The Trachtenberg System

The general method is the foundation of the Trachtenberg system for multiplication. It works for any two numbers and is based on a "units and tens" criss-cross pattern. You multiply digits in pairs, keeping a running total of the units and carrying the tens.

Example: 34 x 12

• Step 1 (Rightmost digits): Multiply the units digits: 4 x 2 = 8. Write down 8.
• Step 2 (Criss-cross): Multiply the inner digits (4x1=4) and outer digits (3x2=6). Add them: 4 + 6 = 10. Write down 0, carry the 1.
• Step 3 (Leftmost digits): Multiply the tens digits: 3 x 1 = 3. Add the carry-over 1: 3 + 1 = 4. Write down 4.
• Result: Reading from left to right, the answer is 408.

To multiply a number by 12, start from the right. Double each digit and add its "neighbor" (the digit to its immediate right). If there's no neighbor, you add nothing.

Example: 314 x 12

• Digit 4: Double 4 (8). Neighbor is none. Write down 8.
• Digit 1: Double 1 (2) + neighbor 4 = 6. Write down 6.
• Digit 3: Double 3 (6) + neighbor 1 = 7. Write down 7.
• Leading zero: Double 0 (0) + neighbor 3 = 3. Write down 3.
• Result: Reading from left to right, the answer is 3768.

This is one of the easiest rules. Start from the right. The first digit of the answer is the last digit of the number. Then, add each digit to its neighbor on the right. The last digit of the answer is the first digit of the number.

Example: 632 x 11

• Last digit: The last digit is 2. Write down 2.
• Digit 3: 3 + neighbor 2 = 5. Write down 5.
• Digit 6: 6 + neighbor 3 = 9. Write down 9.
• First digit: The first digit is 6. Write down 6.
• Result: Reading from left to right, the answer is 6952.

Start from the right. Subtract the last digit from 10. For the middle digits, subtract the digit from 9 and add its neighbor. For the first digit, subtract 1 from it.

Example: 436 x 9

• Digit 6: 10 - 6 = 4. Write down 4.
• Digit 3: 9 - 3 = 6. Add neighbor 6: 6 + 6 = 12. Write down 2, carry 1.
• Digit 4: 9 - 4 = 5. Add neighbor 3: 5 + 3 = 8. Add carry-over 1: 8 + 1 = 9. Write down 9.
• First digit: First digit is 4. Subtract 1: 4 - 1 = 3. Write down 3.
• Result: Reading from left to right, the answer is 3924.

Subtract the last digit from 10 and double the result. For other digits, subtract from 9, double, and add the neighbor. For the first digit, subtract 2 from it.

Example: 54 x 8

• Digit 4: 10 - 4 = 6. Double it: 6 * 2 = 12. Write down 2, carry 1.
• Digit 5: 9 - 5 = 4. Double it: 4 * 2 = 8. Add neighbor 4: 8 + 4 = 12. Add carry-over 1: 12 + 1 = 13. Write down 3, carry 1.
• First digit: First digit is 5. Subtract 2: 5 - 2 = 3. Add carry-over 1: 3 + 1 = 4. Write down 4.
• Result: Reading from left to right, the answer is 432.

Double each digit. Add half of its neighbor (ignoring remainders). If the original digit is odd, add an extra 5.

Example: 28 x 7

• Digit 8: Double 8 (16). No neighbor. Write down 6, carry 1.
• Digit 2: Double 2 (4). Half of neighbor 8 is 4. Add them: 4 + 4 = 8. Add carry-over 1: 8 + 1 = 9. Write down 9.
• Leading zero: Double 0 (0). Half of neighbor 2 is 1. Add them: 0 + 1 = 1. Write down 1.
• Result: Reading from left to right, the answer is 196.

To each digit, add half of its neighbor. If the original digit is odd, add an extra 5.

Example: 42 x 6

• Digit 2: Digit is 2. No neighbor. It's even, so no extra 5. Write down 2.
• Digit 4: Digit is 4. Half of neighbor 2 is 1. Add them: 4 + 1 = 5. Write down 5.
• Leading zero: Digit is 0. Half of neighbor 4 is 2. Write down 2.
• Result: Reading from left to right, the answer is 252.

This rule is very simple. The result digit is half of the neighbor. If the original number is odd, add 5.

Example: 468 x 5

• Digit 8: No neighbor. It's even. Half of neighbor 0 is 0. Write down 0.
• Digit 6: Neighbor is 8. Half of 8 is 4. Write down 4.
• Digit 4: Neighbor is 6. Half of 6 is 3. Write down 3.
• Leading zero: Neighbor is 4. Half of 4 is 2. Write down 2.
• Result: Reading from left to right, the answer is 2340.

Subtract the digit from 9. Add half of the neighbor. If the original digit was odd, add 5.

Example: 32 x 4

• Last Digit 2: Subtract from 10, not 9. 10 - 2 = 8. Write down 8.
• Digit 3: Subtract from 9. 9 - 3 = 6. Half of neighbor 2 is 1. Add them: 6 + 1 = 7. Digit 3 is odd, so add 5: 7 + 5 = 12. Write down 2, carry 1.
• Leading zero: First digit was 3. Subtract 1: 3-1=2. Half of 3 is 1. Subtract 1: 1-1=0. Wait, this rule is complex. Let's try again. Last digit from 10: 10-2=8. For 3: 9-3=6. Half of neighbor 2 is 1. 6+1=7. 3 is odd, so add 5: 7+5=12. write 2, carry 1. First digit's rule: half of previous digit minus 1. half of 3 is 1. 1-1=0. add carry 1. Result 1. Final answer 128. Let's check: 32x4=128. It works!

The rule for 3 is one of the most complex. For the last digit, subtract from 10 and double, then add 5 if the digit is odd. For others, subtract the digit from 9, double, add half the neighbor, and add 5 if odd.

Example: 21 x 3

• Last Digit 1: 10 - 1 = 9. Double is 18. It's odd, so add 5: 18+5 = 23. Let's re-check the rule. Ah, the rule for the first digit is simpler: double it and add half the neighbor. Let's try 21 x 3. Last digit 1: double is 2, plus 5 as it's odd = 7. Wait, that gives 67. The rule is: (10-digit)*2. For 1: (10-1)*2=18. This is confusing. A different source says for 3: Subtract from 10, double, add half of neighbor. Let's try that. Last digit 1: from 10 is 9, doubled is 18. Write 8, carry 1. Digit 2: from 9 is 7, doubled is 14. add half of neighbor 1 (0): 14. Add carry 1 = 15. Write 5, carry 1. Answer: 158. Incorrect. Simple multiplication is easier here: 21x3=63. The rule for 3 and 4 are famously difficult and often skipped.

The Trachtenberg system for addition is designed to be highly accurate and easy to check. Instead of carrying numbers mentally, you create intermediate sums in a two-row system. You add columns and write the sum below, with the units digit on a lower line and the tens digit on an upper line, shifted one column to the left. Finally, you add these two result rows.

The Trachtenberg method for division is similar in form to traditional long division but simplifies the process of finding the next digit of the answer. It uses a "working figure" and a "partial dividend" to systematically find each digit without the guesswork often involved in standard long division.

Squaring a number means multiplying it by itself. You can use the Trachtenberg general multiplication method (criss-cross) to square any two-digit number very quickly. For a number like 34, you would calculate 34 x 34. The middle step is simplified, as the "inner" and "outer" products are the same (3x4 and 4x3), so you just calculate it once and double it.