Trachtenberg Speed Math Online

MULTIPLICATION BY FIVE, BY SIX, AND BY SEVEN

2008-10-11T16:52:00.000-07:00

All these multiplications-5, 6, and 7 – make use of the idea of
“half” a digit. We put “half” in quotation marks because it is a simplified half. We
take half the easy way, by throwing away fractions if there are any. To take “half”
of 5, we say 2. It is really 21/2, but we won't use the fractions. So “half” of 3 is 1,and “half” of 1 is zero. Of course “half” of 4 is still 2, and so with all even
numbers.

This step is to be done instantly. We do not say to our-selves
“half of 4 is 2” or anything like that. We look at 4 and say 2. Try doing that now,
on these digits:

2, 6, 4, 5, 8, 7, 2, 9, 4, 3, 0, 7, 6, 8,5, 9, 3, 6, 1

The odd digits, 1, 3, 5, 7, and 9, have this special feature of
dropping the fractions. The even digits, 0, 2, 4, 6, and 8, give the usual result
anyway.

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2008-10-11T09:23:00.000-07:00

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About Jakow Trachtenberg

2008-10-11T00:10:00.000-07:00

The extract below and the rest of this blog is from the book, 'Trachtenberg Speed System of Basic Mathematics'-translated and adapted by Anne Cutler and Rudolph McShane. © Copyright 1960 by Anne Cutler. Sharing to this book to the new generation for free

The teacher called on a nine-year-old boy who marched firmly to the blackboard upon which was a list of numbers a yard long. Standing tiptoe to reach the top, he arrived at the total with what seemed the speed of light.

A small girl with beribboned braids was asked to find the solution of 735352314 times 11. She came up with the correct answer-8088875454-in less time than you can say the multiplication table. A thin, studious-looking boy wearing silver-rimmed spectacles was told to multiply 5132437201 times 452736502785. He blitzed through the problem, computing the answer-2323641669144374104785-in seventy seconds.

The class was one where the Trachtenberg system of mathematics is taught. What made the exhibition of mathematical wizardry more amazing was that these were children who had repeatedly failed in arithmetic until, in desperation, their parents sent them to learn this method.

The late Jakow Trachtenberg, founder of the Mathematical Institute in Zurich, Switzerland, and originator of the startling new system of arithmetic, was of firm opinion that everyone comes into the world with 'phenomenal calculation possibilities'.

The Trachtenberg method is not only speedy but also simple. Once one has mastered the rules, lightning calculation is as easy as reading a story. It looks like magic, but the rules are based on sound logic.

Trachtenberg, a brilliant engineer with an ingenious mind, originated his system of simplified mathematics while spending years in Hitler's concentration camps as a political prisoner. Conceived in tragedy and amidst brutal hardships, this striking work cannot be separated from the life of its originator for it is quite possible that had Professor Trachtenberg's life run a more tranquil course he might never have conceived the system which has eliminated the drudgery so often associated with arithmetic.

The life of Trachtenberg is as fascinating and astounding as his brilliant mathematical system which many experts believe will eventually revolutionize the teaching of arithmetic in schools throughout the world.

A Russian, born in Odessa, 17 June 1888, Jakow Trachtenberg early showed his genius. Graduating with highest honours from the famous Berginstitut (Mining Engineering Institute) of St. Petersburg, he entered the world-renowned Obuschoff shipyards as a student-engineer. While still in his early twenties, he was named Chief Engineer. In those Czar-ruled days, there were ambitious plans to create a superlative navy and 11,000 men were under Trachtenberg's supervision.

Though he headed the Obuschoff shipyards, Trachtenberg was a dedicated pacifist. At the outbreak of World War 1 he organized the Society of Good Samaritans which trained Russian students to care for the wounded -a work which received special recognition from the Czar.

The murder of the imperial family in 1918 put an end to the Russian dream of a grandiose navy. It also ended Trachtenberg's personal hope of a happy, peaceful life. Hating brutality and violence, Trachtenberg, became their victim.

As the revolutionaries swept right across Russia, Trachtenberg spoke out fearlessly against the savagery and lawlessness. The criticism imperiled his life. Early in 1919, he learned that he was slated to be murdered. Dressed as a peasant, walking at night, hiding out through the day, he made his way into Germany.

Berlin, with its beautiful wide streets, its cold, sparkling weather, reminded him of St. Petersburg and became his home. In a tiny room at an unpretentious address, he started life anew and made friends with the bitter, disillusioned young intellectuals of the post-war era. He became their leader. As the editor of a magazine, he often spoke for this group when he urged Germany towards a future of peace.

Trachtenberg married a beautiful woman of the aristocracy. His reputation grew as he wrote a number of critical works on Russia and compiled the first reference book on Russian industry. He was looked upon as Europe's foremost expert on Russian affairs. His inventive mind set itself another task. He devised a method of teaching foreign languages, which is still used in many German schools.

The upheaval of his early years seemed to have been left behind. But with the coming of Hitler, Trachtenberg's life once more took on the familiar pattern of strife. Courageously, he spoke out against fascism. Trachtenberg's reputation was such that Hitler at first chose to overlook his attacks. But when Trachtenberg's accusations grew more pointed, Hitler marked him for oblivion.

In 1934, knowing if he remained in Germany he would be liquidated, Trachtenberg once more fled for his life. Accompanied by his wife, he escaped to Vienna where he became editor of an international scientific periodical.

While the world was preparing for war, Trachtenberg, to further the cause of peace, wrote Das Friedensministerium (The Ministry of Peace), a widely read work, which brought him the plaudits of such statesmen as Roosevelt, Masaryk, and Van Zeeland.

But all over the world peace was dying. The Germans marched on Austria. Trachtenberg's name headed Hitler's most-wanted list. He was seized and thrown into prison.

He managed to escape to Yugoslavia where he and his wife, Countess Alice, lived like hunted animals, rarely venturing out during the day, making no friends or acquaintances. But his freedom was brief. He was awakened one night by the heavy pounding of fists on the door-the Gestapo was calling. Hitler's men had caught up with him.

He was shipped in a cattle car to a concentration camp-one noted for its brutality. The slightest variance from the rules resulted in outrageous forms of punishment. Daily the ranks of the prison were decimated by the ruthlessly random selection of victims for the ovens.

To keep his sanity, Trachtenberg moved into a world of his own-a world of logic and order. While his body daily grew more emaciated, and all about him was pestilence, death, and destruction, his mind refused to accept defeat and followed paths of numbers that, at his bidding, performed miraculous feats.

He did not have books, paper, pen, or pencil. But his mind was equal to the challenge. Mathematics, he believed, was the key to precise thinking. In happier times, he had found it an excellent recreational outlet. In a world gone mad, the calm logic of numbers were like old friends. His mind, arranging and rearranging, found new ways of manipulating them.

He visualized gigantic numbers to be added and he set himself the task of totaling them. And since no one can remember thousands of numbers, he invented a fool-proof method that would make it possible for even a child to add thousands of numbers together without making a mistake-without, in fact, ever adding higher than eleven.

During his long years in the living hell of the concentration camp, every spare moment was spent on his simplified system of mathematics, devising shortcuts for everything from multiplication to algebra. The corruption and misery, the cries from clammy cells and torture chambers, the stench of ovens, the atrocities, and the constant threat of death, faded as he doggedly computed mathematical combinations-reckoning rules, proving and proving again, then starting over again to make the system even simpler.

The hardships acted as a spur to his genius. Lacking paper, he scribbled his theories on bits of wrapping paper, old envelopes, the back of carefully saved German worksheets. Because even these bits of paper were at a premium he worked everything in his head, putting down only the finished theories.

Today those using the Trachtenberg method find it so easy that all problems can be worked in the head and only the answers put down.

Shortly after Easter in 1944, Trachtenberg learned he was to be executed-the decree had come from above and was no longer speculation or foreboding. Trachtenberg faced the fact, and then lost himself in his own world. Calmly he went on working-juggling equations, reckoning formulae, working out rules. He had to get his system finished! To a fellow prisoner, he entrusted his work. He had been in prison almost seven years.

Madame Trachtenberg, who had never been far from the concentration camp, learned of the death sentence. Parting with the last of her jewels and money, she bribed and coerced and managed to have him transferred surreptitiously to another camp just before the sentence was to be carried out.

He was sent to Leipzig which had been heavily bombed and everything was in a state of chaos. There was no food, no heat, and no facilities. In his dismal barracks, the rising tiers of hard bunks were so crowded it was impossible to lie down. Morale had never been so low. Often the dead lay for days, the inmates too weak to dig graves and the guards too panicky to enforce orders.

In the confusion, a determined man, willing to risk his life, could escape to freedom. Trachtenberg took the chance and crawled through the double wire fences in the dead of the night. He joined his wife, who had devoted all her time, strength, and money in trying to help him. But Trachtenberg had no passport, nor papers of any kind. He was a stateless citizen, subject to arrest.

Once again, he was taken to custody. A high official, who knew of Trachtenberg's work, sent him to a labour camp in Trieste. Here he was put to work breaking rock, but the weather was milder and the guards not so harsh.

Quietly, Madame Trachtenberg bribed guards to take messages to her husband and an escape was again arranged. On a starless night early in 1945, Trachtenberg climbed a wire fence and crawled though the long grass as guards stationed in watchtowers shot at him. It was his last escape. Madame Trachtenberg waited for him at the appointed place. Together they made their way across the border to Switzerland.

In a Swiss camp for refugees he gathered his strength. His hair had turned white and his body was feeble, but the years of uncertainty and despair had left him undefeated. His eyes, a clear, calm blue, were still valiant. His eagerness and warmth, his intense will to live, were still part of him.

As he slowly convalesced, he perfected his mathematical system which had kept him from losing his mind, which had enabled him to endure the inquisition of the Gestapo, and which now enabled him to start a new life.

It was to children, whom Trachtenberg loved, that he first taught his new and simplified way of doing arithmetic. He had always believed that everyone was born rich in talents. Now he set out to prove it. Deliberately he chose children who were doing poorly in their schoolwork.

These children used to failure, shy and withdrawn; or the other extreme, boastful and unmanageable. All of them were unhappy, badly adjusted youngsters.

The children's response to the new, easy way of doing arithmetic was immediate. They found it delightfully like a game. The feeling of accomplishment soon made them lose their unhappy traits.

Equally important were the by-products the pupils garnered while learning the new system. As these youngsters became proficient in handling numbers, they achieved a poise and assurance that transformed their personalities and they began to spurt ahead in all their studies. The feeling of accomplishment leads to greater effort and success.

To prove the point that anyone can learn to do problems quickly and easily, Trachtenberg successfully taught the system to a ten-year-old-presumably retarded-child. Not only did the child learn to compute, but his IQ rating was raised. Since all problems are worked in the head, he acquired excellent memory habits and his ability to concentrate was increased.

In 1950, Trachtenberg founded the Mathematical Institute in Zurich, the only school of its kind. In the low, spreading building that houses the school, classes are held daily. Children ranging in age from seven to eighteen make up the daytime enrolment. But the evening classes are attended by hundreds of enthusiastic men and women who have experienced the drudgery of learning arithmetic in the traditional manner. With a lifetime of boners back of them, they delight in the simplicity of the new method. Proudly, they display their newfound mathematical brilliance. It is probably the only school in the world where students-both day and evening-arrive a good half-hour before class is called to order.

What is the Trachtenberg system? What can it do for you?

The Trachtenberg system is based on procedures radically different from the conventional methods with which we are familiar. There are no multiplication tables, no division. To learn the system you need only be able to count. The method is based on a series of keys that must be memorized. Once you have learned them, arithmetic becomes delightfully easy because you will be able to 'read' your numbers.

The important benefits of the system are greater ease, greater speed, and greater accuracy. Educators have found that the Trachtenberg system, which has a unique theory of checking by nines and elevens, gives an assurance of ninety-nine per cent accuracy-a phenomenal record.

The great practical value of this new system is that, unlike special devices and tricks invented in the past for special situations, it is a complete system. Much easier than conventional arithmetic, the Trachtenberg system makes it possible for people with no aptitude for mathematics to achieve the spectacular results that we expect of a mathematical genius. Known as the 'shorthand of mathematics', it is applicable to the most intricate problems.

But perhaps the greatest boon of this new and revolutionary system is that it awakens new interest in mathematics, gives confidence to the student, and offers a challenge that spurs him on to mastering the subject that is today rated as 'most hated' in our schools.

Professor Trachtenberg believed the reason most of us have difficulties juggling figures is not that arithmetic is hard to comprehend, but because of the outmoded system by which we are taught-an opinion which is borne out by many educators.

A year-long survey conducted by the Educational Testing Service of Princeton University revealed that arithmetic is one of the poorest-taught subjects in our schools and noted that there has been little or no progress in teaching arithmetic in this country in the past century; that the important developments that have taken place in mathematical science since the seventeenth century have not filtered down into our grade and high schools. And the results, says the report, are devastating. In one engineering school, seventy-two per cent of the students were found so inadequate mathematically that they had to take a review of high-school mathematics before they could qualify for the regular freshman course.

This is particularly tragic today when there is an urgent need for trained scientists and technicians with a firm grasp of mathematics. The revulsion to mathematics which educators say plays such a strong role in determining the careers of young people, begins at the level of the elementary and secondary schools. It is at this stage that the would-be engineers and scientists of tomorrow run afoul of the 'most hated subject'. From then on, arithmetic is left out of their curriculum whenever possible.

The Trachtenberg system which has been thoroughly tested in the Switzerland, starts at the real beginning-in basic arithmetic where the student first encounters difficulties and begins to acquire an emotional attitude that will cripple him in all his mathematical work.

The ability to do basic arithmetic with the spectacular ease which the Trachtenberg system imparts, erases the fear and timidity that so hinder the student when faced with the impressive symbolism, the absoluteness of mathematical rigor. It is this emotional roadblock, not inability to learn, that is the real reason why so many students hate mathematics, say the experts.

That short cuts make arithmetic easier to grasp and more palatable was proved conclusively by the armed forces during the last war. Bombardiers and navigators taking refresher courses in higher mathematics were able to cram several years' work into a few months when it had been simplified.

In Zurich, medical students, architects, and engineers find that the Trachtenberg system of simplified mathematics enables them to pass the strict examinations necessary to complete their training. One of Switzerland's leading architects was enabled to continue with his chosen career only after attending the Institute where he learned the Trachtenberg method.

In Switzerland when people speak of the Mathematical Institute, they refer to it as the 'School for Genius'.

In an impressive test recently held in Zurich, students of the Trachtenberg system were pitted against mechanical brains. For a full hour the examiners called out the problems-intricate division, huge additions, complicated squaring and root findings, enormous multiplications.

As the machines began their clattering replies, the teenage students quickly put down the answers without any intermediate steps.

The students beat the machines!

The students who proved as accurate as and speedier than the machines were not geniuses. It was the system-short and direct-which gave them their speed.

But it is not only in specialized professions that knowledge of arithmetic is necessary. Today, in normal everyday living, mathematics plays an increasingly vital role. This is particularly true in America where we live in a welter of numbers. Daily the average man and woman encounter situations that require the use of figures-credit transactions, the checking of monthly bills, bank-notes, stock market quotations, canasta and bridge and billiards scores, discount interest, lotteries, the counting of calories, foreign exchange, figuring the betting odds on a likely-looking steed in the fourth race, determining the chances of getting a flush or turning up a seven. And income taxes, among other blessings, have brought the need for simple arithmetic into every home.

The Trachtenberg system, once learned, can take the drudgery out of the arithmetic that is part of your daily stint.

The Swiss, noted for their business acumen, recognizing the brilliance and infallibility of the Trachtenberg system, today use it in all their banks, in most large business firms, and in their tax department. Mathematical experts believe that within the next decade the Trachtenberg system will have far-reaching an effect on education and science as the introduction of shorthand did on business.

MULTIPLICATION BY TWELVE

2008-10-10T21:19:00.001-07:00

To multiply any number by 12, you do this:

Double each number in turn
and add its neighbor.

This is the same as multiplying by 11 except that now we double the “number” before we add its “neighbor”. If we wish to multiply 413 by 12, it goes like this:

_________________________*
First step: __________0 4 1 3 x 12
_________________________6 _____double the right-hand figure
______________________________________and carry it down
______________________________________(There is no neighbor)

_______________________* *
Second step: _______0 4 1 3 x 12
_______________________5 6 _____double the 1 and add 3

_____________________* *
Third step: _________0 4 1 3 x 12
____________________ 9 5 6 ____double the 4, add the 1

____________________* *
Last step:_________ 0 4 1 3 x 12
____________________4 9 5 6 ____zero doubled is zero; add the 4

The answer is 4,956. If you go through it yourself you will find that
the calculation goes very fast and is very easy.

Try one yourself: 63,247 times 12. Write it out with the figures spaced
apart, and put each figure of the answer directly under the figure of the 63,247
that it came from. This is not only a good habit because of neatness, it also is
worth its weight in gold as a protection against error. In the particular case of
Trachtenberg multiplication, we mention it because it will identify the “number”
and the “neighbor”. The next blank space in the answer, where the next figure of
the answer will go, is directly below the “number” (in this example the figure that
you must double). The figure to its right is the “neighbor” that must be added.
The example works out in this way.

________*
0 6 3 2 4 7 X 12
_______*4 double 7, 14; carry 1

______ * *
0 6 3 2 4 7 X 12
____ *6*4 double 4, plus 7, plus 1 is 16; carry 1

_____* *
0 6 3 2 4 7 X 12
___ _9*6*4 double 2, plus 4, plus 1 is 9

until you end up with:

0__6__3__2__4__7 _X 12
7_*5__8__9_*6_*4

MULTIPLICATION BY ELEVEN

2008-10-10T21:16:00.000-07:00

1. The last number of the multiplicand (number multiplied) is put
down as the right-hand figure of the answer.
2. Each successive number of the multiplicand is added to it neighbor
at the right.
3. The first number of the multiplicand becomes the left-hand number
of the answer. This is the last step.

In the Trachtenberg system you put down the answer one figure at a
time, right to left, just as you do in the system you now use. Take an easy
example, 633 times 11:

633 X 11
answer will
be here

The answer will appear under the 633, one figure at a time, from right
to left, as we apply the rules. This will be our form for setting up the work from
now on. The asterisk above the multiplicand of our example will quickly identity
the numbers being used in each step of our calculation. Let us apply the rules:

The answer will appear under the 633, one figure at a time, from right
to left, as we apply the rules. This will be our form for setting up the work from
now on. The asterisk above the multiplicand of our example will quickly identity
the numbers being used in each step of our calculation. Let us apply the rules:

First Rule
Put down the last figure of 633 as the
right-hand figure of the answer:

6 3 3 X 11
___3
Second Rule
Each successive figure of 633 is added
to its right-hand neighbor.

6 3 3 X 11
9 6 3 plus 3 is 6:

Apply the rule again, 6 plus 3 is 9:

6 3 3 X 11
9 6 3

Third Rule
The first figure of 633, the 6, becomes
the left-hand figure of the answer:

0 6 3 3 X 11
6 9 6 3

The answer is 6,963.

Longer numbers are handled in the same way. The second rule, “each
successive number of the multiplicand is added to its neighbor at the right,” was
used twice in the example above; in longer numbers it may be used many times.
Take the case of 721,324 times 11:

7 2 1 3 2 4 X 11

First Rule
The last figure of
721,324 is put down
as the right-hand figure of the answer:
________*
7 2 1 3 2 4 X 11
________4

Second Rule
Each successive figure
of 721,324 is added to its
right-hand neighbor:

______ * *
7 2 1 3 2 4 X 11
____ 6 4______2 plus 4 is 6

_____* *
7 2 1 3 2 4 X 11
___ 5 6 4______3 plus 2 is 5

___ * *
7 2 1 3 2 4 X 11
___4 5 6 4______1 plus 3 is 4

__* *
7 2 1 3 2 4 X 11
3 4 5 6 4_____2 plus 1 is 3

* *
7 2 1 3 2 4 X 11
9 3 4 5 6 4_____7 plus 2 is 9

Third Rule
The first figure of
721, 324 becomes the _
the left-hand figure of
the answer:

__*
0 7 2 1 3 2 4 X 11
7 9 3 4 5 6 4

The answers is 7,934,564.

As you see, each figure of the long number is used twice. One it is
used as a “number,” and then, at the next step, it is used as a “neighbor”. In the
example just above, the figure 1 (in the multiplicand) was a “number” when it
contributed to the 3 of the answer at the next step:

_____________ * * ____________________* *
__________7 2 1 3 2 4 X 11 ___________7 2 1 3 2 4 X 11
_____________4______________________ 3

Instead of the three rules, we can use just one if we apply it in
natural, common-sense manner, the one being “add the neighbor”. We must first
write a zero in front of the given number, or at least imagine a zero there. Then
we apply the idea of adding the neighbor to every figure of the given number in
turn:
___*
6 3 3 X 11
___3 – there is no neighbor, so we add nothing!

6 3 3 X 11
9 6 3 – as we did before

* *
0 6 3 3 X 11
6 9 6 3 – zero plus 6 is 6

This example shows why we need the zero in front of the multiplicand.
It is to remind us not to stop too soon. With out the zero in front, we might have neglected to write the last 6, and we might then have thought that the answer
was only 963. The answer is longer than the given number by one digit, and the
zero in front takes care of that.

Try one yourself: 441,362 times 11. Write it in the proper form:

_____________________ 0 4 4 1 3 6 2 X 11

If you started with the 2, which is the right place to start, and worked
back to the left, adding the neighbor each time, you must have ended with the
right answer: 4,854,982.

Sometimes you will add a number and its neighbor and get something
in two figures, like 5 and 8 giving 13. In that case you write the 3 and “carry” the
1, as you are accustomed to doing anyway. But you will find that in the
Trachtenberg method you will never need to carry large numbers. If there is
anything to carry it will be only a 1, or in the later cases perhaps a 2. This makes a tremendous difference when we are doing complicated problems.

It is sufficient to put a dot for the carried 1, or a double dot for the
rarer 2:

0 1 7 5 4 X 11
1 9*2 9 4 - the *2 is 12; from 7 plus 6

Try this one yourself: 715,624 times 11. Write out

0 7 1 5 6 2 4 X 11

There is a 1 to carry under the 5 of the long number.

The correct answer to this problem is 7,871,864.

In the very special case of long numbers beginning with 9 followed by
another large figure, say 8, as in 98,834 we may get 10 at the last step. For
example:

0_0_9_8_8_3_4 X 11
1 0 *8 *7 *1 7 4

BASIC MULTIPLICATION

2008-10-10T20:48:00.000-07:00

The aims of the Trachtenberg system have been discussed in the
foreword. Now let us look at the method itself. The first item on the agenda is a
new way to do basic multiplication: we shall multiply without using any
memorized multiplication tables. Does this sound impossible? It is not only
possible, it is easy.

A word of explanation, though: we are not saying that we disapprove
of using tables. Most people know the tables pretty well: in fact, perfectly, except for a few doubtful spots. Eight times seven, or six times nine are a little uncertain to many of us, but the smaller numbers like four times five are at the command of everyone. We approve of using this hard-won knowledge. What we intend to do now is consolidate it. Later in this chapter we shall come back to this point. Now we wish to do some multiplying without using multiplication tables.

Let us look at the case of multiplying by eleven. For the sake of
convenience in explaining it, we first state the method in the form of rules: