Here we are going to look at Trachtenberg's rules for multiplication without tables. We have already looked at multiplication by 11, and multiplication by 0, 1, 2 and 10 are too simple to worry about, so we will be looking here at multiplication by 3, 4, 5, 6, 7, 8, 9 and 12.
With the addition calculation in Lesson 2 we assumed a column of zero on the left, and in these calculations we assume a leading zero at the left of the multiplicand.
We can also assume the rightmost digit has a trailing zero as a neighbor.
For these exercises, the 'halve' operation means "half the digit, rounded down", i.e. discard any remainder (so half of 5 is 2, not 2.5). Also, when told to add half of the neighbor, always add 5 if the current digit if it is odd.
To find the answer we work one digit at a time starting at the least significant (the right-hand) digit and moving left.
Double each digit and add the neighbor (the digit on the right), writing down the units result and carrying any second digit (which will be a one or a two).
Example:
372 x 12 = 4464
1. Double the first digit (2) and add its imaginary neighbor (0) giving 4, so write down 4.
2. Double the second digit (7) and add its neighbor (2) giving 16, so write down 6 and carry 1.
3. Double the third digit (3) and add its neighbor (7) plus the carry giving 14, so write down 4 and carry 1.
4. Double the imaginary left-hand zero (giving zero - duh) and add its neighbor (3) plus the carry, giving 4 and write it down.
Add half of the neighbor to each digit, then, if the current digit is odd, add 5.
Example:
372 Ă— 6 = 2232
1. Take the first digit (2) and add half its imaginary zero, giving 2 of course, so write it down.
2. Take the second digit (7) and add half its neighbor (half of 2 is 1) giving 8, then add 5 because 7 is odd, giving 13, so write down 3 and carry 1.
3. Take the third digit (3) and add half its neighbor (half of 7 is 3 here), add 5 since our digit is odd and add the carry, giving 12, so write 2 and carry 1.
4. Take the imaginary left-hand zero, add half its neighbor (half of 3 is 1 here) and add the carried 1, giving 2, so write it down.
Double each digit, add half of its neighbor, if the digit is odd, add 5.
Example:
372 x 7 = 2604
1. Double the first digit (2, giving 4), add half its imaginary neighbor (0), and write down 4.
2. Double the second digit (7, giving 14), add half its neighbor (half of 2 is 1), plus 5 because 7 is odd, giving 20, so write down 0 and carry 2.
3. Double the third digit (3, giving 6), add half its neighbor (half of 7 being 3), plus 5 because 3 is odd, plus the carried 2, giving 16, so write 6 and carry 1.
4. double the imaginary zero is 0, plus half 3 (1), plus the carried 1, gives 2.
With the rules for 9, 8, 4, and 3, the first digit is subtracted from 10 and the remaing digits are subtracted from nine.
Subtract the right-most digit from 10, subtract the remaining digits from 9 and add the neighbor; for the leading zero, subtract 1 from the neighbor.
Example:
372 x 9 = 3348
1. Subtract the first digit, 2, from 10 and write down the result, 8.
2. Subtract the second digit, 7, from 9, giving 2 and add the neighbor (2), and write the result, 4.
3. Subtract the third digit, 3 from 9, giving 6 and add the neighbor (7), giving 13, so write down 3 and carry 1.
4. Finally subtract 1 from 3, giving 2, add the carried 1, and write the result, 3.
Subtract right-most digit from 10, subtract the remaining digits from 9, double the result in both instances and add the neighbor; for the leading zero, subtract 2 from the neighbor.
Example
372 x 8 = 2976
1. Subtract the first digit, 2, from 10, double the result (twice 8 is 16) add its imaginary neighbor (0), write the 6 and carry 1.
2. Subtract the second digit, 7, from 9, double the result (twice 2 is 4) add its neighbor (2) plus the carried 1, giving 7.
3. Subtract the third digit, 3, from 9, double the result (twice 6 is 12) add its neighbor (7), giving 19, so write down 9 and carry 1.
4. For the final imaginary zero, subtract 2 from its neighbor (3 - 2 = 1) plus the carried 1, gives 2.
Subtract the right-most digit from 10, subtract the remaining digits from 9, add half of the neighbor, plus 5 if the digit is odd; for the leading 0, halve of the neighbor and subtract 1.
Example:
372 x 4 = 1488
1. Subtract the first digit, 2, from 10, and write down the result (8).
2. Subtract the second digit, 7, from 9, add half its neighbor (2) plus 5 since 7 is odd, giving 8.
3. Subtract the third digit, 3, from 9, add half its neighbor (7) plus 5 since 3 is odd, giving 14, so write 4 and carry 1.
4. For the final imaginary zero, subtract 1 from its half its neighbor (half of 3 is 1, subtract 1 =0) plus the carried 1, gives 1.
Subtract the rightmost digit from 10, subtract the remaining digits from 9, and in both cases double the result, then add half of the neighbor, plus 5 if the digit is odd; for the leading zero, subtract 2 from half of the neighbor.
Example:
372 x 3 = 1116
1. Subtract the first digit, 2, from 10, double the result, write down the 6 and carry the 1.
2. Subtract the second digit, 7, from 9, double the result, add half its neighbor (2) plus 5 since 7 is odd, plus the carry, giving 11, so write the 1 and carry 1.
3. Subtract the third digit, 3, from 9, double the result, add half its neighbor (7) plus 5 since 3 is odd, plus the carry, giving 21, so write 1 and carry 2.
4. For the final imaginary zero, subtract 2 from its half its neighbor (half of 3 is 1, subtract 2 = -1) plus the carried 2, gives 1.
Take half of the neighbor, then, if the current digit is odd, add 5.
Example:
372 x 5 = 1860
1. The neighbor of the first digit (2) is an imaginary zero, so half of it is also zero, and 2 is even, so we just write 2.
2. The neighbor of the second digit (7) is a 2 and half of it is 1, plus 5 (since 7 is odd) gives 6 to write down.
3. The neighbor of the third digit (3) is a 7 and half of it is 3, plus 5 (since 3 is odd) gives 8 to write down.
4. For the final imaginary zero, its neighbor is 3, half of which is 1, so we write that down.