The Rule for 3: A Unique Pattern

The Trachtenberg rule for multiplying by 9 is a clever method that relies on subtraction and addition, completely avoiding the need to think about multiplication tables.

The Rule: A Three-Part Process

Working from right to left (with a leading zero):

For the rightmost digit: Subtract it from 10.
For the middle digits: Subtract the digit from 9, then add its right-hand neighbor.
For the leading zero (final digit): Subtract 1 from its neighbor.

Example: Multiply 423 by 9

Add a leading zero: 0423.

Digit 3: $10 - 3 = 7$ . Write 7.
Digit 2: $(9 - 2) + 3 = 10$ . Write 0, carry 1.
Digit 4: $(9 - 4) + 2 + 1~(\text{carry}) = 8$ . Write 8.
Digit 0: $4 - 1 = 3$ . Write 3.

The final answer is 3,807.

🔢Multiplication by 9 Lab

Enter any number to see the step-by-step Trachtenberg process for multiplying by 9.

🧠Quick-Fire Quiz!

Frequently Asked Questions

Why do you subtract from 10 for the first digit, but 9 for the others?

This is a core concept in Trachtenberg's subtraction methods. It's a way to handle 'borrowing' without thinking about it. Subtracting the units digit from 10 is the base operation. Subtracting all subsequent digits from 9 accounts for the 'borrow' that would have occurred from that position in traditional math.

Is this rule related to the common 'casting out nines' trick?

Yes, in a way. Both are based on the mathematical properties of the number 9 in a base-10 system. The fact that 9 is '10 minus 1' is what allows for these kinds of subtraction-based shortcuts.

What is the final step 'neighbor - 1' for the leading zero?

This is the final step of the algorithm. After adding a leading zero, you apply the 'subtract from 9, add neighbor' rule to it: (9 - 0) + neighbor = 9 + neighbor. This would incorrectly inflate the answer. The 'neighbor - 1' is a simplified and corrected final step that gives the proper leftmost digit of the product.