Traditional Abacus and Bead Arithmetic/Division/Special division tables

Principle Suppose we have to perform a large number of divisions by 36525, which could be the case if we do calendar calculations. Then we can simplify the task by creating a specialized division table for this divisor. Following what is stated in the chapter: Guide to traditional division, we will start by calculating the following three Euclidean divisions: +Creating a special division table for 36525100000÷36525200000÷36525300000÷36525QuotientRemainderQuotientRemainderQuotientRemainder226950517375807800 Which can be summarized in the following specialized division table: +Division by 36525 table1/36525>2+269502/36525>5+173753/36525>8+07800 And now we can use this table to do divisions without touching the multiplication table. For example, how many Julian centuries of 36525 days can fit in 1 000 000 days? + 1000000/36525 Abacus Comment ABCDEFGHIJKLM 36525 1000000 Use rule: 1/36525>2+26950 on column G 36525 2000000 change 1 in G into 2       +26950 add 26950 to H-L 36525 2269500 Use rule: 2/36525>5+17375 on column H 36525 2569500 change 2 in H into 5        +17375 add 17375 to I-M 36525 2586875 revise up       +1        -36525 36525 2650350 revise up       +1        -36525 36525 2713825 Done! 1000000÷36525=27, remainder 13825 And we have done a division by a five-digit divisor without using the multiplication table! Two-digit division tables In the past, special division tables were used for divisors between 11 and 99A history of chinese mathematics Martzloff Jean-Claude 2006 Springer 978-3-540-33782-9 221. + Traditional division rules for 2-digit divisors11121314151617181919+018+047+097+026+106+045+155+105+0521222324252627282914+164+124+084+044+003+223+193+163+1329+119+028+168+088+007+187+117+046+2631323334353637383913+073+043+012+322+302+282+262+242+2226+146+086+025+305+255+205+155+105+0539+219+129+038+288+208+128+047+347+2741424344454647484912+182+162+142+122+102+082+062+042+0224+364+324+284+244+204+164+124+084+0437+137+066+426+366+306+246+186+126+0649+319+229+139+048+408+328+248+168+0851525354555657585911+491+481+471+461+451+441+431+421+4123+473+443+413+383+353+323+293+263+2335+455+405+355+305+255+205+155+105+0547+437+367+297+227+157+087+016+526+4659+419+329+239+149+058+528+448+368+2861626364656667686911+391+381+371+361+351+341+331+321+3123+173+143+113+083+053+022+662+642+6234+564+524+484+444+404+364+324+284+2446+346+286+226+166+106+045+655+605+5558+128+047+597+527+457+387+317+247+1769+519+429+339+249+159+068+648+568+4871727374757677787911+291+281+271+261+251+241+231+221+2122+582+562+542+522+502+482+462+442+4234+164+124+084+044+003+723+693+663+6345+455+405+355+305+255+205+155+105+0557+036+686+626+566+506+446+386+326+2668+328+248+168+088+007+687+617+547+4779+619+529+439+349+259+169+078+768+6881828384858687888911+191+181+171+161+151+141+131+121+1122+382+362+342+322+302+282+262+242+2233+573+543+513+483+453+423+393+363+3344+764+724+684+644+604+564+524+484+4456+146+086+025+805+755+705+655+605+5567+337+267+197+127+056+846+786+726+6678+528+448+368+288+208+128+047+847+7789+719+629+539+449+359+269+179+088+8891929394959697989911+091+081+071+061+051+041+031+021+0122+182+162+142+122+102+082+062+042+0233+273+243+213+183+153+123+093+063+0344+364+324+284+244+204+164+124+084+0455+455+405+355+305+255+205+155+105+0566+546+486+426+366+306+246+186+126+0677+637+567+497+427+357+287+217+147+0788+728+648+568+488+408+328+248+168+0899+819+729+639+549+459+369+279+189+09 Some examples Dividing by numbers that start with 1 is awkward, the following table may be used to divide by 19https://sites.google.com/view/jccabacus/abacus-articles/advanced-abacus-algorithms/tide-abacus?authuser=0 Tide Abacus jccAbacus Cabrera Jesús 2021 4 August 2021. +Division by 19 table1915+05 + Division by 99 table99 11+01 22+02 33+03 44+04 55+05 66+06 77+07 88+08 99+09 + 9801÷99 Abacus Comment ABCDEFGHI 9801   99 Dividend AD, divisor HI 9891   99 A: Rule 9/99>9+09 9899   99 B: Rule 8/99>8+08 +1 revising up  -99 99     99 Done! No remainder, quotient: 99 Dividing by 𝝅 is common in applications, here are the tables for two approximations of this irrational number. +Division by 𝝅 tables3143141613+05813+0575226+11626+1150439+17439+17256 Finally, the division by 666 table. +Division by 666 table66611+33423+00234+33646+00457+33869+006 However, It is not advisable to divide by this number; results can be unpredictable… and uncontrollable! In any case, remember the advice: I say to you againe, doe not call up Any that you can not put downe; by the Which I meane, Any that can in Turne call up somewhat against you, whereby your Powerfullest Devices may not be of use. H. P. Lovecraft - The Case of Charles Dexter Ward (1941):)👿 Further reading Specially Crafted Division Tables 2020 Murakami, Masaaki 算盤 Abacus: Mystery of the Bead http://totton.idirect.com/soroban/masaaki/Specially%20Crafted%20Division%20Tables.pdf PDF August 13, 2021 https://web.archive.org/web/20210801083501/http://totton.idirect.com/soroban/masaaki/Specially%20Crafted%20Division%20Tables.pdf August 1, 2021 Division by powers of two Dealing with overflow deep