+ PRODUCT #D IS SUBTRACT$ F ! #E ON ;H4 ,? #B MULTIPLI$ 0! #E ( #BE GIVES ! PRODUCT ( #AJ1 : IS SUB- TRACT$ F ! #AJ ON ;HI4 ,? CLE>S ! ABACUS ( XS DIVID5D1 L1V+ ! CORRECT QUOTI5T #I ON COLUMN ;F4 ,4CUS.N ,! #B ( #BE IS EQUAL 6#B ( #BBE4 ,S1 2F MAK+ A F9AL DECI.N Z 6QUO- TI5T FIGURE PLACE;T1 WE _H 6-P>E ! SECOND DIGIT (! DIVISOR )! SECOND DIGIT (! DIVID5D4 ,FIVE ( #BE IS GRT] ?AN ! SECOND #B ( #BBE4 ,S1 ? TELLS U T \R F/ QUOTI5T FIGURE M/ 2 SET ON ! COLUMN IMMLY 6! LEFT (! DIVID5D4 ,9 ? EXAMPLE1 X 0 COLUMN ;F4 ,"! W !N 2 NO SEP>,N 2T QUOTI5T FIGURE & DIVID5D4 ,COLUMN ;F1 !N1 IS ! M>K] COLUMN F : WE C.T (F POSI- ;NS4 ,\R F/ QUOTI5T FIGURE 0 #G4 ,WE MULTIPLI$ #G 0#B4 ,! RESULT+ PRODUCT 0 #AD4 ,? 0 ! F/ MULTIPLIC,N ( #G &1 "!=E1 X 0 SUBTRACT$ F ! F/ POSI;N4 ,9 SUBTRACT+ ! #D ( #AD1 WE1 ( C\RSE1 _H 6USE "O (! SECRETS = SUB- TRAC;N4 ,WE !N MULTIPLI$ ? #G 0! #E ,US+ ! ,CRANM] ,ABACUS #HG ( #BE4 ,? 0 ! SECOND MULTIPLIC,N ( #G1 &! RESULT+ PRODUCT ( #CE 0 SUB- TRACT$ F ! SECOND POSI;N4 ,? LEFT #EJ Z \R DIVID5D ON ;HI4 ,9 \R NEXT -P>ISON1 ! #B ( #BE IS SMALL] ?AN ! #E ( #EJ4 ,? M1NS T ! QUOTI5T FIGURE RESULT+ F ? -P>ISON IS 6BE SEP>AT$ F ? DIVID5D DIGIT 0"O UNUS$ COLUMN4 ,TWO GOES 96#E TWO "TS4 ,S1 C.T+ 6! LEFT F COLUMN ;H1 ON : ! F/ DIGIT ( \R DIVID5D IS N[ SET1 WE SEE T COL- UMN ;G W 2 ! SEP>,N COLUMN1 & COLUMN ;F : ALR 3TA9S ! QUOTI5T FIGURE #G M/ 2 US$ =! PLACE;T (! QUOTI5T FIG- URE #B4 ,! A4I;N (A NEW QUOTI5T FIG- URE #B 6! #G ALR EXI/+ ON ;F COR- RECTS #G UPW>D1 *ANG+ X 6A #I4 ,MUL- TIPLY+ ! NEW QUOTI5T FIGURE #B 0! F/ DIGIT ( \R DIVISOR1 ! #B ( #BE GIVES U A PRODUCT ( #D4 ,? IS ! F/ MULTI- PLIC,N ( \R QUOTI5T FIGURE1 & ? PRODUCT #D IS SUBTRACT$ F ! F/ POSI- ;N4 ,L X IS SUBTRACT$ F ! SECOND COLUMN (! F/ POSI;N1 OR1 9 O!R ^WS1 F ! #E ON COLUMN ;H4 ,NEXT1 WE MUL- TIPLI$ ! QUOTI5T FIGURE #B 0! #E ( #BE1 GIV+ U A PRODUCT ( #AJ4 ,? IS ! SECOND MULTIPLIC,N ( \R QUOTI5T FIG- URE1 & "!=E1 ! PRODUCT #AJ M/ 2 SUB- TRACT$ F ! SECOND POSI;N4 ,T5 2+ A TWO-DIGIT NUMB] M/ 2 SUBTRACT$ F BO? COLUMNS ( ? POSI;N1 OR F COLUMNS ;HI1 CLE>+ ! BO>D & L1V+ #I Z ! COR- RECT QUOTI5T ON COLUMN ;F4 ,F ! =EGO+ EXAMPLE1 Y %D 2 A# 6AP- PRECIATE ! SELF-CORRECTIVE NATURE ( UPW>D CORREC;N4 ,= ? R1SON1 IF Y >E 9 D\BT Z 6:AT QUOTI5T FIGURE 6USE1 X W 2 6YR ADVANTAGE DELIB]ATELY 6*OOSE A QUOTI5T FIGURE "KN 6BE TOO L[4 ,0S DO+1 Y W TAKE ADVANTAGE (! B5EFITS ( AUTOMATIC UPW>D CORREC;N4 ,D[NW>D ,CORREC;N ,! CORREC;N D[NW>D (A QUOTI5T FIG- URE F.D 6BE TOO L>GE IS ANO!R CASE4 ,B ALL CALCUL,NS P]=M$ ON ! ABACUS1 X IS AUTOMATIC1 B X IS .N SELF-COR- ,US+ ! ,CRANM] ,ABACUS #HI RECTIVE Z 9 UPW>D CORREC;N4 ,"H IS H[ D[NW>D CORREC;N IS D"O4 ,:5 Y SET A QUOTI5T FIGURE ON ! ABACUS1 Y !N MULTIPLY X 0V>I\S DIGITS (! DIVISOR1 & SUBTRACT ! RESULT+ PRODUCTS F ! DIVID5D1 BE>+ 9 M9D ! RULES = PO- SI;N+4 ,N[1 IF ! QUOTI5T FIGURE Y H *OS5 IS TOO L>GE1 "! W -E A "T DUR+ ? PROCESS ( MULTIPLIC,N & SUBTRAC;N :5 Y W 4COV] ? FACT4 ,X MAY 2 T YR 4COV]Y W -E AF Y H MULTIPLI$ ! 9COR- RECT QUOTI5T FIGURE 0"O OR M DIGITS (! DIVISOR4 ,:5 ? HAPP5S1 Y M/ F/ REDUCE ! QUOTI5T FIGURE 0SUBTRACT+ F X ! DI6];E 2T ! 9CORRECTLY HI< QUO- TI5T FIGURE & YR NEW TRIAL QUOTI5T FIGURE4 ,? NUMB] : Y SUBTRACT IS CALL$ ! .CORRECTIVE .FIGURE1 & MO/ P L 6REDUCE QUOTI5T FIGURES 0A CORREC- TIVE FIGURE ( #A = EA* TRIAL4 ,!N MULTIPLY ? CORRECTIVE FIGURE 0^? DIGITS (! DIVISOR : Y H ALR MULTI- PLI$ 0! 9CORRECT QUOTI5T FIGURE1 &= : SUBTRAC;NS H BE5 MADE4 ,! PRODUCTS OBTA9$ 0MULTIPLY+ ! CORRECTIVE FIG- URE 0^! DIGITS (! DIVISOR >E RETURN$ 6! DIVID5D 9 A3ORD.E )! RULES ( PO- SI;N+1 ALR DESCRIB$4 ,ONCE Y H D"O ?1 3T9UE Z 2F1 MULTIPLY+ ! REVIS$ QUOTI5T FIGURE 0! REMA9+ DIGITS (! DIVISOR1 & G ON Z IF NO?+ UNUSUAL _H HAPP5$4 ,AN EXAMPLE WD S]VE 6MAKE ?+S CLE>4 ,LET'S DIVIDE #GBI 0#BG4 ,! CORRECT ANSW] W 2 #BG1 B WAT* :AT HAPP5S4 ,SET ! DIVISOR #BG ON ;AB & ! DIVID5D #GBI ON ;GHI1 AG ;I 2+ A UNIT COLUMN4 ,N[1 -P>E ! #B ( #BG )! #G ( #GBI4 ,TWO GOES 96#G ?REE "TS4 ,SET ! QUOTI5T FIGURE #C ON COLUMN ;E4 ,MULTIPLY ? QUOTI5T FIGURE #C 0! #B ( #BG4 ,?REE 0#B IS #F4 ,S SUB- TRACT #F F COLUMN ;G4 ,NEXT MULTIPLY ? QUOTI5T FIGURE #C 0#G ( #BG4 ,?REE 0#G IS #BA4 ,? #BA IS 6BE SUBTRACT$ F ;G & ;H4 ,B #BA IS L>G] ?AN ! #AB ON ;G & ;H4 ,S X IS AT ? PO9T T WE H 4COV]$ \R QUOTI5T FIGURE 6BE TOO HI<4 ,S SUBTRACT # F ! #C ON COL- UMN ;E4 ,? *ANGES ? #C 6A #B4 ,\R ,US+ ! ,CRANM] ,ABACUS #IA CORRECTIVE FIGURE IS #A4 ,MULTIPLY ? CORRECTIVE FIGURE #A 0! #B ( #BG & SET ! PRODUCT #B ON COLUMN ;G4 ,NEXT MULTIPLY ! REVIS$ QUOTI5T FIGURE #B 0! #G ( #BG & SUBTRACT #AD F ;G & ;H4 ,N[ -P>E ! #B ( #BG )! #AH ON ;G & ;H4 ,TWO W G 96#AH N9E "TS4 ,S SET #I Z \R QUOTI5T FIGURE ON COLUMN ;F4 ,N9E MULTIPLI$ 0! #B ( #BG IS #AH4 ,S SUBTRACT #AH F ;GH4 ,N9E MULTI- PLI$ 0#G IS #FC1 : M/ 2 SUBTRACT$ F ;HI4 ,SIXTY-?REE IS L>G] ?AN ! #I REMA9+ ON COLUMN ;I4 ,"!=E1 WE H ONCE AG 4COV]$ T \R QUOTI5T FIGURE IS TOO HI<4 ,SUBTRACT #A F ! #I ON ;F *ANG+ X 6AN #H4 ,MULTIPLY ! COR- RECTIVE FIGURE #A 0! #B ( #BG & SET ! PRODUCT #B ON ;H4 ,N[1 MULTIPLY ! NEW QUOTI5T FIGURE #H 0! #G ( #BG4 ,EIG] ?AN #BI O3URR+ ON ;HI1 WE "K \R QUOTI5T FIGURE IS / TOO HI<4 ,ONCE AG1 SUB- TRACT #A F ! #H ON COLUMN ;F1 *ANG+ X 6A #G4 ,MULTIPLY ? CORRECTIVE FIG- URE #A 0! #B ( #BG & SET ! PRODUCT #B ON COLUMN ;H4 ,N[ MULTIPLY ! RE- VIS$ QUOTI5T FIGURE #G 0! #G ( #BG4 ,SEV5 0#G IS #DI & ? C 2 SUBTRACT$ F ! #DI ON ;HI1 CLE>+ ! ABACUS & L1V+ ! CORRECT QUOTI5T #BG ON COLUMNS ;EF4 ,4CUS.N ,AG1 WE SEE ! IMPORT.E ( QUOTI5T FIGURE PLACE;T1 &( ADH];E 6! RULES ( POSI;N+4 ,9 \R F/ -P>ISON1 ! DIVISOR FIGURE #B 0 SMALL] ?AN ! DIVID5D FIGURE #G1 : M1NS "O UNUS$ COLUMN M/ SEP>ATE QUOTI5T FIGURE F DIVID5D4 ,9 ! SECOND -P>ISON1 ! DIVISOR DIGIT #B 0 L>G] ?AN ! #A ( #AH4 ,S1 ? M1NS T ! QUOTI5T FIGURE M/ 2 PLAC$ IMMLY 6! LEFT (! DIVID5D4 ,9 ALL ?REE CASES ( MULTIPLIC,N (! CORRECTIVE FIGURE 0^? DIGITS (! DIVISOR : _H BE5 MULTIPLI$ 0! 9CORRECT QUOTI5T FIGURE1 ! NUMB]S 6BE RETURN$ 6! DIVID5D 7 "O-DIGIT ,US+ ! ,CRANM] ,ABACUS #IC NUMB]S4 ,? M1NS T !Y 7 RETURN$ 6! SECOND COLUMN (! POSI;N 6: !Y 2L;G$4 ,N[1 LET'S REVIEW J A "P ( T EXAM- PLE & MAKE AN OBS]V,N OR TWO4 ,SET UP ! ABACUS ) #BG ON ;AB1 #B ON ;E1 & #AHI ON ;GHI4 ,? IS ! PO9T AT : WE H ALR E/ABLI%$ ! CORRECT;S ( \R F/ QUOTI5T FIGURE #B & >E R1DY 6MAKE \R SECOND SET ( -P>ISONS4 ,-P>E ! #B ( #BG )! #AH ( #AHI4 ,TWO 96#AH IS #I4 ,S SET ! QUOTI5T FIGURE #I ON COL- UMN ;F4 ,N[ MULTIPLY ? QUOTI5T FIG- URE #I 0! #B ( #BG4 ,N9E 0#B IS #AH4 ,S SUBTRACT #AH F ;GH4 ,N[ MULTIPLY ? #I 0! #G (! DIVISOR #BG4 ,N9E 0#G IS #FC4 ,B Y _C SUBTRACT #FC F #I4 ,S F> Z ! ABACUS IS 3C]N$1 !N1 Y H N YET MULTIPLI$ #I 0#G 2C NO RECORD+ ( S* MULTIPLIC,N & XS IMPOSSI# SUB- TRAC;N HAS BE5 MADE4 ,N[ WE "K T ! CORRECT QUOTI5T FIGURE = COLUMN ;F IS #G4 ,S1 9/1D ( REDUC+ ! QUOTI5T FIGURE #I 0A CORRECTIVE FIGURE ( #A EA* "T1 LET'S SUBTRACT #B F ! #I & *ANGE X 6! CORRECT QUOTI5T FIGURE #G AT "O /ROKE4 ,WE MULTIPLY ! CORREC- TIVE FIGURE #B 0! #B ( #BG4 ,? GIVES U A PRODUCT ( #D4 ,\R M>K] COLUMN IS ;F & ? IS ! F/ MULTIPLIC,N (! COR- RECTIVE FIGURE1 S X M/ 2 SET ON ! F/ POSI;N4 ,F\R IS A "O-DIGIT NUMB]1 S X M/ 2 SET ON ! SECOND COLUMN ( ? POSI;N1 9 O!R ^WS1 ON COLUMN ;H4 ,N[ WE >E R1DY 6MULTIPLY ! NEW QUOTI5T FIGURE #G 0! #G ( #BG4 ,SEV5 0#G IS #DI4 ,WE SUBTRACT #DI F ;HI1 CLE>+ ! ABACUS & L1V+ ! CORRECT QUOTI5T #BG ON ;EF4 ,OBS]VE1 AG1 ! IMPORT.E ( QUOTI5T FIGURE PLACE;T &( MULTIPLIC,N (! CORRECTIVE FIGURE 0^? DIGITS : H BE5 MULTIPLI$ 0! UN9T5;N,Y HI< QUOTI5T FIGURE1 &! RETURN (! RESULT+ PRODUCT ( ? CORRECTIVE FIGURE MULTIPLIC,N 6! DIVID5D 9 A3ORD.E ) RULES ( POSI;N+4 ,US+ ! ,CRANM] ,ABACUS #IE ,! ,TR1T;T ( ,Z]OS ,IF WE >E "W+ )A DIVISOR : 3TA9S "O OR M Z]OS 9 ! MI4LE (! NUMB]1 ^! DIGITS >E TR1T$ J Z ANY O!R NUMB]1 T IS1 :5 ! QUOTI5T FIGURE IS MULTIPLI$ 0! V>I\S DIGITS (! DIVISOR1 XS MUL- TIPLIC,N 0A #J RESULTS 9 A PRODUCT ( #J4 ,:5 ? HAPP5S1 ? #J O3UPIES XS POSI;N 9 A3ORD.E )! ALR E/ABLI%$ RULES ( POSI;N+4 ,Z]O IS A "O-DIGIT NUMB]4 ,S1 IF WE FOLL[ ! TE*NIQUE DESCRIB$ E>LI] ( US+ ! 9DEX F+] & MI4LE F+] (! "R H& 6M>K (F POSI;NS1 ! #J1 2+ A "O-DIGIT NUMB]1 W O3UR "U ! MI4LE F+] 9 XS APPROPRIATE POSI;N4 ,"W+ "? AN 5TIRE EXAMPLE W N 2 NEC 6MAKE ? CLE>4 ,LET'S J SET A QUOTI5T FIGURE ( #G ON COLUMN ;E1 & RETA9 \R DIVISOR 9 MEMORY4 ,\R DIVISOR IS #CJ1JJF4 ,F/1 WE WD MULTIPLY ! QUO- TI5T FIGURE #G 0#C GIV+ U #BA4 ,? IS ! F/ MULTIPLIC,N ( #G & WD 2 SUB- TRACT$ F ! F/ POSI;N1 COLUMNS ;FG4 ,SEV5 0#J IS #J--SECOND MULTIPLICA- ;N2 "!=E1 SUBTRAC;N M/ 2 F ! SECOND POSI;N4 ,PLAC+ ! 9DEX F+] ON COLUMN ;F &! MI4LE F+] ON COLUMN ;G1 WE SAY 8POSI;N "O04 ,MOV+ ? PAIR ( F+]S "O PLACE 6! "R1 WE SAY 8POSI;N TWO04 ,! #J IS SUBTRACT$ F ! COLUMN ON : ! MI4LE F+] RE/S1 9 ? CASE COLUMN ;H4 ,! ?IRD MULTIPLIC,N ( #G IS 0! SEC- OND #J (! DIVISOR1 ! PRODUCT 2+ #J4 ,? IS SUBTRACT$ F ! SECOND COLUMN (! ?IRD POSI;N1 COLUMN ;I4 ,AG1 #G 0#J IS #J--F\R? MULTIPLIC,N2 "!=E1 SUB- TRAC;N IS F ! F\R? POSI;N4 ,SEV5 0#F IS #DB--FIF? MULTIPLIC,N2 "!=E1 ! #DB WD 2 SUBTRACT$ F ! FIF? POSI;N4 ,N[1 3SID]3 ;E IS ! M>K] COLUMN4 ,S1 6F9D ! F\R? & FIF? POSI;NS1 WE WD AG PLACE ! 9DEX F+] ON ;F1 MI4LE F+] ON COLUMN ;G--POSI;N "O4 ,"O PLACE 6! "R IS POSI;D TWO2 "O PLACE 6! "R1 POSI;N ?REE2 "O PLACE 6! "R1 POSI;N F\R4 ,"!=E1 ! #J RESULT+ I ! F\R? MULTIPLIC,N IS SUBTRACT$ F ,US+ ! ,CRANM] ,ABACUS #IG COLUMN ;J4 ,MOV+ \R 9DEX F+] & MI4LE F+] OV] "O PLACE 6! "R GIVES U ! FIF? POSI;N4 ,Y W REMEMB] T ! FIF? MULTIPLIC,N ( #G 0 0#F4 ,SEV5 0#F IS #DB4 ,S ! #D ( #DB IS SUBTRACT$ F ! COLUMN ON : ! 9DEX F+] RE/S4 ,AG1 ? IS COLUMN ;J1 ! SAME COLUMN F : WE J SUBTRACT$ \R #J4 ,! #B ( #DB IS SUB- TRACT$ F COLUMN ;K1 ON : YR MI4LE F+] IS RE/+4 ,? TE*NIQUE ( US+ ! 9DEX F+] & MI4LE F+] 6ID5TIFY POSI;NS IS ALL "R =A 2G9N]4 ,B1 )! -+ ( SKILL & SPE$1 Y W F9D YRF AUTOMATIC,Y TRAVEL+ AL;G ! ABACUS ) NO 3SCI\S EF=T ( POSI;N C.T+4 ,:5 ? HAPP5S1 ! PROCESS ( DI- VI.N W 2G9 6BE AUTOMATIC = Y4 ,9 OR- D] 6R1* ? GOAL ( AUTOMATIC OP],N (! ABACUS1 Y %D TAKE ADVANTAGE ( E OP- PORTUN;Y = PRACTICE4 ,DIVIDE YR TEL- EPH"O NUMB] 0TWO- OR ?REE-DIGIT NUM- B]S4 ,AN 9T]E/+ EX]CISE IS 6DIVIDE #IHG1FED1CBA 0ALL (! NUMB]S F #A 6#AJJ1 RECORD+ YR QUOTI5T EA* "T4 ,2F RECORD+ X1 ( C\RSE1 Y %D MULTI- PLY X 6V]IFY XS CORRECT;S1 &= "S AD- DI;NAL PRACTICE 9 MULTIPLIC,N4 ,ONLY "? EX]CISES ( ? TYPE1 C Y GA9 ! SKILL : Y W NE$ = MAK+ USE (! ABACUS 9 YR DAILY LIFE4 ,F ,PROCESS 6,APPLIC,N ,IF Y H BE5 FAI?;L 6? PO9T1 Y %D 2 -=T$ 6"K T Y H N[ LE>N$ ALL (! PROC- ESSES ( >I?METIC Z !Y >E P]=M$ ON ! ,CRANM] ,ABACUS4 ,:AT FOLL[S NEXT IS A 4CUS.N ( "S (! WAYS 9 : ^! PROC- ESSES MAY 2 PUT 6"W = Y4 ,! ,EXTRAC;N ( ,ROOTS ,"! >E S"EAL ME?ODS 0: SQU>E1 CUBE1 F\R?1 & O!R ROOTS C 2 EXTRACT$ US+ ! ABACUS4 ,^! ME?ODS 3SI/ (A -B9,N ( SUBTRAC;N & DIVI.N4 ,A DE- SCRIP;N ( ^! ME?ODS WD REQUIRE CON- SID]A# SPACE & MII?MS LOOK UP ! LOGA- RI?M (! NUMB] ^: ROOT Y WANT 6EX- TRACT4 ,!N1 DIVIDE ? LOG>I?M 0! NUM- B] (! ROOT4 ,! QUOTI5T W 2 ! LOGA- RI?M (! ACTUAL ROOT Y >E SEEK+4 ,= EXAMPLE1 ! SQU>E ROOT ( #GBI IS #BG4 ,SUPPOSE Y D N "K ?1 & WANT 6EXTRACT ! SQU>E ROOT ( #GBI4 ,LOOK UP ! LOG- >I?M ( #GBI1 & DIVIDE X 0#B4 ,! QUO- TI5T W 2 ! LOG>I?M ( #BG1 ! SQU>E ROOT ( #GBI4 ,! SIX? ROOT ( #GBI IS #C4 ,DIVID+ ! LOG ( #GBI 0#F W PRO- DUCE1 Z A QUOTI5T1 ! LOG>I?M ( #C1 ETC4 ,TA#S ( LOG>I?MS >E AVAILA# 9 ,BRL F ! ,AM]ICAN ,PR9T+ ,H\SE =! ,BL1 #AHCI ,FRANK=T ,AV5UE1 ,L\ISVILLE #F1 ,K5T-CKY4 ,! ,TR1T;T ( ,FRAC;NS ,-MON FRAC;NS1 S* Z #A/B1 #C/E1 #E/H1 ETC41 OR MIX$ NUMB] FRAC;NS1 S* Z #B-C/H1 #G-C/AF1 _C 2 H&L$ 0! ,CRANM] ,ABACUS 9 _! ORIG9AL =M4 ,NEI C !Y 2 H&L$ 0ANY O!R CALCULAT+ MA*9E 9 G5]AL USE4 ,H["E1 ! >I?METIC -PUT,N ASSOCIAT$ ) FRAC;NS C 2 H&L$ V NICELY ON ! ABACUS4 ,Y W1 ( C\RSE1 NE$ EI A SECOND ABACUS OR ,BRL WRIT+ EQUIP;T =! PURPOSE ( RECORD+ YR "W4 ,"O EXAMPLE %D 2 5 6GIVE Y ! IDEA4 ,SUPPOSE Y WANT 6MULTIPLY #E/H BY #B/C4 ,S* A PRO#M IS WRITT5 GRAPH- IC,Y3 #E/H 0#B/C EQUALS #E 0#B/#H BY #C4 ,WE >E !N A# 6MULTIPLY ! NUM]A- TORS #E 0#B1 RECORD ! RESULT+ PROD- UCT #AJ1 &1 AG1 WE >E A# 6MULTIPLY ! DENOM9ATORS #H 0#C1 & RECORD \R RE- SULT+ PRODUCT #BD4 ,REPRES5T+ ? GRAPHIC,Y1 WE WD H #E/H 0#B/C EQUALS #E 0#B/#H 0#C EQUALS #AJ/BD EQUALS #E/AB4 ,? EXAMPLE IS 5 6GIVE Y ! IDEA1 B XS SIMPLIC;Y MAY L1D Y 6"Q ,US+ ! ,CRANM] ,ABACUS #AJA ! VALUE (! USE (! ABACUS ) FRAC;NS4 ,IF Y H -E 6S* A 3CLU.N1 WE L1VE Y 6POND] ! PRO#M #D-CG/FD 0#CC-BB/HE4 ,)9 A SPACE ( TWO OR ?REE M9UTES1 A SKILL$ OP]ATOR (! ABACUS %D 2 A# TO -PLETE ? PRO#M1 EXPRESS+ X EI Z A MIX$-NUMB] FRAC;N OR Z A DECIMAL FRAC;N4 ,?US1 ! >I?METIC ASSOCIAT$ )! TR1T;T ( -MON FRAC;NS & MIX$-NUM- B] FRAC;NS C 2 D"O QKLY & E6ICI5TLY )! ABACUS1 :5 6D S IS NEC4 ,/ ANO!R WAY 6TR1T FRAC;NS IS TO 3V]T !M 6DECIMAL EQUIVAL5TS4 ,"O- HALF WD 2 EXPRESS$ Z #.E2 #A/D Z #.BE2 #C/H Z #.CGE4 ,! MIX$ NUMB] FRAC;N #B-C/H 2COMES #B.CGE4 ,CALCUL,NS ,3TA9+ ,DECIMALS ,9 ! SEC;N ON A4I;N1 WE ALR M5;N$ T ! A4I;N ( DECIMALS IS D"O 9 A /RAID MANN]4 ,! UNIT M>K IS US$ Z A DECIMAL PO9T1 & BO? A4I;N & SUB- TRAC;N C 2 H&L$ ) NO NE$ ( FUR!R EXPLAN,N4 ,! MULTIPLIC,N & DIVI.N ( NUMB]S 9 : DECIMALS O3UR PRES5TS ON- LY "O PRO#M1 & T IS ! LOC,N (! DECI- MAL PO9T 9 ! PRODUCT ( QUOTI5T4 ,DIGITS ,V]SUS ,DECIMALS ,6? PO9T 9 \R "W )! ABACUS1 WE H C5T]$ \R ATT5;N ^U SPECIFIC COLUMNS4 ,WE H 3C]N$ \RVS )! CORRECT PLACE;T ( DIGITS .^U .SPECIFIC .COLUMNS4 ,! DECIMAL PO9T IS .N A DIGIT1 & 9 "W+ ) DECIMALS WE M/ 3C]N \RVS N ONLY )! COLUMNS !MVS1 B )! >1S .2T COLUMNS4 ,A DECIMAL PO9T IS A M>K ( SEP>,N4 ,DIGITS 6! LEFT (! DECIMAL PO9T >E 3SID]$ 6BE :OLE NUMB]S4 ,DIGITS 6XS "R >E 3SID]$ 6BE FRAC;NAL NUMB]S4 ,?US1 9 ! NUMB] #B.E ! #B : IS 6! LEFT (! DECIMAL IS A :OLE NUMB] #B4 ,B1 ! #E 6! "R (! DECIMAL PO9T IS #.E OR ! -MON FRAC;N #E/AJ4 ,SIMI- L>LY1 #.BE IS 3SID]$ 6BE ! -MON FRAC;N #BE/AJJ4 ,9 ! NUMB] #F.BE #F IS A :OLE NUMB] FOLL[$ 0! DECIMAL ,US+ ! ,CRANM] ,ABACUS #AJC PO9T ( SEP>,N & #BE OR ! -MON FRAC;N #A/D4 ,S1 #F.BE 2COMES #F-A/D4 ,"!- =E1 9 "W+ ) DECIMALS1 WE C N"E SAY T ! DECIMAL PO9T IS .ON COLUMN ;E1 OR .ON COLUMN ;H1 ETC4 ,WE M/ SAY ! DECIMAL PO9T IS .TO .! .LEFT ( COL- UMN ;E1 .TO .! ."R ( COLUMN ;E1 OR .2T COLUMNS ;D & ;E1 OR .2T COLUMNS ;H & ;I4 ,NOTICE T ! UNIT M>KS ON YR ABACUS FALL 2T COLUMNS & D N O3UR ON ! COLUMNS !MVS4 ,? %D HELP Y REMEMB] ! IMPORT.E ( ? MATT]4 ,DECIMALS D N O3UR .^U COLUMNS1 !Y O3UR .2T COL- UMNS4 ,MULTIPLIC,N ( ,DECIMALS ,9 \R E>LI] 4CUS.N ( MULTIPLIC,N1 WE ASK$ Y 6SET ! MULTIPLIC& ON YR ABACUS 9 S* A WAY T XS LA/ DIGIT OC- CURR$ ON A UNIT COLUMN4 ,? PROC$URE IS IMPORTANT N ONLY 9 HELP+ Y LOCATE ! LA/ DIGIT (! PRODUCT1 B AL 9 "W+ CALCUL,NS 9VOLV+ DECIMALS4 ,LOCAT+ ! DECIMAL PO9T 9 ! PRODUCT IS SIMPLIC- ;Y XF1 NO MATT] H[ 9VOLV$ ! CALCUL,N SEEMS 6BE1 IF Y W SET ! LA/ :OLE- NUMB] DIGIT (! MULTIPLIC& ^U A UNIT COLUMN1 & IF Y FOLL[ TWO SIMPLE PRO- C$URES4 ,WE W \TL9E ^! PROC$URES IN- DIVIDU,Y1 & FOLL[ EA* ) AN APPROPRI- ATE EXAMPLE4 ,F/1 :5 ! MULTIPLI] IS A MIX$-DECIMAL FRAC;N1 T IS1 A NUMB] 3TA9+ BO? :OLE-NUMB] DIGITS &A DECI- MAL FRAC;N1 ! DECIMAL PO9T MOVES 6! "R F XS ORIG9AL SETT+ 9 ! MULTIPLIC& 0Z _M PLACES Z "! >E :OLE NUMB] DIG- ITS 9 ! MULTIPLI]1 PLUS "O1 2C WE >E "W+ )! PROCESS ( MULTIPLIC,N4 ,= \R EXAMPLE1 #CHD MULTIPLI$ 0#B.H1 ! MULTIPLIC& IS #CHD4 ,S1 ? IS SET ON ! ABACUS 9 S* A WAY T ! #D ( #CHD O3URS ON A UNIT COLUMN1 T IS1 A COL- UMN IMMLY 6! LEFT ( "O (! UNIT M>KS4 ,LET'S J CALL ? COLUMN ;G4 ,NEXT1 MULTIPLY #CHD 0#BH 9 A /RAID MANN]1 =GETT+ =! MO;T T ! MULTIPLI] IS #B.H4 ,:5 WE MULTIPLY #CHD 0#BH1 \R PRODUCT IS #AJ1GEB4 ,N[1 6LOCATE ,US+ ! ,CRANM] ,ABACUS #AJE ! DECIMAL PO9T 9 ? PRODUCT1 LET'S PLACE \R F+] ^U ! UNIT M>K IMMLY 6! LEFT ( : WE _H ORIG9,Y SET ! #D ( #CHD4 ,9 O!R ^WS1 PLACE YR F+] ON ! UNIT M>K : FALLS 2T ;G & ;H4 ,? IS YR />T+ PLACE1 & X REPRES5TS ! DECI- MAL PO9T (! MULTIPLIC&1 2C ! :OLE NUMB] #CHD C 2 ?"\ 6BE #CHD.JJ4 ,N[1 ! DECIMAL PO9T 9 ! PRODUCT IS MOV$ 6! "R (! ORIG9AL SETT+ (! DECIMAL PO9T 9 ! MULTIPLIC& 0A NUMB] ( PLACES EQUAL 6! NUMB] ( :OLE-NUMB] DIGITS 9 ! MULTIPLI]1 PLUS "O M PLACE 2C ? IS ! PROCESS ( MULTIPLI- C,N4 ,F YR />T+ PLACE 2T COLUMNS ;G & ;H1 MOVE "O PLACE 6! "R4 ,! MULTI- PLI] 0 #B.H4 ,S1 X _H J "O :OLE-NUM- B] DIGIT1 "NLY #B4 ,NEXT1 MOVE "O M PLACE 6! "R 2C WE >E "W+ )! PROC- ESS ( MULTIPLIC,N4 ,! F/ PLACE 6: Y MOV$ 0 2T COLUMNS ;H & ;I1 &! SECOND PLACE1 ! "O REQUIR$ 0! PROCESS ( MULTIPLIC,N1 IS 2T COLUMNS ;I & ;J4 ,"!=E1 ? IS ": ! DECIMAL PO9T O3URS1 2T COLUMNS ;I & ;J4 ,LOOK+ AT \R PRODUCT1 WE SEE T ! DECIMAL O3URS 2T ! #E ON COLUMN ;I &! #B ON COLUMN ;J1 MAK+ \R PRODUCT #A1JGE.B4 ,IF1 9/1D ( #CHD1 \R MULTIPLIC& _H BE5 #C.HD1 WE WD H FOLL[$ PRECISELY ! SAME PROC$URE1 EXCEPT WE WD H SET ! MULTIPLIC& 9 S* A WAY T ! #C ( #C.HD O3URR$ ON ! UNIT COLUMN4 ,WE WD H !N C.T$ TWO PLACES 6! "R 9 PRE- CISELY ! SAME WAY4 ,IF YR PRO#M IS 5T]$ CORRECTLY ^U ! ABACUS1 LOC,N (! DECIMAL PO9T IS 3TROLL$ 0! NUMB] ( :OLE-NUMB] DIGITS 9 ! MULTIPLI]1 & S F> Z ! MULTIPLIC& IS 3C]N$1 Y C SET X & =GET X--UNTIL ! "T -ES 6F9D YR />T+ PLACE4 ,SUPPOSE N[ T \R MULTIPLI] _H BE5 #.BH 9/1D ( #B.H4 ,9 S* A CASE1 WE WD H F.D \R />T+ PLACE1 T IS1 ! UNIT M>K 2T COLUMNS ;G & ;H1 & S9CE "! WD 2 NO :OLE-NUMB] DIGITS 9 ! MULTI- PLI]1 WE WD MOVE J "O PLACE 6! "R TO ALL[ =! PROCESS ( MULTIPLIC,N4 ,? WD H PLAC$ ! DECIMAL 2T COLUMNS ;H & ;I ,US+ ! ,CRANM] ,ABACUS #AJG & GIV5 U A PRODUCT ( #AJG.EB4 ,! SECOND PROC$URE =! MULTIPLIC,N ( DECIMALS 3C]NS ! SITU,N 9 : ! MUL- TIPLI] IS A PURE DECIMAL FRAC;N ^: F/ SIGNIFICANT FIGURE IS SEP>AT$ F ! DECIMAL PO9T 0"O OR M Z]OS4 ,9 ? SITU,N1 ! DECIMAL PO9T (! PRODUCT MOVES "O PLACE 6! "R ( XS ORIG9AL SETT+ 9 ! MULTIPLIC& 6ALL[ =! PROC- ESS ( MULTIPLIC,N1 & !N 6! LEFT 0A NUMB] ( PLACES EQUAL 6! NUMB] ( Z]OS : SEP>ATE ! DECIMAL F XS F/ SIGNIFI- CANT DIGIT4 ,LET'S USE ! SAME EXAM- PLE1 ) #CHD Z \R MULTIPLIC&1 & #.JJJBH Z ! MULTIPLI]4 ,WE WD SET #CHD S T ! #D O3URS ON A UNIT COL- UMN1 : WE W AG CALL COLUMN ;G4 ,WE WD !N MULTIPLY X 0#BH4 ,? "T WE WD MAKE NO ALL[.E =! Z]OS IMMLY FOLL[+ ! DECIMAL 9 ! MULTIPLI]1 S F> Z ! POSI;N 3CEPT IS 3C]N$4 ,J MULTIPLY #CHD 0#BH4 ,? WD GIVE U A PRODUCT ( #AJ1GEB4 ,N[ LET'S F9D ! DECIMAL PO9T4 ,PLACE YR F+] ON ! />T+ PLACE1 ! UNIT M>K 2T COLUMNS ;G & ;H4 ,N[ MOVE "O PLACE 6! "R 6ALL[ =! PROCESS ( MULTIPLIC,N1 9 O!R ^WS 2T COLUMNS ;H & ;I4 ,N[1 S9CE \R MULTIPLI] IS #.JJJBH1 WE H ?REE Z]OS SEP>AT+ ! DECIMAL PO9T F ! #B ( #BH4 ,S1 F \R PLACE 2T COLUMNS ;H & ;I1 WE MOVE 6! LEFT ?REE PLACES2 F/ PLACE1 BACK 6! />T+ PLACE 2T ;G & ;H2 SECOND PLACE1 2T ;F & ;G2 ?IRD PLACE1 2T COLUMNS ;E & ;F1 & ? IS ": ! DECIMAL PO9T O3URS4 ,\R DECIMAL PO9T !N IS IMMLY 6! LEFT ( COLUMN ;F1 GIV+ U ! PROD- UCT ( #.AJGEB4 ,N[1 SUPPOSE T ! MULTIPLIC& 7 A DECIMAL FRAC;N HAV+ _M Z]OS 2T DECI- MAL PO9T & F/ SIGNIFICANT DIGIT1 = EXAMPLE1 #.JJJCHD4 ,WE WD F9D A UNIT M>K 6USE Z A />T+ PO9T1 & L1VE ! F/ ?REE COLUMNS IMMLY 6XS "R 9 ! Z]O 3DI;N1 SETT+ ! SIGNIFICANT DIGITS #CHD ON ! F\R?1 FIF?1 & SIX? COLUMNS 6! "R ( \R />T+ DECIMAL PO9T4 ,WE WD MULTIPLY /RAID1 & D \R C.T+ F ! ORIG9AL />T+ DECIMAL PO9T1 FOLL[+ ! ,US+ ! ,CRANM] ,ABACUS #AJI RULE PREVI\SLY DESCRIB$4 ,S "H1 TOO1 S F> Z ! MULTIPLIC,N IS 3C]N$1 IF PROP] 5TRY IS MADE (! MULTIPLIC&1 Y C SET X1 & =GET X4 ,! ,DECIMAL 9 ,DIVI.N ,X IS J Z EASY 6LOCATE ! DECIMAL PO9T 9 A QUOTI5T Z X 0 6F9D X 9 ! PRODUCT ( MULTIPLIC,N4 ,9 ! PROCE- DURES 6BE DESCRIB$1 ! DIVID5D IS 6! MULTIPLIC& :AT ! DIVISOR IS 6! MUL- TIPLI]1 & ": WE MOV$ 6! "R 9 MULTI- PLIC,N1 WE W MOVE 6! LEFT 9 DIVI.N4 ,! PROC$URE1 !N1 IS ?3 ,SET ! DIVI- D5D 9 S* A WAY T XS LA/ :OLE-NUMB] DIGIT O3URS ON A UNIT COLUMN4 ,!N1 DIVIDE X 0! DIVISOR 9 A /RAID WAY4 ,6F9D ! DECIMAL PO9T 9 ! QUO- TI5T1 PLACE YR F+] ON YR />T+ PO9T1 T IS1 ! ORIG9AL DECIMAL SETT+ (! DIVID5D4 ,NEXT1 MOVE 6! LEFT Z _M PLACES Z "! >E :OLE NUMB] DIGITS 9 ! DIVISOR1 PLUS "O A4I;NAL PLACE 2C ? IS ! PROCESS ( DIVI.N4 ,? W REPRES5T ! LOC,N (! DECIMAL PO9T 9 ! QUOTI5T4 ,LET'S D AN EXAMPLE3 #F1BEE DIVID$ 0#D.E4 ,^U YR ABACUS SET ! DIVID5D #F1BEE 9 S* A WAY T ! LA/ #E O3URS ^U A UNIT COLUMN4 ,LET'S CALL ? COL- UMN ;J4 ,N[ DIVIDE X 0! DIVISOR #DE 9 A /RAID WAY4 ,YR QUOTI5T W 2 #ACI4 ,B1 N[ WE H 6F9D ! DECIMAL PO9T =! QUOTI5T4 ,PLACE YR F+] ON ! />T+ PO9T1 ! UNIT M>K 2T COLUMNS ;J & ;K4 ,! DIVISOR IS #D.E4 ,? M1NS WE H "O :OLE-NUMB] DIGIT 6! LEFT (! DECIMAL 9 ! DIVISOR1 PLUS "O A4I;NAL PLACE 2C ? IS ! PROCESS ( DIVI.N4 ,! F/ PLACE 6: WE MOV$ 0 2T COLUMNS ;I & ;J2 ! SECOND PLACE1 2T COLUMNS ;H & ;I1 & ? IS ": ! DECIMAL IS4 ,EX- AM9+ ! ABACUS1 !N1 WE F9D \R QUOTI5T 6BE #A1CIJ.JJ4 ,IF1 9/1D ( #F1BEE1 \R DIVID5D _H BE5 #F.BEE1 WE WD H SET X 9 S* A WAY T ! #F ( #F1BEE WD H FALL5 ^U COL- UMN ;J1 ! UNIT COLUMN4 ,X IS IMPOR- TANT T Y 5T] ! DIVID5D CORRECTLY1 ,US+ ! ,CRANM] ,ABACUS #AAA B1 HAV+ 5T]$ X1 Y NE$ H NO FUR!R 3C]N ) X1 2C Y >E "W+ ) DECIMALS1 ?AN Y WD IF Y 7 "W+ A /RAID PRO#M 9 :OLE-NUMB] DIVI.N4 ,J Z 9 ! CASE (! MULTIPLIC&1 ONCE Y SET X1 Y C =GET X4 ,! SECOND PROC$URE =! DIVI.N ( DECIMALS IS 3C]N$ )A DIVISOR : IS A DECIMAL FRAC;N HAV+ "O OR M Z]OS 2T ! DECIMAL PO9T &! F/ SIGNIFICANT DIGIT4 ,SUPPOSE1 ? "T1 WE DIVIDE #F1BEE 0#.JJDE4 ,WE 5T] \R DIVID5D J Z 2F1 & DIVIDE 0#DE 9 A /RAID MANN]4 ,!N1 PROC$URE IS Z FOLL[S3 ,2G9N+ ) \R />T+ PO9T1 WE MOVE 6! LEFT "O PLACE =! PROCESS ( DIVI.N1 & 6! "R 0A NUMB] ( PLACES EQUAL 6! NUMB] ( Z]OS 2T ! DECIMAL &! F/ SIG- NIFICANT DIGIT (! DIVISOR4 ,S1 PLACE YR F+] ON YR />T+ PO9T1 ! UNIT M>K 2T ;J & ;K4 ,G "O PLACE 6! LEFT =! PROCESS ( DIVI.N1 &1 S9CE ! DIVISOR IS #.JJDE1 WE H TWO Z]OS 2T DECIMAL & F/ SIGNIFICANT DIGIT4 ,S1 WE G 6! "R TWO PLACES4 ,? PLACES ! DECIMAL PO9T 2T COLUMNS ;K & ;L4 ,? GIVES U A QUOTI5T ( #A1CIJ1JJJ.JJ4 ,IF \R DIVID5D 7 A DECIMAL FRAC;N HAV+ "O OR M Z]OS 2T ! DECIMAL PO9T &! F/ SIGNIFICANT DIGIT1 WE 5T] X J Z IF WE 7 "W+ A PRO#M 9 MULTIPLIC,N1 T IS1 WE WD SELECT \R UNIT M>K1 &1 IF ! DIVID5D 7 #.JJFBEE1 WE WD USE ! F/ TWO COLUMNS 6! "R (! UNIT M>K TO REPRES5T ! Z]OS1 & FOLL[ X 0#FBEE4 ,DIVI.N WD 2 9 A /RAID WAY1 & LOC,N (! DECIMAL PO9T WD 2G9 F ! />T+ PO9T1 J Z WE H PREVI\SLY \TL9$4 ,! ,ABACUS Z A ,CAL5D> ,2C (! BULKI;S ( ,BRL CAL5D>S1 & 2C (! NE$ 6REPLACE !M E YE>1 X MI1 T IS1 X C 2 US$ 6DET]M9E ! "D (! WEEK ON : A GIV5 DATE W O3UR AT ANY "T 9 ! FUTURE4 ,X C AL 2 US$ 6DET]M9E ! "D ^U : A DATE : HAS ALR PASS$ HAS O3URR$1 = EXAM- ,US+ ! ,CRANM] ,ABACUS #AAC PLE1 YR BIR?"D1 OR W$D+ ANNIV]S>Y1 OR "S DATE IMPORTANT 6Y OR 6A FR4 ,"H'S H[ X'S D"O4 ,SUPPOSE ? IS ,?URS"D1 ! F/ ( ,M>*1 & Y WANT 6"K ON :AT "D ,NOVEM- B] #AE W FALL4 ,WE W CALL ,NOVEMB] #AE ! T>GET DATE1 ,NOVEMB]1 ! T>GET MON?1 ,M>* #A1 ! PRES5T DATE1 & ,M>*1 ! PRES5T MON?4 ,ON ! ABACUS Y 5T] ! TOTAL NUMB] ( "DS 9 ! PRES5T MON?4 ,F ? Y SUBTRACT ! NUMB] (! PRES5T DATE1 & 6? FIGURE1 Y ADD A S]IES ( NUMB]S EQUAL 6! NUMB] ( "DS 9 ! 9T]V5+ MON?S4 ,? IS1 ! MON?S SEP>AT+ ! PRES5T MON? F ! T>GET MON?4 ,:5 Y R1* ! T>GET MON?1 5T] A NUMB] EQUAL 6! T>GET DATE4 ,? W GIVE Y ! TOTAL NUMB] ( "DS 2T PRES5T DATE & T>GET DATE4 ,DIVIDE ? NUMB] 0#G & ? W N ONLY GIVE Y ! NUMB] ( 9T]V5+ WEEKS1 B1 IF "! IS A REMA9D]1 X W REPRES5T ! DI6];E 2T ! PRES5T "D &! T>GET "D4 ,LET'S "W "? "O PRO#M TO GIVE Y ! IDEA4 ,? IS ,?URS"D1 ,M>* #A1 & \R T>GET DATE IS ,NOVEMB] #AE4 ,M>* IS ! PRES5T MON?4 ,S1 ON YR AB- ACUS1 5T] #CA = ,M>*4 ,SUBTRACT F ? ! PRES5T DATE1 : IS #A4 ,? L1VES #CJ4 ,N[1 6? #CJ ADD ! NUMB] ( "DS = EA* (! 9T]V5+ MON?S4 ,= ,APRIL1 WE ADD #CJ2 = ,MAY1 #CA2 = ,JUNE1 #CJ2 = ,JULY1 #CA2 = ,AUGU/1 #CA2 = ,SEP- TEMB]1 #CJ2 = ,OCTOB]1 #CA2 &1 = ,NOVEMB]1 \R T>GET MON?1 WE ADD #AE1 ! NUMB] ( \R T>GET DATE4 ,\R ABACUS %D N[ %[ A TOTAL ( #BEI4 ,? M1NS T "! >E #BEI "DS F ,M>* #A 6,NOVEMB] #AE4 ,N[1 DIVIDE ? #BEI 0#G4 ,Z X HAPP5S1 ! NUMB] DIVIDES EV5LY & \R QUOTI5T IS #CG1 ) NO REMA9D]4 ,S9CE "! IS NO REMA9D]1 X M1NS T \R PRES5T "D & \R T>GET "D >E ! SAME "D (! WEEK1 "NLY ,?URS"D4 ,IF \R REMA9D] _H BE5 #E1 WE WD C.T =W>D #E "DS F ,?URS"D3 ,FRI"D1 #A2 ,SATUR"D1 #B2 ,SUN"D1 #C2 ,MON"D1 #D2 ,TUES"D1 #E1 & 9 S* A CASE \R T>GET DATE WD H FALL5 ON A T>GET "D ( ,TUES"D4 ,IF Y WANT 6LOOK AH1D A NUMB] ( ,US+ ! ,CRANM] ,ABACUS #AAE YE>S1 ! PROC$URE IS ! SAME1 EXCEPT T Y D ! A4I;NAL /EP ( MULTIPLY+ #CFE 0! NUMB] ( YE>S J 6SAVE YRF "S TR\#4 ,9 S* A CASE1 2 SURE 69CLUDE AN EX- TRA "D = E L1P YE> CROSS$ 0YR CAL- CUL,NS4 ,N[1 WE W \TL9E ! PROC$URE = F9D+ A DATE : HAS ALR PASS$1 B WE W N "[ "? AN EXAMPLE1 S9CE1 0N[1 Y %D "U/& \R T]M9OLOGY1 & H GA9$ 5 9SIGET YE>1 ,OCTOB]1 \R T>GET MON?1 & #AH1 \R T>GET DATE4 ,F/1 DE- T]M9E ! T>GET DATE =! PRES5T YE>1 T IS1 ,OCTOB] #AH1 9 ! CURR5T YE>4 ,!N1 5T] ! NUMB] (! CURR5T YE> ON YR ABACUS & F ? SUBTRACT ! NUMB] (! T>- GET YE>4 ,MULTIPLY ! REMA9D] 0#CFE4 ,? W REPRES5T ! NUMB] ( NON-,L1P ,YE> "DS 2T ! T>GET DATE 9 ! T>GET YE> &! T>GET DATE 9 ! PRES5T YE>2 IS 2T ,OCTOB] #AH1 #AIAJ & ,OCTOB] #AH (! PRES5T YE>4 ,NEXT1 6? FIGURE ADD ! NUMB] ( ,L1P ,YE> "DS4 ,WE SU7E/ T Y D X 9 ? WAY4 ,ON ! APPRO- PRIATE COLUMN ( YR ABACUS1 MOVE UP C.T]S "O AT A "T1 Z Y SAY ! V>I\S ,L1P ,YE>S3 #AIAB1 #'AF1 #'BJ1 #'BD1 #'BH1 #'CB1 #'CF1 #AIDJ1 ETC4 ,:5 Y H 5T]$ ! LA/ ,L1P ,YE> "D1 Y W H ! TOTAL NUMB] ( "DS 2T ,OCTOB] #AH1 #AIAJ1 & ,OCTOB] #AH (! PRES5T YE>4 ,DIVIDE ? 0#G1 & ? W GIVE Y ! NUMB] ( WEEKS4 ,%D "! 2 A REMA9D]1 Y WD C.T BACKW>DS1 T IS1 IF ,OCTOB] #AH ( ! PRES5T YE> IS ON ,W$NES"D1 & Y H A REMA9D] ( #A1 Y WD C.T BACK J #A "D 6,TUES"D4 ,IF Y _H A REMA9D] ( #E1 & ,OCTOB] #AH (! PRES5T YE> HAPP5S 6BE A ,SATUR"D1 Y WD C.T BACK #E "DS F ,SATUR"D3 ,FRI"D1 #A2 ,?URS"D1 #B2 ,W$NES"D1 #C2 ,TUES"D1 #D2 ,MON"D1 #E4 ,9 S* A CASE1 ,OCTOB] #AH1 #AIAJ WD H FALL5 ON A ,MON"D4 ,9 CASE Y WD L 6*ECK YR RESULTS1 ,OCTOB] #AH1 #AIAJ 0 ACTU,Y A ,TUES"D4 ,US+ ! ,CRANM] ,ABACUS #AAG ,3CLU.N ,9 PRES5T+ ! =EGO+ 9=M,N1 \R AIM HAS N BE5 6T1* Y >I?METIC4 ,9/1D1 WE H TRI$ 6MAKE AVAILA# 6Y ! M1NS ( EN- JOY+ ! SPE$1 E6ICI5CY1 3V5I;E1 & IN- DEP5D;E1 : ! ,CRANM] ,ABACUS AF=DS BL P4 ,IF Y H R1D ? "W ?OR\G4 #AAC ,S#H4 #BJH ,S#I4 #H1GFE ,S#AJ4 #C1BAJ ,S#AA4 #A1IGF ,S#AB4 #AIH ,S#AC4 #IIE ,S#AD4 #DGJ ,S#AE4 #IC ,S#AF4 #HHI ,S#AG4 #IIB ,S#AH4 #II ,S#AI4 #F1CAB ,S#BJ4 #HFD1AIG1ECB ,DIVI.N ,D#A4 #CG ,D#B4 #CG ,D#C4 #CG ,D#D4 #BD ,D#E4 #ABC ,D#F4 #ABC ,D#G4 #EAE ,D#H4 #C1CFG ,D#I4 #BCF ,D#AJ4 #ACG ,D#AA4 #C ,D#AB4 #C1CFG ,D#AC4 #C1CFG ,D#AD4 #BA ,D#AE4 #BG ,D#AF4 #AD ,,! ,,5D