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Trachtenberg speed multiplication: exploring why it works

Last November I wrote about the Trachtenberg system of speed mathematics and showed one of the techniques - multiplying an arbitrarily large number by 12 with great ease. In this article I want to show you why the technique for multiplying by 12 works, share two more of the Trachtenberg multiplication techniques, and give you some direction as to how you can develop your own multiplication techniques.

In that first article about Trachtenberg multiplication I taught you to multiply by 12 by "doubling the digit and adding its neighbor (on the right)." I gave an example of multiplying 346 by 12:

Start with the right-most digit, 6. Double the 6 and add the neighbor (none in this case). We get 12. Write down 2 from 12, and carry the 1.

Move left to the next digit, 4. Double the 4 to get 8, add the neighbor, 6, to get 14, and add the carry, to get 15. Write down the 5, and carry the 1.

Move left to the next digit, 3. Double the 3 to get 6. Add the neighbor, 4, and we get 10. Add the carry, to get 11. Write down the 1, and carry the 1.

Move left to the non-existent digit. Double it to get 0, and add the neighbor, 3, which gives us 3. Finally, add the carry to get 4. Write down 4.

Our answer is 4152.

So, why does this work? Let's look at what happens when we multiply 346 by 12. We would write the following:


What you may notice is that the way the partial products 692 and 346 are lined up corresponds to how Trachtenberg multiplication for 12 works. The last digit is 2, which is the last digit of 6x2, just like in Trachtenberg. Then, to get the next digit of the answer, 5, we add 9 to 6. Well, 9 comes from multiplying 4 by 2 and adding the carry of 1. That's the same as doubling the 4 via Trachtenberg. The 6 we add to 9 comes from multiplying 6x1, which is the "add your neighbor" part of Trachtenberg. So, 6+9=15 and 5 becomes out next digit and 1 is the carry.

To get the next digit of the answer, 1, we add 6 and 4 from the partial product and add the carry of 1 to get 11. The 6 comes from multiplying 3x2 (doubling 3 in Trachtenberg) and the 4 is 4x1 (or "adding the neighbor.")

The final digit of the answer, 4, comes from multiplying 3x1 and adding the carry of 1. This is the same as doubling the non-existent digit to the left of the 3, and adding the neighbor, 3, plus the carry of 1, to get 4.

I hope that you can see that there's a direct correspondence between the familiar method of writing partial products one beneath the other, with the second partial product offset one place to the left, and what Trachtenberg is doing. If this is not clear please let me know and I'll try to explain it better.

Now that you see how Trachtenberg approaches multiplication by 12 can you work out how to multiply by 13? If it's not immediately clear how to proceed, make up some simple examples, like 111x13 and 122x13. The approach to multiplying by 13 will be almost identical to the approach to multiplying by 12, with just one difference. Can you figure out what it is?

Once you see how to quickly multiply by 12 and 13, how would you multiply by 11, or 14?

Now, I'm going to teach you the Trachtenberg rule for multiplication by 7 and leave you with the exploration of figuring out why this approach works. The rule for multiplying by 7 says "Double each digit. Add half its neighbor. If the digit is odd, add 5." Let's try an example: 124x7:

Start with the right-most digit, 4. Double the 4 to get 8. There is no neighbor to add half of. 4 is also not odd. So, the last digit of the answer is 8.

The next digit is 2. 4 is not odd. Double the 2 to get 4. Add half of 2's neighbor 4. Half of 4 is 2. So, we double the digit 2 to get 4, then add half of 4 to that and we get 6. That's the next digit of the answer.

The next digit is 1. 1 is odd. We double the 1 to get 2. We add half its neighbor, half of 2 is 1, and get 3. Since the digit we're working with, 1, is odd we add 5, so the next digit in our answer is 8.

Finally, we move one more place to the left, where there's no digit, so we call it 0. We double 0, add half of 1 (rounded down) to that and get 0. So, 0 is the leftmost digit of our answer, which we can ignore. So, the product is 868.

I'm going to leave you the challenge of figuring out why this rule works. I'll give you three hints:

  • 7=5+2
  • Multiplying by 5 is the same as multiplying by 10 and taking half of what you get.
  • 6x5=30 and 3 is half of 6; 7x5=35 and 3 is half of 7 if you round down.
Comments (2) Trackbacks (1)
  1. Another multiplication or squaring a number ending in “5”. For instance 125, write down the 5 squared is 25 to this add 12 squared is 144 + 12=156. So, our answer is 15625.

  2. Hi team,

    Does the trachenberg math software have an application for either Apple iPAD or the Amazon Fire?

    I am considering getting one of them. I would rather use one of them than the computer for playing with your math software.

    I have a break from school in May and hope to get the software then.

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