Trachtenberg system of multiplication by 11 and 12
Basic concept is each digit in multiplicand is used twice once as a number and next as neighbor.
Multiplication by 11
Following are 3 basic rules to multiply by 11:
- The last digit of the multiplicand (rightmost digit) is put down as rightmost digit.
- Each successive digit is added to its neighbor at the right.
- The first digit (leftmost digit) of the multiplicand becomes the leftmost digit of the answer. Carry of previous addition will be added to this leftmost digit.
742 * 11
First rule: right most digit:
Second rule: number and neighbors addition:
Third rule: leftmost digit
Second rule ie “each successive digit of the multiplicand is added to its neighbor at the right” used multiple times based on the length of the multiplicand. In above example, it is done twice.
In short , the rule is “add the neighbor”. First write or imagine a zero in front of the leftmost digit. Then apply the add the neighbor to every digit of given number.
In case of carry, we can put a dot for carry 1 and double dot for carry 2. This is helpful to not to miss carry addition:
Multiplication by twelve
This method is similar to the method of multiplication by eleven except that before adding the digit/number to the neighbor , double it and then add to neighbor.
This can be done in a single line:
Please try and comment :)