251:
259:
As you solve each digit of the answer you then subtract
Product Pairs (UT pairs) and also NT pairs (Number-Tens) from the Partial Dividend to find the next Partial Dividend. The Product Pairs are found between the digits of the answer so far and the divisor. If a subtraction results in a negative number you have to back up one digit and reduce that digit of the answer by one. With enough practice this method can be done in your head.
234:
134:. Without pencil or paper he developed a system of easily memorized operations that allows one to perform mental arithmetic computations very quickly. These methods have been successfully taught to children who had consistently failed at arithmetic. Not only did their mathematical ability improve but as they became more proficient in handling numbers their confidence in all areas of study improved.
96:
31:
258:
Division in the
Trachtenberg System is done much the same as in multiplication but with subtraction instead of addition. Splitting the dividend into smaller Partial Dividends, then dividing this Partial Dividend by only the left-most digit of the divisor will provide the answer one digit at a time.
305:
The 'halve' operation has a particular meaning to the
Trachtenberg system. It is intended to mean "half the digit, rounded down" but for speed reasons people following the Trachtenberg system are encouraged to make this halving process instantaneous. So instead of thinking "half of seven is three
276:
A method of adding columns of numbers and accurately checking the result without repeating the first operation. An intermediate sum, in the form of two rows of digits, is produced. The answer is obtained by taking the sum of the intermediate results with an L-shaped algorithm. As a final step, the
267:
In other words, 8 minus (2 from the answer times 3 from the divisor) minus (tens digit of 2 times 1) is the 2 placed on the
Working Row. Dropping down the next digit from the Dividend makes the number in the Working Row 22. Then 22 minus(units digit of 2 times 1) minus(tens digit of 2 times 6) is
264:
In the diagram at right begin by dropping the first number of the dividend down to the
Partial Dividend row. The left digit of the divisor, 3, goes into 8 twice so 2 is the first digit of the answer. Then move up and right from the 8 by subtracting the NT pair to get 2. Drop down the 2 from the
241:
Professor
Trachtenberg called this the 2 Finger Method. The calculations for finding the fourth digit from the example above are illustrated at right.The arrow from the right-most digit of the multiplier will always point to the digit of the multiplicand directly above the digit of the answer you
160:
The rest of this article presents some of the methods devised by
Trachtenberg. The most important algorithms are the ones for general multiplication, division and addition. Also, the system includes some specialized methods for multiplying by small numbers between 3 and 12 that is no more involved
142:
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
176:
The method for general multiplication is to achieve multiplication of a*b with low space complexity, i.e. as few temporary results as possible to be kept in memory. By solving the equation in columns instead of rows we can sum the intermediate steps without having to write them down for later
187:
Trachtenberg defined this algorithm with a kind of pairwise multiplication where 2 digits are multiplied by 1 digit, essentially only keeping the middle digit of the result. By performing the above algorithm with this pairwise multiplication, even fewer temporary results need to be held.
777:
The book's copyright is by Ann Cutler, a journalist in New York City at the time. The other person involved in the translation to
English, Rudolph McShane, is a mathematician who lived in New Orleans at the time of publication and also worked on restricted USA government projects
242:
wish to find, with the other arrows each pointing one digit to the right. If an arrow points to a space with no digit there is no calculation for that arrow. As you solve for each digit you will move each of the arrows over the multiplicand one digit to the left.
767:
by Jakow
Trachtenberg, A. Cutler (Translator), R. McShane (Translator), Rudolph Mcshane (Translator) was published by Doubleday and Company, Inc. Garden City, New York in 1960. The book contains specific algebraic explanations for each of the above operations.
277:
checking method that is advocated removes both the risk of repeating any original errors and allows the precise column in which an error occurs to be identified at once. It is based on a check (or digit) sums, such as
309:
And remember that whenever the rule calls for adding half of the neighbor, always add 5 if the current digit is odd. This makes up for dropping 0.5 in the next digit's calculation.
787:
774:
The algorithms/operations for multiplication etc. can be expressed in other more compact ways that the book doesn't specify, despite the chapter on algebraic description.
295:
The answer must be found one digit at a time starting at the least significant digit and moving left. The last calculation is on the leading zero of the multiplicand.
284:
For the procedure to be effective, the different operations used in each stage must be kept distinct, otherwise there is a risk of interference.
184:
Ordinary people can learn this algorithm and multiply 4 digit numbers in their heads, writing down the final result, with the last digit first.
161:
than doubling or halving a one digit number. The chapter on addition demonstrates an effective method of checking calculations known as
76:
Finished writing a draft article? Are you ready to request review of it by an experienced editor for possible inclusion in
Knowledge?
974:
820:
180:
In general, for each position n in the final result, we sum for all i = 0 to n (the right-most digit of each number is position 0):
182:
Units digit of the product b(digit at i) x a(digit at(n-i)) + Tens digit of the product b(digit at i) x a(digit at (n-i-1)).
990:
70:
306:
and a half, so three" it's suggested that one thinks "seven, three". This speeds up calculation considerably.
49:
987:
102:
926:
878:
830:
792:
292:
When performing any of these multiplication algorithms, the following "steps" should be applied.
131:
124:
116:
970:
816:
278:
162:
120:
376:
Prefix two zeros to the multiplicand because the multiplier 12 has two digits → 00316 × 12
951:
939:
903:
891:
855:
843:
44:
40:
17:
302:, i.e., the digit on its right. The rightmost digit's neighbor is the trailing zero.
250:
214:
To find the fourth digit of the answer, start at the fourth digit of the multiplicand:
202:
To find the second digit of the answer, start at the second digit of the multiplicand:
66:
52:. It serves as a testing spot and page development space for the user and is
690:(9 - 9) x 2 + Half of 2 (1) + 5 (since 9 is odd) + 1 (carried) = 7. Write 7.
233:
265:
dividend to get 22. Then move down by subtracting UT pairs to get 19.
143:
excellent memory habits and his ability to concentrate was increased.
128:
771:
Most of the information in this article is from the original book.
637:(9 - 3) + Half of 4 (2) + 5 (since 3 is odd) = 13. Write 3, carry 1.
471:(3 x 2) + Half of 0 (0) + 5 (since 3 is odd) = 11. Write 1, carry 1.
211:
The second digit of the answer is 8 and carry 1 to the third digit.
230:
The fourth digit of the answer is 6 and carry 2 to the next digit.
249:
232:
432:
3 + half of 4 (2) + 5 (since 3 is odd) = 10. Write 0, carry 1.
426:
7 + half of 0 (0) + 5 (since 7 is odd) = 12. Write 2, carry 1.
90:
25:
228:
7 + 3 + 2 + 4 + 5 + 4 = 25 + 1 carried from the third digit.
673:
For the leading zero, subtract 2 from half of the neighbor.
477:(5 x 2) + Half of 2 (1) + 5 (since 5 is odd) = 16. Write 6.
127:
in order to keep his mind occupied while being held in a
617:
For the leading 0, subtract 1 from half of the neighbor.
79:
59:
752:
Half of 7's neighbor is 2, + 5 (since 7 is odd), is 7.
746:
Half of 3's neighbor is 0, + 5 (since 3 is odd), is 5.
584:(9 - 4) x 2 + 5 + 1 (carried) = 16. Write 6, carry 1.
687:(10 - 2) x 2 + Half of 0 (0) = 16. Write 6, carry 1.
564:
For the leading zero, subtract 2 from the neighbour.
221:
The units digit of 8 x 4 + the tens digit of 8 x 5,
216:
The units digit of 9 x 3 + the tens digit of 9 x 4,
204:
The units digit of 9 x 5 + the tens digit of 9 x 6,
693:(9 - 4) x 2 + Half of 9 (4) = 14. Write 4, carry 1.
509:
For the leading zero, subtract 1 from the nieghbor.
474:(2 x 2) + Half of 3 (1) + 1 (carried) = 6. Write 6.
226:
The units digit of 7 x 5 + the tens digit of 7 x 6.
919:The Trachtenberg Speed System of Basic Mathematics
871:The Trachtenberg Speed System of Basic Mathematics
813:The Trachtenberg Speed System of Basic Mathematics
765:The Trachtenberg Speed System of Basic Mathematics
788:Swami Bharati Krishna Tirtha's Vedic mathematics
149:Trachtenberg Speed System of Basic Mathematics.
140:
729:Half of 2's neighbor, the trailing zero, is 0.
435:0 + half of 3 (1) + 1 (carried) = 2. Write 2.
696:Half of 4 (2) - 2 + 1 (carried) = 1. Write 1.
640:Half of 3 (1) - 1 + 1 (carried) = 1. Write 1.
429:4 + half of 7 (3) + 1 (carried) = 8. Write 8.
123:. It was developed by the Ukrainian engineer
8:
48:. A user sandbox is a subpage of the user's
268:the 19 placed on the Working Dividend row.
480:(0 x 2) + Half of 5 (2) + 1 (carried) = 3.
755:Half of the leading zero's nieghbor is 3.
735:Half of the leading zero's neighbor is 2.
532:(9 - 2) + 1 + 1 (carried) = 9. Write 9.
803:
602:Subtract the right-most digit from 10.
581:(9 - 5) x 2 + 6 = 14, Write 4, carry 1.
526:(9 - 3) + 0 + 1 (carried) = 7. Write 7.
498:Subtract the right-most digit from 10.
387:(1 × 2) + 6 + 1 (carried) = 9. Write 9.
947:
935:
924:
899:
887:
876:
851:
839:
828:
655:Subtract the rightmost digit from 10.
631:(10 - 6) + Half of 0 (0) = 4. Write 4.
197:To find the first digit of the answer:
815:. Doubleday and Company. p. 13.
658:Subtract the remaining digits from 9.
634:(9 - 4) + Half of 6 (3) = 8. Write 8.
605:Subtract the remaining digits from 9.
553:Subtract the remaining digits from 9.
501:Subtract the remaining digits from 9.
7:
921:. Doubleday and Company. p. 85.
873:. Doubleday and Company. p. 81.
529:(9 - 1) + 3 = 11. Write 1, carry 1.
523:(10 - 0) + 0 = 10. Write 0, carry 1.
550:Subtract right-most digit from 10.
384:(6 × 2) + 0 = 12. Write 2, carry 1.
934:Cite has empty unknown parameter:
886:Cite has empty unknown parameter:
838:Cite has empty unknown parameter:
24:
587:4 - 2 + 1 (carried) = 3. Write 3.
101:Link to article I am editing is
94:
58:Create or edit your own sandbox
29:
288:Other multiplication algorithms
578:(10 - 6) x 2 + 0 = 8. Write 8.
199:The units digit of 9 x 6 = 4.
1:
81:Submit your draft for review!
948:|coauthors(translator)=
917:Trachtenberg, Jakow (1960).
900:|coauthors(translator)=
869:Trachtenberg, Jakow (1960).
852:|coauthors(translator)=
811:Trachtenberg, Jakow (1960).
683:Working from right to left:
627:Working from right to left:
574:Working from right to left:
519:Working from right to left:
991:Category:Mental calculation
714:Take half of the neighbor,
54:not an encyclopedia article
1006:
749:Half of 4's neighbor is 1.
732:Half of 4's neighbor is 1.
666:Add half of the neighbor,
610:Add half of the neighbor,
453:Add half of the neighbor,
408:Add half of the neighbor,
969:. Doubleday and Company.
393:(0 × 2) + 3 = 3. Write 3.
390:(3 × 2) + 1 = 7. Write 7.
209:The units digit of 8 x 6.
113:The Trachtenberg System
718:if current digit is odd
467:Working right to left:
422:Working right to left:
380:Working right to left:
333:Working right to left:
254:Setting up for Division
670:5 if the digit is odd.
614:5 if the digit is odd.
457:5 if the digit is odd.
255:
238:
172:General multiplication
145:
119:, somewhat similar to
253:
236:
115:is a method of rapid
965:Cutler, Ann (1962).
412:if the digit is odd.
330:3,425 × 11 = 37,675
988:Category:Arithmetic
535:2 - 1 = 1. Write 1.
516:2,130 × 9 = 19,170
349:0 + 3 = 3. Write 3.
346:3 + 4 = 7. Write 7.
343:4 + 2 = 6. Write 6.
340:2 + 5 = 7. Write 7.
337:5 + 0 = 5. Write 5.
946:Unknown parameter
936:|unused_data=
898:Unknown parameter
888:|unused_data=
850:Unknown parameter
840:|unused_data=
793:Jakow Trachtenberg
663:Double the result.
558:Double the result.
450:Double each digit.
364:Double each digit.
256:
239:
132:concentration camp
125:Jakow Trachtenberg
117:mental calculation
561:Add the neighbor.
506:Add the neighbor.
464:523 x 7 = 3,661.
367:Add the neighbor.
354:Multiplying by 12
323:Add the neighbor.
313:Multiplying by 11
298:Each digit has a
279:Casting out nines
163:Casting out nines
121:Vedic mathematics
109:
108:
89:
88:
65:Other sandboxes:
63:
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704:Multiplying by 5
645:Multiplying by 3
592:Multiplying by 4
540:Multiplying by 8
488:Multiplying by 9
440:Multiplying by 7
398:Multiplying by 6
272:General addition
246:General division
152:
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71:Template sandbox
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742:743 x 5 = 3715
706:
700:
680:492 x 3 = 1476
647:
624:346 * 4 = 1384
594:
571:456 x 8 = 3648
542:
490:
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419:347 × 6 = 2082
400:
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374:316 × 12 = 3792
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237:2 Finger method
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194:123456 x 789
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725:42 x 5 = 210
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985:
967:Instant Math
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67:Main sandbox
64:
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177:reference.
799:References
45:Netherstar
950:ignored (
927:cite book
902:ignored (
879:cite book
854:ignored (
831:cite book
50:user page
39:the user
782:See also
740:Example:
723:Example:
678:Example:
622:Example:
569:Example:
514:Example:
462:Example:
417:Example:
372:Example:
328:Example:
300:neighbor
192:Example:
37:This is
69:|
41:sandbox
973:
819:
716:plus 5
410:plus 5
709:Rule:
650:Rule:
597:Rule:
545:Rule:
493:Rule:
445:Rule:
403:Rule:
359:Rule:
318:Rule:
16:<
971:ISBN
952:help
940:help
904:help
892:help
856:help
844:help
817:ISBN
760:Book
668:plus
612:plus
455:plus
223:plus
218:plus
206:plus
129:Nazi
103:here
60:here
281:.
151:p13
43:of
944:;
931::
929:}}
925:{{
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877:{{
848:;
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829:{{
56:.
979:.
954:)
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938:(
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894:)
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62:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
