Talk:Cube root

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[edit] Infinitely nested cube roots

What is the point of this section ? This is an encyclopedia, not a bunch of random stuff from high school math problems. I think this part should be deleted.

[edit] Related image

Plot of y = ∛x by Mathematica.
Plot of y = ∛x by Mathematica.

User:Shen has replaced the old picture with a newer one (that is probably better). Ah well... I don't like images (especially ones I made) to disappear into oblivion. :-) — Ambush Commander(Talk) 17:03, 29 January 2006 (UTC)

Aha, yeah... Sorry about the confusion. I'll get round to telling people if I do it again (really) Shen 21:29, 29 January 2006 (UTC)
Uh... no need to be sorry. :-D If it's a better picture, it's a better picture. — Ambush Commander(Talk) 21:34, 29 January 2006 (UTC)
Plot of y = ∛x,the new image.
Plot of y = ∛x,the new image.

Both the new and the old image suffer from the same problem, the numbers on the axes are too small. Would be nice if the picture is redrawn with biggerfont sizes. Oleg Alexandrov (talk) 03:29, 30 January 2006 (UTC)

Well, part of the problem is the resizing of the plots. — Ambush Commander(Talk) 03:36, 30 January 2006 (UTC)
Good point. The first picture does not scale the axes, and is more natural. Unless the new picture fixes that and the font sizes, I would suggest using the old one. Oleg Alexandrov (talk) 21:07, 30 January 2006 (UTC)
I gave it some bigger fonts (increased the size of everything by 2). It should look a lot nicer! Shen 21:11, 30 January 2006 (UTC)
Thanks, looks good! Oleg Alexandrov (talk) 02:23, 31 January 2006 (UTC)

[edit] The Cube Root and Root-Solving History

It is certainly disturbing to realize that the extremely simple arithmetical method for approximating the Cube Root (and Nth. root) shown at: Arithmetical High-Order root-solving methods do not appear in any book on numbers since ancient times up to now. —Preceding unsigned comment added by 200.109.112.12 (talk • contribs)

How does this have any relevance to the article at hand? — Edward Z. Yang(Talk) 02:54, 21 October 2006 (UTC)

I found a cube root method in a book on Vedic Mathematics. There is an algebraic digit schedule. The method is simple enough to be done by mental math. If one has already memorized the cubes in the first twelve digits then you can obtain the first and last digit on sight of the perfect cube. If you have the digit schedule then you can calculate the middle digits. Example: four Cube Root digits: HJKL ==> Start with the units digit in the root, L, as the units digit in the cube root of L3 in the number.

A Hindu woman traveled to San Jose, California about 20 years ago and beat a computer at the 23rd root of a 50?-digit number in a demonstration of mental math and Vedic algorithms. Apparently she had the general formula. More later. Larry R. Holmgren 09:40, 1 March 2007 (UTC)

This method is only useful on TV shows. Most numbers have no integer cube root so the algorithm cannot be applied. Thus it is only a curiosity. It cannot be extended to work for any N-th root (N odd) as claimed above: for N=11 there are 8 possible endings for eleventh roots of ...7056: namely 0006, 1256, 2506, 3756, 5006, 6256, 7506, 8756. --Alpertron 12:59, 30 March 2007 (UTC)


It is for sure that the issue on the extremely simple and general high-order arithmetical root-solving methods shown in the aforementioned webpage: http://mipagina.cantv.net/arithmetic/rmdef.htm are of so much relevance to the Cube-Root's article and its history, indeed. The history of root-solving is plenty of Tricks and Patches (as for example: those Vedic tricks you have brought to light), however the true is that no natural arithmetical methods habe been developed --since Babylonian times up to now-- by far similar to these new general high-order methods shown in the aforementioned webpage, and frankly this is so worrying. Author: D. Gomez.


[edit] CUBE ROOTS

Cube Roots of exact cubes, mainly by inspection and argumentation, page 316-317. Place-value analysis of a three-digit cube root, cba, page 318. Analytic sorting of the expansion into place-values, the three-digit cube root (cba) schedule of subtractions for cube roots of a given number, page 319, 321. First divide out any factors of 8, or 64, or 512, from the number to remove ambiguity in digit selection, page 323. Knowing the cubes of the first 10 counting numbers enables one to match the units digit of the given number (the perfect cube) to the cube root. A similar analysis is done to infer the next digit, b, by seeing the right-most digit in the cube. If the remainders grow too large, go back and increase the quotient digit. The following step's deductions will be larger.

[edit] Digit schedule of subtractions for a three-digit cube root, cba

The units place is determined by a3. 
(Subtract it to eliminate the units digit.) 
The tens place is determined by 3a2b. 
The hundreds place is formed by 3ab2 + 3a2b. 
The thousands place is formed by b3 + 3b2c. 
The ten-thousands place is given by 3ac2 + 3b2c. 
The hundred-thousands place is contributed by 3bc2. 
The millions place is constituted by c3.
Example one:

What is the cube root of the perfect cube —The preceding unsigned comment was added by Larry R. Holmgren (talk • contribs) 00:56, 6 March 2007 (UTC).

[edit] Four-digit schedule of subtractions for cube roots HJKL, pages 321, 323, 325.

To determine the units place, L, examine the units digit in the cube. 
Deduct L3.  
The tens digit is K.  Evaluate and analyse 3L2K. 
Deduct 3L2K. 
For J, the hundreds digit, evaluate and analyse 3L2J + 3LK2. 
Deduct 3L2J + 3LK2. 
For H, the thousands digit, evaluate and analyse 
3L2H + 6LKJ + K3. 
And 3L2H + 6LKJ + K3 is the portion to deduct.

Four-digit cube root, HJKL Method

1. Subtract (L3) to eliminate the units digit. We have L by the final digit pattern in the cube.

2. Subtract (3L2K) to eliminate the penultimate digit to find the cube root digit K.

3. Subtract (3L2J+3LK2) to eliminate the pre-penultimate digit to find the cube root digit J.

4. Subtract (3L2H+6LKJ+K3) to eliminate the prior digit to find the cube root digit H.

Example: Find an incompletely given perfect cube. 
Find the cube root of a 12-digit perfect cube 
whose last four digits are 6741.  
Thus, on sight, we known that 
the number of cube root digits = 12/3 = 4; Root digits H = ? and L = 1.  
Framework:  ___,___,___6,741. = (HJKL)3  
                                                                                                            
For L : L=1, Therefore L3 = 1.  Subtract it.           ………6741                                            
For K:  Deduct 3L2K = 3K (ends in 4).                  ……-……1                               
The 3rd multiple of 8 ends in 4. K=8. Deduction:3K=24. ……-…24  
For J:  Deduct 3L2J+3LK2 = 3J+192 (ends in 5).        ………65x  
So, 3J ends in 3, J=1. Deduction = 3+192=195.       ……-…195  
For H: Deduct 3L2H+6LKJ+K3 = 3H+48+512 = 3H+560.      ………7x  
(ends in 7). So, 3H ends in 7.                     …-……587 
The 9th multiple of 3 ends in 7. So H=9.              ………x
(No need to deduct 27+560=587.                                  
Therefore, (HJKL)3 = (9181)3 = ___,___,__6,741.

[edit] General cube roots

General cube roots can be calculated using a long division framework and rightward, column-wise division. The sequence of digits for a three digit schedule, for cube roots, abc, given the number (100a+10b+1c)3 + R, page 327:

The millions place is determined by a3.  
The hundred-thousands place is determined by 3a2b.  
The ten-thousands place is formed by 3ab2 + 3a2b.  
The one-thousands place is formed by b3 + 3b2c.  
The hundreds place is given by 3ac2 + 3b2c.  
The tens place is contributed by 3bc2.  
The units place is constituted by c3.

The Dividends, Quotients, and Remainders: The divisor in cube roots is 3c3. The units digit, c, is obtained by inspection of the first group of digits in the cube after the digits of the given number are grouped in threes. Since we know (or have a reference list of) the first ten cubes we may see the nearest cube below the first group at sight.

The first D, Q, and R are available on sight.  
Use the following adjustments to the gross dividend at each step.  
From the second dividend no deduction is made. 
From the third dividend subtract 3ab2. Note the change.  
From the fourth dividend subtract 6abc + b3. Note the change.  
From the fifth dividend subtract 3ac2 + 3b2c. 
From the sixth dividend subtract 3bc2. 
From the seventh dividend subtract c3.

The divisor should not be small, try two groups (4, 5, or 6 digits) as the first group, page 330. The four-digit schedule for cube roots, abcd, given the cube (1000a+100b+10c+1d)3, or an incomplete cube with a remainder. pages 337-338.

[edit] The analytic digit schedule:

First digit, billions = a3 
Second digit, hundred-millions = 3a2b. 
Third digit, ten-millions place = 3ab2 + 3a2c 
Fourth digit, one-millions place = 6abc + b3 + 3a2d 
Fifth digit, hundred-thousands place = 6abd + 3ac2 + 3b2c 
Sixth digit, ten-thousands place = 6acd + 3bc2 + 3b2d 
Seventh digit, one-thousands place = 6bcd + 3ad2 + c3 
Eighth digit, hundreds place = 3bd2 + 3c2d 
Ninth digit, tens place = 3cd2 
Tenth digit, ones place = d3

The divisor in cube roots is 3a3. The deduction schedule for the long division, rightward, column-wise method for cube roots.

For the first dividend, Q1 and R1 are obtained by inspection, a3. 
From the second dividend Q2 and R2 by simple division, no other subtraction! 
From the third gross dividend subtract 3a2b.  
From the fourth gross dividend subtract 6abc + b3  
From the fifth gross dividend subtract 6abd + 3ac2 + 3b2c  
From the sixth gross dividend subtract 6acd + 3bc2 + 3b2d 
From the seventh gross dividend subtract 6bcd + 3ad2 + c3 
From the eighth gross dividend subtract 3bd2 + 3c2d 
From the ninth gross dividend subtract 3cd2 
From the tenth gross dividend subtract d3

An alternative method is to cut the four-digit schedule to three-digits by equating d=0, page 338, and substituting (c+d) for c, page 340.

No Vedic sutra mentioned for cube roots by the author of Vedic Mathematics: Sixteen Simple Mathematical Formulae from the Vedas, by Swami Sankaracarya (1880-1960), Motilal Banarsidass Indological Publishers and Booksellers, Varnasi, India, 1965; reprinted in Delhi, India, 1975, 1978. 367 pages.

Examples and several cube root digit schedules are given. 5th root, 7th root, and nth root general formula only hinted at by the binomial expansion of that power. Page 340, Vedic Mathematics, 1965, 1978. Larry R. Holmgren 20:03, 3 March 2007 (UTC)

How much space is appropriate for examples? Larry R. Holmgren 19:02, 4 March 2007 (UTC)

[edit] "The cube root" vs. "A cube root"

The introductory paragraph begins: "In mathematics, the cube root of a number, denoted \sqrt[3]{x} or x1/3, is the number a such that a3 = x. All real numbers have exactly one real cube root and 2 complex roots, and all nonzero complex numbers have 3 distinct complex cube roots."


As there are generally three cube roots, shouldn't we replace the with a? The new text would read: "In mathematics, a cube root of a number, denoted \sqrt[3]{x} or x1/3, is a number a such that a3 = x. All real numbers have exactly one real cube root and 2 complex roots, and all nonzero complex numbers have 3 distinct complex cube roots." DRE 18:01, 20 February 2007 (UTC)

Sounds good. I've went ahead and made the change. — Edward Z. Yang(Talk) 00:18, 4 March 2007 (UTC)
Technically, the complex numbers are the set of numbers in the form a + bi where a and b are real numbers. So, all real numbers are also complex numbers, if b = 0. Thus, perhaps it would be to say "exactly one real cube root and 2 nonreal roots..." StatisticsMan 20:03, 21 April 2007 (UTC)
Except when a is also zero. In this case all three roots are real. --Alpertron 13:18, 30 April 2007 (UTC)
Well, distinct cube roots, anyway. — Edward Z. Yang(Talk) 19:13, 30 April 2007 (UTC)

[Query by Joe user] 18:30, 22 October 2007 (UTC)

I have another question about the introductory paragraph. How many cube-roots does zero have?

Also, some explanation of the 3D plot would be appreciated. The vertical axis is clearly the imaginary part of the complex cube-root. Which of the two horizontal axes is the real number whose cube-root is being evaluated? Obviously the remaining axis is the one that is real (part of the complex) cube-root.

[End query by Joe user] 18:30, 22 October 2007 (UTC)