Using an Abacus/Modern multiplication

You should achieve some basic skills in addition and subtraction before going further with multiplication. Otherwise you will be calling for frustration. Introduction The basic concept of multiplication for natural numbers is that of a repeated addition. For example, to multiply 47 by 23 it is only necessary to add 23 47 times or add 47 23 times; we can do it with our abacus: + Multiplication as repeated addition Abacus Comment ABCDEFHIJ   .  .  . Unit rod  +1   +47 Add 1 to C and 47 to IJ   1    47  +1   +47 Add 1 to C and 47 to IJ   2    94  +1   +47 Add 1 to C and 47 to IJ   3   141    ... Continue in the same way 19 times...!  22  1034  +1   +47 Add 1 to C and 47 to IJ  23  1081 End. 23×47=1081   .  .  . Unit rod Where we repeat 23 times the sum of 47 to the IJ columns while we add 1 to C to have a "counter" at our disposal. But this is desperately slow! A more effective way to do the same thing can be the following: thumb|Mechanical calculators of the past used the procedure on the left to multiply + A more efficient multiplication Abacus Comment ABCDEFHIJ   .  .  . Unit rod  +1   +47 Add 1 to C and 47 to IJ   1    47  +1   +47 Add 1 to C and 47 to IJ   2    94  +1   +47 Add 1 to C and 47 to IJ   3   141 +1   +47 Add 1 to B and 47 to HI  13   511 +1   +47 Add 1 to B and 47 to HI  23  1081 End. 23×47=1081   .  .  . Unit rod Where this time, after adding 47 three times to IJ (and 1 to C) we have moved one column to the left and we have started adding 47 to columns HI (and 1 to B). Adding 47 in HI is equivalent to adding 470 = 10×47 to HIJ (10 to BC) drastically reducing the number of operations to be carried out, because after doing it twice only we reach 23 in the counter BC and 1081 in GHIJ, the final result. This way of multiplying was the usual one in mechanical calculators that appeared at the end of the 19th century and that continued in use until the 1970s. But this is still excessively slow. Think that the abacus as we know it now allows adding very quickly, but that before its invention Chinese mathematicians used counting rods which are extraordinarily slow to handle. It is not surprising therefore that Chinese mathematicians, seeking to abbreviate calculations, eventually invented the decimal multiplication table, as we know it, a few centuries before our era. The multiplication table thumb|right|180px|The Tsinghua Bamboo Slips, Chinese Warring States era decimal multiplication table of 305 BC This is the decimal multiplication table as we learn it in school: + Decimal multiplication table × 1 2 3 4 5 6 7 8 9 1 1 2 3 4 5 6 7 8 9 2 2 4 6 8 10 12 14 16 18 3 3 6 9 12 15 18 21 24 27 4 4 8 12 16 20 24 28 32 36 5 5 10 15 20 25 30 35 40 45 6 6 12 18 24 30 36 42 48 54 7 7 14 21 28 35 42 49 56 63 8 8 16 24 32 40 48 56 64 72 9 9 18 27 36 45 54 63 72 81 But living in the computer age, the most likely thing is that we will soon start using an electronic calculator and in adulthood we will do little multiplication by hand. Often many of us, even mathematicians, do not have the multiplication table "fresh" in memory and this can be bad news for you: if you want to multiply (and divide) efficiently with an abacus, you necessarily have to refresh the multiplication table in your memory! Using the multiplication table we can solve the multiplication problem in the form: i.e. we only have to retrieve the partial products: from the multiplication table and add them in the correct places, as we do with paper and pencil 47 ×23 ----- 21 12 (×10) 14 (×10) + 8 (×100) ----- 1081 This is absolutely parallel to the multiplication method that we are going to follow with the abacus. The modern multiplication method When we multiply two numbers and , we call both numbers factors and product to the result , but it is also common to call multiplicand to one of the factors and multiplier to the other. Nevertheless, when it comes to multiplying with the abacus: Multiplicand It is the number that we are going to manipulate on the abacus and that will guide us to obtain the partial products in an orderly manner and to align them correctly for their addition in the correct positions. Multiplier It is the factor that we are not going to manipulate on the abacus. in fact it is not mandatory to even enter it (but it is convenient). It will usually be the factor of the two with the fewest digits. Multiplication arrangement There are two ways of entering both factors in the abacus that can be considered practically equivalent; Each of them has its own advantages and disadvantages. The same can be said of the division that we will study in the next chapter. Feel free to experiment with both arrangements. Traditional Chinese arrangement The multiplicand is located to the left of the abacus and the multiplier far enough away from the multiplicand. At least as many columns as digits have the multiplier plus two or better three must be left free. Example + Traditional Chinese arrangementmultiplicand: 345, multiplier: 67ABCDEFGHIJKLM 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px34567 or in table form: + Traditional Chinese arrangementmultiplicand: 345, multiplier: 67 Abacus Comment ABCDEFGHIJKLM 345        67 Traditional Japanese arrangement This is the reverse way. The multiplier is on the left and the multiplicand on the right, leaving at least two empty columns in between. We need to have at least as many free columns to the right of the multiplier as the number of digits in the multiplier plus one. + Traditional Japanese arrangementmultiplicand: 345, multiplier: 67ABCDEFGHIJKLM 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px67345 or in table form: + Traditional Japanese arrangementmultiplicand: 345, multiplier: 67 Abacus Comment ABCDEFGHIJKLM 67  345 This is the form that has been most popular in JapanThe Japanese Abacus: its Use and Theory Kojima Takashi 1954 Charles E. Tuttle Co., Inc. Tokyo 978-0-8048-0278-9 https://archive.org/details/japaneseabacus00taka and also ended up being imported to China. It is also the form that we will use in this book. 1-digit × 1-digit multiplication Of course this is so trivial that we don't need an abacus, but it serves to introduce the rest of the examples. Suppose we have to multiply , let's take 7 as a multiplier, 8 as multiplicand and adopt the Japanese arrangement just explained; that is, we start from: + 7×8=56ABCDEFG 30px 30px 30px 30px 30px 30px 30px 30px 30px78 + 7×8=56 Abacus Comment ABCDEFG 7  8 Setting up the problem    +56 Multiply D×A and add it to EF 7  856 7  856 Clear D 7   56 Result: 7×8=56 + Final resultABCDEFG 30px 30px 30px 30px 30px 30px 30px 30px 30px756 Yes, you are right; it is you who did the multiplication, not the abacus. In the following example, the abacus begins to show its usefulness. 1-digit × 2-digit multiplication Let us multiply , the multiplicand will be 83. + 7×83ABCDEFGH 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px7 83 + 7×83 Abacus Comment ABCDEFGH 7  83 Setting up the problem     +21 Multiply E by A and add it to FG 7  8321 7  8321 Clear E 7  8 21    +56 Multiply D by A and add it to EF 7  8581 7  8581 Clear D 7   581 Result: 7×83=581 + 7×83 ResultABCDEFGH 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px7 581 At least, the abacus has served to add the two partial products in FG and EF. 2-digit × 2-digit multiplication Now, let us multiply . + 79×83ABCDEFGHI 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px7983 + Caption text Abacus Comment ABCDEFGHI 79  83 Setting up the problem      +21 Multiply F by A and add it to GH       +27 Multiply F by B and add it to HI 79  83237 79  83237 Clear F 79  8 237     +56 Multiply E by A and add it to FG      +72 Multiply E by B and add it to GH 79  86557 79  86557 Clear E 79   6557 Result: 79×83=6557 + 79×83 ResultABCDEFGHI 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px796557 Multi-digit multiplication Generalizing what was seen in the previous examples: For each digit of the multiplicand, starting from the right Multiply the current digit of the multiplicand by the digits of the multiplier (from left to right), adding the first partial product to the two columns to the right of the current digit of the multiplicand, and the rest of the products by successively shifting one column to right every time. Clear the current multiplicand digit. Let us see it with the following example: : + 799×835ABCDEFGHIJKL 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px799835 + 799×835 Abacus Comment ABCDEFGHIJKL 799  835 Setting up the problem        +35 Multiply H by A and add it to IJ         +45 Multiply H by B and add it to JK          +45 Multiply H by C and add it to KL 799  8353995 799  8353995 Clear H 799  83 3995       +21 Multiply G by A and add it to HI        +27 Multiply G by B and add it to IJ         +27 Multiply G by C and add it to JK 799  8327965 799  8327965 Clear G 799  8 27965      +56 Multiply F by A and add it to GH       +72 Multiply F by B and add it to HI        +72 Multiply F by C and add it to IJ 799  8667165 799  8667165 Clear F 799   667165 Result: 799×835=667165 + 799×835 ResultABCDEFGHIJKL 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px799667165 Embedded zeroes + 3075×2707ABCDEFGHIJKLMNO 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px30752707 + 3075×2707 Abacus Comment ABCDEFGHIJKLMNO 3075  2707 Set up problem          +21 Multiply JxA, add it to KL            +49 Multiply JxC, add it to MN!             +35 Multiply JxD, add it to NO 3075  270721525 3075  270721525 Clear J 3075  27  21525        +21 Multiply HxA, add it to IJ          +49 Multiply HxC, add it to KL!           +35 Multiply HxD, add it to LM 3075  272174025 3075  272174025 Clear H 3075  2 2174025       +06 Multiply GxA, add it to HI         +14 Multiply GxC, add it to JK!          +10 Multiply GxD, add it to KL 3075  2 8324025 3075  2 8324025 Clear G 3075    8324025 Result: 3075×2707=8324025 + 3075×2707 ResultABCDEFGHIJKLMNO 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px30758324025 The unit rod and decimals Please, review all the examples seen so far and check that, in all cases: The column of the units of the product is located columns to the right of the column of the units of the multiplicand; where is the number of digits of the multiplier. This is a general rule for the multiplication of natural numbers following the modern method of multiplication that we are studying. It is convenient to keep this rule in mind since the product could have zeros at the end, as in the case ; which could confuse you. For instance + 32×1625ABCDEFGHIJKL 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px321625u In the above diagram, the unit rod of multiplicand is column H (signaled with a white dot on the bar). After multiplication, the abacus shows: + 32×1625 Result: 52000ABCDEFGHIJKL 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px3252000u You need to know that the unit rod of the result is rods to the right of H (i.e. in J) to correctly read the result 52000. We can extend this rule to decimal numbers: The column of the units of the product is located columns to the right of the column of the units of the multiplicand; where is the number of digits of the multiplier to the left of its decimal point (which could be negative!). The following table shows the values for some multipliers: +Multipliern32.723.2710.32700.00327 -2 Let us multiply ; The unit rod of multiplicand is F. + 0.0032×16.25ABCDEFGHIJKL 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px321625 and for the multiplier , we have + 0.0032×16.25 Result: 0.052ABCDEFGHIJKL 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px 30px320052 so that the unit rod of the product is rods to the right of F, i.e. one rod to its left (E) and the result must be read as . References External resources Exercise sheets Uitti, Stephen Soroban Sheets (Multiplication) Soroban https://www.uitti.net/stephen/soroban/multiplication.pl September 21, 2021 The generator Practicing the soroban https://www.sorobanexam.org/ September 21, 2021 Further reading The Japanese Abacus: its Use and Theory Kojima Takashi Multiplication 1954 Charles E. Tuttle Co., Inc. Tokyo 978-0-8048-0278-9 https://archive.org/details/japaneseabacus00taka/page/52/mode/2up Multiplication 2004 Heffelfinger, Totton 算盤 Abacus: Mystery of the Bead http://totton.idirect.com/abacus/pages.htm#Multiplication1 September 25, 2021 https://web.archive.org/web/20210629003824/http://totton.idirect.com/abacus/pages.htm June 29, 2021 Modern division Addition and subtraction