Cube Root by Hand

Date: 01/23/98 at 15:48:37
From: Adrian Van Cauwenberghe
Subject: Cube root

Dear Sir, 

Many years ago I learned to calculate the square root of a number 
(without calculators or computers, just pencil and paper).

An example will tell you how (square root of 222784)

 22.27.84   I   472
 16         I ---------------------------
-----       I
   62.7     I    87    I    942
   60.9     I   X 7    I   X  2
   -------- I   -------------------------
      188.4 I   609    I   1884 
      188.4 I        
      ------
          0

I suppose this is relatively familiar to you.

Later we learned a comparable but of course more complicated way to 
calculate the cube root of a number. I forgot almost completely how. 
What I remember is that we had to split the number up in groups of 
three figures instead of two (example: 274625  becomes 274.625) and 
that we had to take the first group (here 274) and find the 
approximate cube root of that number (here 6 since 6x6x6=216 and 
7x7x7=343). But then I am lost.
 
Can you help me further?
Many thanks in anticipation.

Adrian ( Bruges - Belgium )

Date: 03/03/98 at 16:18:48
From: Doctor Rob
Subject: Re: Cube root

To find a cube root by the "longhand" method, we proceed very much as
we do to find a square root by hand. I intersperse numbered steps 
with an example. We will find the cube root of 113 to two decimal 
places.

1. Draw a cube root symbol, or radical, with the number whose root you
   are seeking underneath.  Start with the decimal point and mark off
   digits in both directions in groups of three. Put a decimal point
   above the radical, and directly above the other decimal point.

                            .
                       3/-----------
                      \/ 113.000 000

2. Start with the first group of 1, 2, or 3 digits. Find the largest
   cube of a single-digit integer less than it. Write the single digit
   above the radical, and its cube under the first group. Draw a line
   under that cube, and subtract it from the first group.

                           4.
                       3/-----------
                      \/ 113.000 000
                          64
                         -------
                          49

3. Bring down the next group below the last line drawn. This forms 
   the current remainder. Draw a vertical line to the left of the 
   resulting number, and to the left of that line put three hundred 
   times the square of the number above the radical, a plus sign, 
   thirty times the number above the radical, a multiplication sign, 
   an underscore character, another plus sign, another underscore 
   character, the exponent 2, an equals sign, and some blank space for 
   the answer.

                           4.
                       3/-----------
                      \/ 113.000 000
                          64
                         -------
    4800+120*_+_^2=???? | 49 000
                   
4. Pick the biggest digit D that would fit into both underscore
   places, and give a number such that D times it is less than the 
   current remainder. Put it above the radical above the last group of 
   digits brought down, and put it in each of the blanks where the 
   underscore characters are. Compute the number given by the 
   expression, and put it after the equals sign. Multiply D times that 
   number, and put that below the current remainder, draw a horizontal 
   line below that, and subtract, to give a new current remainder.

                           4. 8
                       3/-----------
                      \/ 113.000 000
                          64
                         -------
    4800+120*8+8^2=5824 | 49 000
                          46 592
                          ----------
                           2 408

5. If the current answer, above the radical, has the desired accuracy,
   stop. Otherwise, go back to step 3.

Step 3:
                           4. 8
                       3/-----------
                      \/ 113.000 000
                          64
                         -------
    4800+120*8+8^2=5824 | 49 000
                          46 592
                          ----------
691200+1440*_+_^2=?????? | 2 408 000

Step 4:
                          4 . 8   3
                       3/-----------
                      \/ 113.000 000
                          64
                        -------
    4800+120*8+8^2=5824 | 49 000
                          46 592
                          ----------
691200+1440*3+3^2=695529 | 2 408 000
                           2 086 587
                           ---------
                             321 413

Step 5:  Stop.

Thus the cube root of 113 to two decimal places is 4.83.  Checking,
4.83^3 = 112.6786, and 4.84^3 = 113.3799, so the answer is correct.

-Doctor Rob,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/   

Date: 03/14/98 at 07:24:04
From: Adrien Van Cauwenberghe
Subject: Cube root

Dear Sirs,

I want to thank you very much for your help. I would like to 
give you a few comments on the answer.

1. The "determination" of D seems to me a bit arbitrary: try 9, if 9 
is incorrect try 8, if 8 is incorrect, try 7; if 7 is not correct, 
try 6, and so on. In a book that I found recently in a shop for 
second-hand books (edition 1910!), I found large theoretical 
considerations on the square root and the cube root of a number. To 
find the cube root of a number up to step 2 they use a similar method 
as the one you proposed. To determine D they say (for your given 
example): divide 490|00 by 3*4^2 = 3*16 = 48. Result 10, which of 
course is too large. Trying 9: still too large, but as we know 8 is 
correct. In most cases, however, this way of determination of D gives 
the correct D immediately.

2. Another way to control whether D is correct or not is to calculate
(4.8)^3 and to subtract it from the given number 113.000. If the 
result is positive then D is correct. Here also (113.000 - 110.592) 
= 2.408; however, this result will also be positive if D is taken too 
small: nothing is perfect in this world.

Date: 03/18/98 at 12:36:59
From: Doctor Rob
Subject: Re: Cube root

Usually the next digit D is the quotient gotten by dividing the number 
to the left of the vertical line (the divisor), *ignoring D* (or 
setting D = 0), into the current remainder. In this case, you would 
divide 4920 into 49000, and the quotient digit D is guessed to be 9.  
As you progress to later and later stages in the algorithm, this 
method is more and more accurate. This problem is exactly analogous to 
the one encountered when doing long division: what is the next digit?  
You have to estimate the right answer, and adjust if the remainder is 
too big or too small.

-Doctor Rob,  The Math Forum
 http://mathforum.org/dr.math/