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The Rule for 6: Add Half the Neighbor

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The Trachtenberg rule for multiplying by 6 is a simple and elegant process known as the "Add Half the Neighbor" method. It builds the answer digit by digit, making large multiplications manageable.

The Rule: "Add Half the Neighbor"

For each digit, working from right to left:

  1. Start with the digit itself.
  2. Add half of its right-hand neighbor (ignoring fractions).
  3. If the starting digit is odd, add 5.

Example: Multiply 254 by 6

First, add a leading zero: 0254.

  1. Digit 4: 4 (even) + 0 (half of neighbor 0) = 44. Write 4.
  2. Digit 5: 5 (odd) + 2 (half of neighbor 4) + 5 = 1212. Write 2, carry 1.
  3. Digit 2: 2 (even) + 2 (half of neighbor 5) + 1 (carry) = 55. Write 5.
  4. Digit 0: 0 (even) + 1 (half of neighbor 2) = 11. Write 1.

The final answer is 1,524.

🔢Multiplication by 6 Lab

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Frequently Asked Questions

What is the 'neighbor' in this rule?

The 'neighbor' is always the digit immediately to the right of the digit you are currently working on. For the rightmost digit of the number, its neighbor is considered to be 0.

Why do you add 5 only for odd digits?

This is a clever trick to handle decimals without actually using them. It mathematically compensates for the .5 that is dropped when taking 'half of the neighbor' if the neighbor is odd, along with other factors, to keep the calculation correct.

Is it really necessary to add the digit to itself?

Yes, this is the most common point of confusion. You always start with the digit you are working on, and then you add the other parts (half the neighbor, 5 for odd digits, and the carry) to it.