Traditional Abacus and Bead Arithmetic/Division/Division by powers of two

Introduction A fraction whose denominator only contains 2 and 5 as divisors has a finite decimal representation. This allows an easy division by powers of two or five if we have the fractions tabulated (or memorized) where is one of such powers of two or five. For instance, given Then Which can easily be done on the abacus by working from right to left. For each digit of the numerator: Clear the digit Add the fraction corresponding to the working digit to the abacus starting with the column it occupied + Division 137/8 using fractions Abacus Comment  ABCDEF  --+--- Unit rod  137 enter 137 on A-C as a guide    7 clear 7 in C   +0875 add 7/8 to C-F  130875   3 clear 3 in B  +0375 add 3/8 to B-E  104625  1 clear 1 in A +0125 add 1/8 to A-D   17125 Done!  --+--- unit rod We only need to have the corresponding fractions tabulated or memorized, as in the table below. Powers of two In the past, both in China and in Japan, monetary and measurement units were used that were related by a factor of 16Williams, Samuel Wells Morrison, John Robert A Chinese commercial guide 1856 Canton: Printed at the office of the Chinese Repository https://archive.org/details/chinesecommercia00willuoft/page/298/mode/2up 298Specially 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, 2021Kwa Tak Ming The Fundamental Operations in Bead Arithmetic, How to Use the Chinese Abacus San Francisco 1922 Service Supply Co. https://archive.computerhistory.org/resources/access/text/2016/12/B1671.01-05-01-acc.pdf PDF, a factor that begins with one which makes normal division uncomfortable. For this reason, it was popular to use the method presented here for such divisions. Table of fractions + Power of two fractionsDD/2D/4D/8D/16reference_name_A aD/32reference_name_A aD/64reference_name_A a105025012506250312501562521005002501250062500312503150750375187509375046875420100050025001250006250052512506253125156250781256301500750375018750093750735175087543752187510937584020010005000250001250009452251125562528125140625111Unit rod left displacement reference_name_A aUnit rod left displacement. Examples of use + 137/2  ABCD  --+-reference_name_B b  137    7   +35   3  +15  1 +05  --+-reference_name_B b  0685 68.5 + 137/4  ABCDE  --+--reference_name_B b  137    7   +175   3  +075  1 +025  --+--reference_name_B b  03425 34.25 + 137/8  ABCDEF  --+---reference_name_B b  137    7   +0875   3  +0375  1 +0125  --+---reference_name_B b  017125 17.125 + 137/16  ABCDEF  --+---reference_name_B b  137    7   +4375   3  +1875  1 +0625  -+----reference_name_B b  085625 8.5625 + 137/32  ABCDEFG  --+----reference_name_B b  137    7   +21875   3  +09375  1 +03125  -+-----reference_name_B b  0428125 4.28125 + 137/64  ABCDEFGH  --+-----reference_name_B b  137    7 Clear 7 in C   +109375   3Clear 3 in B  +046875  1Clear 1 in A +015625  -+------reference_name_B b  02140625 2.140625 reference_name_B b"+" indicates the unit rod position. Division by 2 in situ The fractions for divisor 2 are easily memorizable and this method corresponds to the division by two "in situ" or "in place" explained by SiqueiraKato Fukutaro's Square Roots Siqueira, Edvaldo Heffelfinger, Totton 算盤 Abacus: Mystery of the Bead http://totton.idirect.com/soroban/katoSq/ August 10, 2021 https://web.archive.org/web/20210801083040/http://totton.idirect.com/soroban/katoSq/ August 1, 2021 as an aid to obtaining square roots by the half-remainder method (半九九法, hankukuho in Japanese, Bàn jiǔjiǔ fǎ in Chinese, see Chapter: Square root), it is certainly a very effective and fast method of dividing by two. Fractions for other denominators are harder to memorize. Being a particular case of what was explained in the introduction above, to divide in situ a number by two we proceed digit by digit from right to left by: clearing the digit adding its half starting with the column it occupied For instance, 123456789/2: + 123456789÷2 in situ Abacus Comment  ABCDEFGHIJ  123456789          9 Clear 9 in I         +45 Add its half to IJ  1234567845         8 Clear 8 in H        +40 Add its half to HI  1234567445        7 Clear 7 in G       +35 Add its half to GH  1234563945       6 Clear 6 in F      +3 Add its half to FG  1234533945      5 Clear 5 in E     +25 Add its half to EF  1234283945     4 Clear 4 in D    +2 Add its half to DE  1232283945    3 Clear 3 in C   +15 Add its half to CD  1217283945   2 Clear 2 in B  +1 Add its half to BC  1117283945  1 Clear 1 in A +05 Add its half to AB.   617283945 Done! The unit rod does not change in this division. Powers of five Table of fractions +Power of five fractionsDD/5D/25D/125D/62510.20.040.0080.001620.40.080.0160.003230.60.120.0240.004840.80.160.0320.0064510.20.040.00861.20.240.0480.009671.40.280.0560.011281.60.320.0640.012891.80.360.0720.0144 Traditional division examples Special division tables