Conference rusure::math

    Doing cube roots in their heads?  Ya gotta be kidding.  Ask 'em what is
    the cube root of 2.
    
    I think it is all gimmick.  Yea, there are some tricks one can do with
    arithmatics, but what is the point other than getting good grades?  
    Calculators should be able to take care of that.  You can teach a third
    grader how to take derivatives on polynomials, but what will be the
    point with learning some artificial rules?
    
    Eugene

    Tricks?  Sure, I used to "Amaze My Friends" in college by taking the
    Cube Root of 15-digit numbers, and generally faster than their
    calculators could (really, I'm not kidding).  I could describe the method, 
    but it only works for integer cube roots.
    
    Ther *are* mentalist tricks for real-valued (OK, of limited decimal
    precision) cube roots and the other things that you mentioned.  You
    probably know them as "Logarithim Tables".
    
    --  Barry

    re .2,
    
    Naw Barry, you ain't gonna be faster than the calculator.  It is just
    that it takes a lot of time to punch in the numbers.  A more fair test
    will be to have someone key in the number first and when you are
    shown a 15 digit number the other guy press the cuberoot key.
    
    I remember reading Richard Feymann's _Surely you must be joking Mr.
    Feymann_.  In it he played a lot of those tricks on arithematic.  I
    still remember the part about a Japanese trying to sell him an
    abacuss. It is amazing what one can do with the numbers, but that 
    requires an understanding of a lot of series and sequences and other 
    miracle stuff. More headache than their worth especially in this 
    electronic age.
    
    Eugene
                                                                  

    
    Thanks for the replies.
    
    OK. Let's forget about the cube roots for the moment. These people
    could also add 3-three digit numbers in their head in the same time it
    would take me to input them into the calculator. And also multiplied 
    2-three digit numbers in the same time.
    	I agree a calculator could do the same but I don't always have my
    calculator with me. I'd like to be able to mentally add up prices of
    groceries (before going to the register and finding out I don't have
    enough cash). Or numbers a salesman throws out quickly. Of course, 
    I could do this given enough time or after rechecking the written
    numbers. But I think it would be useful to have this ability.

    re .4,
    
    If you find it useful, then learn it.  The key word is "useful".  It is
    like getting used to an operating system.  If you don't USE it for more
    than a few weeks, you forget about it.  I used UNIX in college, but
    have forgotten most of it by now.  Of course, it will be easier to pick
    it up again than to learn it from the start, but still I don't know
    much about it if you put me on a UNIX machine right now, and this is
    the point.
    
    Eugene

    Books that I have seen on this subject basically contain two things:

	1) Lots of stuff to memorize.  Some lighting calculators memorize
	   log tables but that is beyond most people.  What is in the
	   range for many people is memorizing the standard arithmetic
	   tables in some other base which is relatively easy to convert
	   to and from decimal.  An obvious one is base 100.  A little
	   easier to memorize but a bit harder to use is base 25.

	2) Lots of special cases, which can be recognized quickly with
	   practice, things like to multiply anything (say x) by any number
	   ending in a 9 (say 10y+9), multiply x by y+1 add a 0 and
	   subtract x.  One which I have actually found useful is applying
	   binomial expansion to square a two digit (or low-3 digit)
	   number: ab^2 = (a^2)00 + b^2 + (2*a*b)0 if you'll excuse my
	   ad hoc notation.

    My advice: buy a calculator watch -- mine will do general exponentials,
    logs, trig functions and convert in and out of hex, octal and binary. 
    That's more than the mental math courses teach, and, translating
    learning time to money, is very much cheaper unless you have a
    phenomenal memory.

				    Topher

    Re .3:
    
>    Naw Barry, you ain't gonna be faster than the calculator.  It is just
>    that it takes a lot of time to punch in the numbers.  A more fair test
    
    Only half true, Eugene.  It's true that that is one of the standard
    tricks of the practicing lightning calculator (and I used lots of
    them), but there was more to than that.
    
    As I recall, 15 digits was a *REAL* stretch for me and I only ever did
    it on one occasion (and I had to practice like the devil for it).  15
    digits took me so long that I could never really beat a calculator. 
    (Of course only a special calculator could handle 15 digits in those
    days).  I could not even hope to do it today.
    
    The other end of the scale was different story however.  To this day I
    can *still* do 6 digit cube roots in my head, with only about a minute
    of prepartion.  And in my college days I could do them *very* fast. 
    In fact, I had a number of friends who would try to beat me by punching
    up the Cube on the calculator, and would then show me the number at
    the same that they hit the cube-root key, and I never failed to beat
    it.  Of course, I did have some help.  The calculators of those days
    (late seventies) for most non-Arithmetic functions (Roots, Exponents, 
    Etc.)  had a noticable delay of 2 to 4 seconds before returning their
    answer.
    
    And yes, a lot of people just assumed that I had memorized all of the
    numbers from 1 to 100 cubed.  But it is much simpler (IMHO) than that,
    I am much to lazy and absent-minded to retain that large a table for
    this long.
    
    --  Barry

    There are some unusual methods that are documented in a (probably
    out of print) book called "The Trachtenberg Speed System of
    Mathematics" by Ya'akov Trachtenberg.  It was old when I first
    read it in 1973.  Libraries would probably have it.

    Ya'akov Trachtenberg had the misfortune of being interred by the
    Nazis in a concentration camp.  To keep himself sane, he developed
    methods for doing arithmetic in his head.  It includes methods for
    addition/subtraction, multiplication, division, and square roots.

    It is possible, for instance, to multiply two numbers of many
    digits each in your head, writing down only one digit at a time
    (right to left).

    It's been quite some time since I've seen the book, but I
    remember that the rules are more than just the typical short cuts. 
    Some of them are truly counter-intuitive.

    You do not have to memorize large multiplication tables (1 - 9 is
    sufficient).  You do not have to memorize addition tables (again,
    0 - 9 is sufficient).  There are no mnemonics involved.

    For instance, when adding a large column of numbers, Trachtenberg
    describes a method of casting out elevens (yes, elevens) instead
    of nines.  It is one of the few algorithms that I remember.  It's
    a bit involved, but if you're interested, send me MAIL and I'll
    post the method. 

    Multiplying a number by 11 or 12 is a piece of cake.  For 11:

    	Beginning with the least significant digit, add it to it's
    	neighbor on the right, writing down the result as the next
    	digit of the answer.

    		1883473928347 x 11 =
    					7 + (nothing) = 7
    					4 + 7		1
    					3 + 4 + carry   8
    					8 + 3           1
    					2 + 8 + carry   1
    					9 + 2 + carry   2
    					3 + 9 + carry   3
    					7 + 3 + carry   1
    					4 + 7 + carry   2
    					3 + 4 + carry   8
    					8 + 3           1
    					8 + 8 + carry   7
    					1 + 8 + carry   0
    					    1 + carry   2

    		Result: 20718213211817

    For 12, same process, except that you double the digit before you
    add to the right neighbor.

    Other algorithms for 2 - 9 involve taking a digit (or half of a
    digit, dropping the remainder) and adding it to its neighbor (or
    half its neighbor).    

    I saw a book by Devi Shekuntala (sp?), a modern calculating prodigy.
    It had lots of tricks for doing mental arithmetic.

    I was given an abacus as a gift when I was a kid and the book of
    instructions was a lot like these books for mental calculations -
    lots of memorized procedures for doing things.  It wasn't very
    appealing.

    Speaking of computational aids of yesteryear, anyone remember the
    Kurta calculators?  A pepper mill kind of gadget that was popular
    with auto rallyists.

    - Jim

    RE .9 
    
    I remember the Kurta (from my auto rallying days)  It always
    amazed me how the really proficient could crank out answers
    with it.  
    
    I remember they came in two sizes (more digits).  It seemed 
    to have been designed with the rallyist in mind (a running 
    accumulation of successive additions for each .10 or .01 mile).
    
    -Joe

Hey, great.
Lets hear some more detail on Shekuntula, Kuntas, etc.

I can add a bit more on Trachtenberg.

I searched this conference about 6 months ago but found nothing on it.
Nor was there even anything on Arithmetic.  I found this a bit surprising.
Gauss rated the subject quite highly in his day. Could it really have fallen
this far from grace?  Its been a while since I was a practising Mathie,
and I thought the world had perhaps moved on from such mundane things!

[Also, nothing on computer arithmetic.  I seem to remember that the evolution
of combinatorial configurations of logic gates designed to produce faster
arithmetical results was a mind blowing story.
Has this path also petered out ... ?]


Anyway, on to Trachtenberg ...

 

I picked up a paperback copy of "The Trachtenberg Speed System of Basic
Mathematics" a few months back.  It is a translation by Ann Cutler and Rudolph
McShane.  Trachtenberg's first name listed as "Jakow".  He was a Russian Jew.


According to the inside cover of the book, the first British Edition was
published in February 1962 by Souvenir Press.  It was reprinted several times
in the sixties, then re-issued in 1984.  The copy which I have is a paperback
re-issue dated 1989.  ISBN is 0 285 62916 6.  Cost was 5.95 pounds.


I picked it up 'cos my kids are at that age when 'sums' forms a major part of
the evening's conversation.  I needed something new to give me the edge for a
change!



I would have to say that I am still undecided as to the merits of the system's
tricks, especially  w.r.t. allowing speedy calculations.
I share the opinion of the previous contributors on this topic.
My times to date haven't yet exceeded the ones using the  standard techniques
taught to me a long time ago!

However, I think that this is more due to a mental reluctance to learn the
(somewhat non-intuitive) rules. Being able to forget the existing ones would
also help.  As with everything in this world, what you get out depends on what
you put in!


The technique is apparently taught quite extensively in Switzerland (where
Trachtenberg founded the Mathematical Institute) and also in the USA.
I have never seen any sign of it in (north) British education.


One of the aims of the system is to allow ALL children to become masters of
calculation, and claims some very spectacular successes.



The method of adding (casting out 11's) means that you never have
to count higher than 9+9 at any stage, so only a relatively small 'table'
needs to be memorised. This helps new learners get to the state where they feel
that they are on top of the figures that much quicker. Early confidence keeps
them going through the stickier bits of the method learning activity.


Another feature in the adding method (extra-Trachtenberg, but incorporated)
is that of "casting out 9's" to check the calculation.  This is done on each
of the original rows or columns of figures, the working and the final result.
Again, I had never come across this.

This is something which would be useful even if you had an electronic
calculator to hand, when you could make a mistake typing in the figures.
The claim is that it guarantees 99% accuracy.
[I am still trying to work out how.]
As is pointed out in the book, most people were never taught to check the
result.  If we did, it would probably be by laboriously performing the whole
calculation several times, accepting the answer when it came out the
same on several successive attempts!



Here is a very small example, adding together four three digit numbers.


                                            digit sum
					    for each row
		
num 1		9	6	7      -------> 4

num 2		7 .	5 .	8 .    -------> 2

num 3		5	7	9 .    -------> 3

num 4		8 .	6 .	5      -------> 1

              ---------------------           ----
                                                1    Digit sum of rows


Wrk 1           7       2	7      -------> 7

Wrk 11          2       2       2      -------> 6

                                                6

                                              ----
					        1   Digit sum of working


         --------------------------
ans      3      1       6       9      -------> 1    Digit sum of answer



First step is to add the columns.
It doesn't matter in which order, or whether up or down.
I chose to do it from the top down.

Add the digits together, but each time they exceed 11, through away 11, hang on
to the remainder and mark a dot beside the row.
Then continue with this remainder, adding it to the digit on the next row down.

The first column goes ....

 9 + 7 = 16 = 5 + 11   think  "5 ."
 5 + 5 = 10
10 + 8 = 18 = 7 + 11   think  "7 ."


This final 7 is written in the "wrk 1" row.
The dots in the column are added and their sum written in the "wrk 11" row.


[Once practised, you don't do the sums or write the dots each time.
 You just learn "7 + 8 = 4 .", and keep a running tally of the dots.]



Once all the columns have been done, the "wrk 1" and "wrk 11" rows are combined
by doing a dog-legged summation.

Start at the right hand end and simply add the two rows together.
	7 + 2 = 9

Move left to the next column.  
Add the two digits togther as before, but also add in the digit in the "wrk 11"
row to the right.
	2 + 2 + 2 = 6

Repeat this to the left, also going in to the fourth (and possibly fifth?)
columns, pretending they have zeros written in them.




The lines are then checked by calculating the digit sums.
Just add the digits together, combining them if the resulting number ever gets
in to mutiple digits.  (9's or sums to 9 can be thrown away.)

So, for example, the number 967 generates a check sum of 4.
	9 + 6 = 15 ----> 1 + 5 = 6   (or we could have thrown away the 9)
	6 + 7 = 13 ----> 1 + 3 = 4 


The three checks should now all come to the same thing.
	a) The check of the (sum of the checks of the rows)
	b) The check of the working ("wrk 1" + "wrk 11" + "wrk 11")
	c) The check of the answer.

If not, you got the sum wrong somewhere.
If OK, you are 99% likely of having got it correct!



The book does open up all sorts of questions in one's mind about the techniques
for calculation learnt already.

	What are their strengths/weaknesses?

	Why do they work?

	How on earth did anyone ever manage to come up with that algorithm
	in the first place?!


For example, how was the calculating method for finding square roots (as
displayed in the following example for 239) devised?

                     1   5  .   4
                 --------------------------
               1 \   2  39  .  00
               1  \  1
               ----\-----
               2 5  \  1  39
                 5   \ 1  25
               -------\-----
               3 0  4  \  14  00
                    4   \ 12  16
               ----------\---------
               3 0  8     \ 1  84


[Its funny how books on the origins of such things seem to be hard to come by.]
Lots on the causal structure of non-positive-definite metric paracompact
orientable differentiable manifolds, but none on how to work out cube roots!]



I found Trachtenberg's method for finding square roots to be the least
different from the calculating techniques which I knew.  It was similar but
subtley different.

Its major improvement is that the numbers involved at each stage remain
small and do not increase in size as the calculation proceeds.
However, once again, it is harder to learn initially.

[Neither method seems particularly intuitive.
Thought: Is this connected in any way with the fact that the square root
operation doesn't have a natural geometrical analogue?]