General Multiplication
The method for general multiplication is a method to achieve multiplications with low space complexity, i.e. as few temporary results as possible to be kept in memory. This is achieved by noting that the final digit is completely determined by multiplying the last digit of the multiplicands. This is held as a temporary result. To find the next to last digit, we need everything that influences this digit: The temporary result, the last digit of times the next-to-last digit of, as well as the next-to-last digit of times the last digit of . This calculation is performed, and we have a temporary result that is correct in the final two digits.
In general, for each position in the final result, we sum for all : . Ordinary people can learn this algorithm and thus multiply four digit numbers in their head - writing down only the final result. They would write it out starting with the rightmost digit and finishing with the leftmost.
Trachtenberg defined this algorithm with a kind of pairwise multiplication where two digits are multiplied by one 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.
Example:
To find the first digit of the answer:
The units digit of .
To find the second digit of the answer, start at the second digit of the multiplicand:
The units digit of plus the tens digit of plus
The units digit of .
.
The second digit of the answer is and carry to the third digit.
To find the fourth digit of the answer, start at the fourth digit of the multiplicand:
The units digit of plus the tens digit of plus
The units digit of plus the tens digit of plus
The units digit of plus the tens digit of .
carried from the third digit.
The fourth digit of the answer is and carry to the next digit.
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 nine will always point to the digit of the multiplicand directly above the digit of the answer you wish to find, with the other arrows each pointing one digit to the right. Each arrow head points to a UT Pair, or Product Pair. The vertical arrow points to the product where we will get the Units digit, and the sloping arrow points to the product where we will get the Tens digits of the Product Pair. 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 until all of the arrows point to prefixed zeros.
Read more about this topic: Trachtenberg System
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