Tutorial del sistema trachtenberg (en ingles)

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Multiplication
In case we want to multiply two numbers we can find a range of opptions following Trachtenberg's system. Suppose we want to multiply a number by... Case 1: Multiplication by 2 1) Add each digit of the multiplicand to itself. 2) Add “carried tens” 3) Write down the result. Case 2: multiplication by 3 1) First step: Subtract the first number of the multiplicand form 10. Double theresult. If the original number is Odd add 5, else avoid that. Write down the result. 2) Middle steps: subtract each middle number from 9. Double the result. Add half the neighbor without the fractions. If the original number is odd, add 5, else avoid doing that. Do this to the last number of the multiplicand. Case 3: multiplication by 4 1) First step: Subtract the first number of the multiplicandfrom 10. If the original number is odd add 5, else don't add anything. Write down the result. 2) Middle steps: Subtract each number from 9. Add half the neighbor without the fractions. If the original number is odd add 5, else don't add anything. Do this to the last number. Add carried tens. Write down the result. Case 4: multiplication by 5 1) Just look at the number to see if it is odd or even.If the original number is even, say 0; but if the original number is odd, say 5 and write it down. 2) Add half of the neighbor without the fractions. Write down the result. 3) NOTE: for this rule there will never be “carried tens”. Case 5: multiplication by 6 1) Take each number of the multiplicand in turn. Add half the neighbor. If the original number is odd add 5, else do nothing. Add “carriedtens”. Write down the result. Case 6: multiplication by 7 1) Take each number of the multiplicand in turn. Double it. Add half the neighbor without the fractions. If the original number is odd add 5, else do nothing. Add “carried tens”. Write down the result. Case 7: multiplication by 8 1) First step: subtract the first number of the multiplicand from 10. Double the result. Write down the result. 2)Middle steps: subtract each following number from 9. Double the result. Add the whole neighbor. Add “carried tens”. Write down the result. Do this to the left-hand number as

well. 3) Last step: once you arrive the last number, reduce it by 2. Add the carried tens and write down the result. Case 8: multiplication by 9 1) First step: subtract the first number of the multiplicand from 10. Writedown the result. 2) Middle steps: subtract each number from 9. Add the whole neighbor. Write down the result. Do this to the left-hand number as well. 3) Last step: say the last number, reduce it by 1. Add “carried tens”. Write down the result.

Two-fingers Method:
Let's say we want to opperate 3112 x 6. The steps to do this with the Units and Tens or the twofingers method is as follows: 1) Westart by placing the U from the UT (Units and Tens) just above the number 2. Multiply 6x2 and place the Units result, carry the tens: UT 3112 x 6 2

→ carry 1

2) Move the UT to the left. That is because the U of the UT is always placed over the position where the next figure of the answer is to appear, that is, where the 7 will be. In this example, the 7 is the pair-product of the units-digitof 06 (from 1 times 6) plus the tens-digit of 12 (from 2 times 6). In the first step, the 2 of 3112 was used as a U. In the second step it was used again, but this time as a T. This always happens. Each figure of the multiplicand is used twice, once under the U of UT and then under the T. UT 3112 x 6 72

→ 6 times 1 = 6 + 1 from the carry = 7

3) Move UT again to the next figure to the left.6 is the pair-product of the units-digit of 06 (1 times 6 is 06), plus the tens-digit of 06 (again 1 times 6 is 6). UT 3112 x 6 672 4) Move UT to the next figure to the left. The 8 is the pair-product of the units-digit of 18 (3 times 6), plus the tens-digit of 06 (1 times 6 is 06). UT 3112 x 6 8672

5) Add a 0 at the end of the multiplicand. Following the same logic, This is the pair-product...
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