Traditional Abacus and Bead Arithmetic/Roots/Cube root
Theory
editLet be the number of which we want to obtain the cube root ; Let's consider its decimal expansion, for example: . Let's separate its digits into groups of three around the decimal point in the following way
or, in other words, let's define the sequence of integers
and let's build the sequence recursively from
and let be the integer part of the cube root of
i.e. is the largest integer whose cube is not greater than . Finally, let us call remainders to the differences
For our example we have:
0 | 0 | 0 | 0 | |
1 | 456 | 456 | 7 | 113 |
2 | 789 | 456789 | 77 | 256 |
3 | 012 | 456789012 | 770 | 256012 |
4 | 300 | 456789012300 | 7701 | 78119199 |
5 | 000 | 456789012300000 | 77014 | 6949021256 |
⋯ |
Let's see that, by construction, grows as (three more digits in each step), in fact the sequence , i.e. 0, 400, 456, 456.789, 456.789012, etc. tends to ( ). By comparison, , as the integer part of the cube root of , grows only as (one digit more each step). As is the largest integer whose square is not greater than we have as above but
by definition of , or
multiplying by
but as grows only as , the second term tends to zero as .
and so that we have
For other numbers, the above factors are: and , where is the number of three-digit groups to the left of the decimal point, negative if it is followed by 000 groups (ex. for , for , etc.).
This is the basis for traditional manual cube root methods.
Procedure
editWe start with .
First digit
edit1 | 1 |
2 | 8 |
3 | 27 |
4 | 64 |
5 | 125 |
6 | 216 |
7 | 343 |
8 | 512 |
9 | 729 |
For . It is trivial to find such that its square does not exceed through the use of the following table of cubes that can be easily retained in memory. In the case of the example it is
Rest of digits
editFor , we have as defined above and we try to build in the way:
where is a one-digit integer ranging from 0 to 9. To obtain we have to choose the greatest digit from 0 to 9 such that:
or
if we write . Expanding the binomial we have
or
The left side of the above expression may be seen as simply the previous remainder with the next three-digits group appended to it. If we evaluate the term on the right for each value of and compare with the left term we have:
0 | 0 | ≤ 113789 | |
1 | 14911 | ≤ 113789 | |
2 | 30248 | ≤ 113789 | |
3 | 46017 | ≤ 113789 | |
4 | 62224 | ≤ 113789 | |
5 | 78875 | ≤ 113789 | |
6 | 95976 | ≤ 113789 | |
7 | 113533 | ≤ 113789 | ⬅ |
8 | 131552 | > 113789 | |
9 | 150039 | > 113789 |
and it is clear that the next figure of our root is a 7 but, how can we proceed in the general case without having to systematically explore every possibility ( )?
Here Knott^{[1]} distinguishes two different approaches:
- Preparing the divisor
- Preparing the dividend
Preparing the divisor
editThis correspond with the above expression
And it is the strategy usually used with paper and pencil and can also be implemented, of course, on the abacus. In the above expression, if we see the left part as dividend and the parentese as divisor, is the first digit of the division
but since we don't know b yet, we approximate it using only the main part of the divisor
This gives us a guess as to what the value of might be, but we need:
- Verify that the value thus obtained is correct, or, where appropriate, correct it up or down as needed.
- Obtain the next remainder to prepare the calculation of the next digit of the root.
You can see an example in Tone nikki blog^{[2]}, see also Modern approaches below.
Preparing the dividend
editStarting again with
we prepare the dividend by dividing (the next three-digits group appended to the previous remainder) by
As usual, we don't know and we can't evaluate the parentheses on the right, but we can get a clue about by approximating the parentheses by its main part and use it as a trial divisor.
so that
After that, we need again:
- Verify that the value thus obtained is correct, or, where appropriate, correct it up or down as needed.
- Obtain the next remainder to prepare for the obtention of the next digit of the root by evaluating .
Please note that:
- Divisor 3 is involved in the prepared dividend and this leads to non-finite decimal fractions.
- The division by not only worsens the above, but also makes the prepared dividend specific to the current step, since the value of evolves with the calculation of the different figures of the result.
This did not occur in the calculation of square roots and, as a consequence, the process of obtaining cube roots is much more complicated and requires a complex cycle of preparation-restoration of the dividend that, following Knott, can be represented by the following scheme:
- a) Divide by .
- b) Divide by 3.
- c) Obtain as the first digit of the division of the above by .
- d) Subtract (Equivalent to subtracting and in ).
- e) Multiply by 3.
- f) Multiply by .
- g) Subtract .
In our example ( ), using traditional division and traditional division arrangement (like Knott does), working the two first digits:
Abacus | Comment |
---|---|
ABCDEFG | |
456789 | Enter number aligning first group with B |
-343 | -7^3=343 |
113789 | First remainder |
7113789 | Enter 7 in A as first root digit and append second group |
7113789 | a) Divide B-F by 7^{1} |
7162554 | b) Divide B-F by 3^{2} |
7541835 | c) Divide B by A (one digit) |
7751835 | d) Subtract 7*7=49 from CD |
77 2835 | e) Multiply CDEF by 3. Add 3✕283 to CDEFG |
77 854 | f) Multiply CDEF by 7. Add 7✕85 to CDEFG |
77 599 | |
-343 | g) Subtract 7^3=343 to CDEFG |
77 256 | New remainder |
... | Root obtained so far: 7.7 |
Notes:
- ^1 a) It is unnecessary to extend the division by 7 beyond the current three-digit group. The 4 in G is a division remainder meaning 4/7.
- ^2 b) The same can be said of division by 3. It is carried out up to column F and the remainder (1) is temporarily added to column G. The value (5) in said column is a strange hybrid meaning 1/3 and 4/7 . It does not matter, it will be reabsorbed in steps e) and f).
Modern approaches
editMembers of the Soroban & Abacus Group modified the technique described by Knott to adapt it to modern soroban use^{[3]}. The result is allegedly faster at the expense of being less compact and requiring an abacus with more rods to store intermediate data. The simplicity of having the result directly substituting the radicand is also lost.
You can also find a compilation of modern methods for both square and cube roots in Tone Nikki (とね日記)^{[2]} by a Japanese blogger (Author's name does not appear to be available).
Examples of cube roots
editThe following examples are all worked using traditional division and traditional division arrangement. Components of the dividend preparation-restoration cycle are labelled with a), b), etc as detailed above.
Cube root of 157464
editAbacus | Comment |
---|---|
ABCDEFG | Cube root of 157464 |
157464 | Enter number aligning first group with B |
-125 | Subtract 5^3=125 from BCD |
32464 | First remainder: 32 |
5 32464 | Enter 5 in A as first root digit and append second group |
5 32464 | a) Divide C-F by 5 (G will be the division remainder) |
5 64924 | b) Divide C-F by 3 |
5216404 | c) Divide B by 5 |
5416404 | d) Subtract 4x4=16 from CD |
54 404 | e) Multiply 40x3 in EFG (adding to remainder in G) |
54 124 | f) Multiply 12x5 in EFG |
54 64 | g) Subtract 4^3=64 from FG |
54 | Remainder 0; Done! Root is 54 |
Clearly, if the remainder is zero and there are no more (not null) groups to add, the number is a perfect cube and we are done. Root is 54.
Cube root of 830584
editAbacus | Comment |
---|---|
ABCDEFG | Cube root of 830584 |
830584 | Enter number aligning first group with B |
-729 | Subtract 9^3=729 from BCD |
101584 | 101: first remainder |
9101584 | Enter 9 in A as first root digit and append second group |
9101584 | a) Divide C-F by 9 (G will be the division remainder) |
9112871 | b) Divide C-F by 3 |
9376232 | c) Divide B by 9 (A) |
9416232 | d) Subtract 4x4=16 from CD |
94 232 | e) Multiply 23x3 in EFG (adding to remainder in G) |
94 71 | f) Multiply 07x9 in EFG |
94 64 | g) Subtract 4^3= 64 from FG |
94 | Remainder 0; Done! Root is 94 |
Root is 94.
Cube root of 666
editAbacus | Comment |
---|---|
ABCDEFG | Cube root of 666 |
666 | Enter 666 in BCD |
+ | (Unit rod) |
-512 | Subtract 8^3=512 from BCD |
154 | First remainder |
8154 | Enter 8 in A as the first root digit |
8154000 | Append 000 as new group |
8154000 | a) Divide B-F by 8 (A) |
8192500 | b) Divide B-F by 3 |
8641662 | c) Divide B by 8 (A) |
8781662 | d) Subtract BxB=49 from CD |
8732662 | e) Multiply C-F by 3 in C-G |
87 9800 | f) Multiply C-F by 8 (A) in C-G |
87 7840 | g) Subtract B^3=343 from EFG |
87 7497 | Root so far 8.7, Remainder 7.497 |
Now we continue using Abbreviated operations. We need to divide the remainder (7497) by three times the square of the current root ( )
Abacus | Comment |
---|---|
ABCDEFGHIJKLM | |
87 7497 | |
87 7497------ | Squaring 87 |
+49 | 7^2 |
+112 | 2*7*8 |
+64 | 8^2 |
87 7497 7569 | multiplying by 3 (adding double) |
+14 | |
+10 | |
+12 | |
+18 | |
87 7497 22707 | dividing 7497/22707, two digits |
... | |
8733 | Root 8.733 (Compare to: ) |
Cube root of 237176659 (three digits)
editAbacus | Comment |
---|---|
ABCDEFGHIJ | Cube root of 237176659 |
237176659 | Enter number aligning first group with B |
-216 | Subtract 6^3=216 from BCD |
21176659 | 21: first remainder |
21176659 | Enter 6 in A as first root digit and append second group |
6 21176659 | a) Divide B-F by 6 (A) |
6 35292659 | b) Divide B-F by 3 |
6117633659 | c) Divide B by 6 (A) |
6157633659 | d) Subtract BxB=1 from CD |
6156633659 | e) Multiply C-F by 3 in C-G |
6116992659 | f) Multiply C-F by 8 (A) in C-G |
6110196659 | g) Subtract B^3=343 from EFG |
6110195659 | Root so far 61, Remainder 10195 |
---------- | |
6110195659 | Append third group |
6110195659 | a) Divide C-H by 61 (AB) |
6116714158 | b) Divide C-H 3 |
6155713678 | c) Divide C by 61 (AB) |
6190813678 | d) Subtract CxC=81 from EF |
619 3678 | e) Multiply D-H by 3 in D-I |
619 1158 | f) Multiply D-H by 61 (AB) in D-J |
619 729 | g) Subtract C^3=729 from HIJ |
619 000 | Done, no remainder! |
---------- | Root is 619 |
Cube root of to eight digits
editThe first triplet 110 is between 64 and 125, so that the cube root of 110 591 is between 40 and 50. First root digit is 4
First digit:
Abacus | Comment |
---|---|
ABCDEFG | Cube root of 110591 |
110591 | Enter number aligning first group with B |
-64 | Subtract 6^3=216 from BCD |
46591 | 46: first remainder |
46591 | Enter 4 in A as first root digit and append second group |
4 46591 | OK 1st digit! |
Second digit:
Abacus | Comment |
---|---|
ABCDEFG | |
4 46591 | a) Divide B-F by 4 (A) |
4116473 | b) Divide B-F by 3 |
4388234 | c) Divide B by 4 (A) |
4868234 | d) Subtract BxB=64 from CD |
48 4234 | e) Multiply C-F by 3 in C-G |
48 1273 | f) Multiply C-F by 4 (A) in C-G |
48 511 | g) Cannot subtract 8^3=512 from EFG! Going back (See note at the end) |
48 511 | -f) Divide C-F by 4 (A) |
48 1273 | -e) Divide C-F by 3 |
48 4234 | -d) Add 8x8=64 in CD |
4868234 | -c) Revise down B |
-1 | |
+4 | |
47T8234 | d) Subtract BxB=49 from CD (T=10) |
4759234 | e) Multiply C-F by 3 in C-G |
4717773 | f) Multiply C-F by 4 (A) in C-G |
47 7111 | g) Subtract B^3=343 from EFG |
47 6768 | OK 2nd digit! Remainder 6768 |
Third digit:
Abacus | Comment |
---|---|
ABCDEFGHIJ | |
47 6768000 | Append 000 to previous remainder |
47 6768000 | a) Divide C-H by 47 (AB) |
4714400000 | b) Divide C-H 3 |
4748000000 | c) Divide C by 47 (AB) |
4795700000 | d) Subtract C^2=81 from EF |
4794890000 | e) Multiply D-H by 3 in D-I |
4792298300 | f) Multiply D-H by 47 (AB) in D-J |
479 689490 | g) Subtract C^3=729 from HIJ |
479 688761 | OK 3rd digit! Remainder 688761 |
Fourth digit:
Abacus | Comment |
---|---|
ABCDEFGHIJKLM | |
479 688761000 | Append 000 to previous remainder |
479 688761000 | a) Divide D-J by 479 |
4791437914194 | b) Divide D-J by 3 |
4794793046394 | c) Divide D by 479 1d |
4799482046394 | d) Subtract 9^2=81 from GH |
4799473946394 | e) Multiply E-J by 3 in E-K |
4799142184194 | f) Multiply E-J by 479 in E-M |
4799 68106330 | g) Subtract -D^3=729 from KLM |
4799 68105601 | Ok 4th digit! Remainder 68105601 |
Now we finish the calculation using abbreviated operations. We need to divide the remainder (68105601) by three times the square of the current root (4799). The first four digits of the result are appended after the ones already obtained; for instance:
Abacus | Comment |
---|---|
ABCDEFGHIJKLM | |
4799 68105601 | Divide E-M by 4799 |
479914191623 | Divide E-M by 4799 |
47992957204 | Divide E-M by 3 |
47999857 | Compare this to |
As we can see, we have obtained a result with 7 correct figures.
Note: We found above that with root 48 we could not subtract , or we had a negative remainder (-1). This might seem unfortunate since it forced us to undo part of the work and correct the new root figure downwards, but in practice what we find is a fortunate result: the small remainder (-1) tells us that 48 was a excellent approximation (by excess) to the root, opening a new way to solve the problem. In fact, what we have is:
or
where we can use
so that
compare to . We could have thus achieved great precision with little effort!
From elementary arithmetic to numerical analysis
editThe abacus is currently studied as a traditional art or as a means to develop numerical and cognitive skills in general, it is not expected that in the computer age it will be used as a calculator to solve real world problems. But if that were the case and you had to solve a large number of cube roots (something unusual) you might want to move from traditional methods or basic arithmetic to modern numerical analysis methods and try the Newton-Raphson method. You can find an adaptation of this method to the abacus in jccAbacus^{[4]}...
Appendix: Cubes of two digits numbers
edit+1 | +2 | +3 | +4 | +5 | +6 | +7 | +8 | +9 | |
---|---|---|---|---|---|---|---|---|---|
10 | 1331 | 1728 | 2197 | 2744 | 3375 | 4096 | 4913 | 5832 | 6859 |
20 | 9261 | 10648 | 12167 | 13824 | 15625 | 17576 | 19683 | 21952 | 24389 |
30 | 29791 | 32768 | 35937 | 39304 | 42875 | 46656 | 50653 | 54872 | 59319 |
40 | 68921 | 74088 | 79507 | 85184 | 91125 | 97336 | 103823 | 110592 | 117649 |
50 | 132651 | 140608 | 148877 | 157464 | 166375 | 175616 | 185193 | 195112 | 205379 |
60 | 226981 | 238328 | 250047 | 262144 | 274625 | 287496 | 300763 | 314432 | 328509 |
70 | 357911 | 373248 | 389017 | 405224 | 421875 | 438976 | 456533 | 474552 | 493039 |
80 | 531441 | 551368 | 571787 | 592704 | 614125 | 636056 | 658503 | 681472 | 704969 |
90 | 753571 | 778688 | 804357 | 830584 | 857375 | 884736 | 912673 | 941192 | 970299 |
This can help you practice two-digit cube roots.
Example:
References
edit- ↑ Knott, Cargill G. (1886), "The Abacus, in its Historic and Scientific Aspects", Transactions of the Asiatic Society of Japan, 14: 18–73
- ↑ ^{a} ^{b} Tone? (2017). "Square root and Cube root using Abacus". とね日記.
{{cite web}}
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ignored (|access-date=
suggested) (help) - ↑ Baggs, Shane; Heffelfinger, Totton (2011). "Cube Roots". 算盤 Abacus: Mystery of the Bead. Archived from the original on August 1, 2021.
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: Unknown parameter|accesdate=
ignored (|access-date=
suggested) (help) - ↑ Cabrera, Jesús (2021). "Newton's method for abacus; square, cubic and fifth roots". jccAbacus.
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