Trachtenberg Speed Mathematics Start Page

Trachtenberg Speed Math is a series of methods that allow rapid computation of results in multiplication, division, addition, etc. This is the starting point for multiplication.

This website is an interactive demonstration of the multiplication methods developed by Trachtenberg. You can use this website to learn the methods, but hand and paper practice will increase your speed once the methods are learned.

You may ask why! Why not just use a multiplication table, simple memorization? The reason is that if you use a multiplication table, you have to memorize 169 rules to get to 12x12, starting from 0x0. You might quickly notice that 25 of those rule can be eliminated by noting that they include one or more zeros, and everyone knows:

'zero times any number is zero'

So what you have done is applied the first Trachtenberg rule to reduce the amount of memorization required. Then you might notice that one times any number is the same, so you can eliminate another 23 facts from memory and replace it with one rule. This is the second Trachtenberg rule:

'one times any number is itself'

So the point is that we have all used rules to replace memorization in multiplication facts. It is just as easy to memorize a rule as it is to memorize a fact. I might argue that it is easier, because a rule can be tested against what you already know, whereas a fact cannot. There is no simple, independent method of verifying that 7x8 = 56. It is also easier to recall and use a smaller set of information, whether facts or rules.

The final reason for learning these rules is that they apply far beyond 12x12. They apply to any number. These 21 rules (of which three you already know), replace 169 facts. Oh, what is the third rule everyone knows?

'two times any number is the number added to itself'

Start by selecting a number from zero to twelve to multiply by. Then choose the number you want to multiply. Click on the 'START!' button and you will be shown the rule to apply for the first step of the result. At each step, your work will be checked for accuracy.

For more explanation of higher level multiplication (multipliers greater than 12), see these examples of Trachtenberg math. Or view a local copy of this paper here.

One more note about multipliers greater than 10. The rules are additive. This means that the rule for 17 is the rule for 7 plus the rule for 10. Since the rule for 10 is pretty easy, this rule of additivity of the rules can be used to get you all the way to 19.


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