In The Trachtenberg Speed System Of Basic Mathematics there are rules for each of the numbers from zero to twelve. Although for the numbers below three the rules are those we already know but are included to show how the rules fit together.

I am going to show you how I remember the Trachtenberg Basic Multiplication System rules. By looking at all the rules together instead of thirteen separate rules the patterns started to emerge and the rules can be put into several groups. From there is was relatively easy to find a simple key that allowed me to recall any rule almost instantly.

## The Rules for Basic Multiplication

The list of Rules are as follows:

Multiplier | Rule(s) |
---|---|

0 | Zero times any number at all is zero |

1 | Use the number (or Copy down the multiplicand unchanged) |

2 | Double the number. |

3 | First step: subtract from 10 and double, and add 5 if the number is odd. Middle step: subtract from 9, double and add 5 if the number is odd, and add “half” the neighbor. Last step: take “half” the left-hand digit of multiplicand and reduce by 2. |

4 | First step: subtract from 10, and add 5 if the number is odd. Middle step: subtract from 9 and add “half” the neighbor, plus 5 if the number is odd. Last step: take “half” the left-hand of the multiplicand and reduce by 1. |

5 | Use “half” the neighbor, plus 5 if the number is odd |

6 | Add 5 to the number only if it is odd; Add “half” the neighbor. |

7 | Double the number and add 5 if the number is odd, and add “half” the neighbor |

8 | First step: subtract from 10 and double. Middle step: subtract from 9, double and add the neighbor. Last step: reduce left hand digit of multiplicand by 2 |

9 | First step: subtract from 10 Middle steps: subtract from 9 and add the neighbor. Last step: reduce left hand digit of multiplicand by 1 |

10 | Use the neighbor |

11 | Add the neighbor |

12 | Double the number and add the neighbor |

Remember the "half" mentioned here is a true half for the even numbers but for the odd numbers we are referring to just the integer part of the half.

For example: half of 7 is 3.5 so we are only interested in the 3.

An odd number can be represented by:

2* n* + 1

eg; 7 = 2** n** + 1 where

**= 3 as 2 x 3 + 1 = 7**

*n*We are just interested in the value that is ** n **for the odd numbers.

## Is There a Pattern?

A first glance at the rules and some people are likely to think "I can't remember all of that!" however, there is a pattern here and once it becomes clear the rules get a whole lot easier. In fact, if I told you that you only need to learn 5 of these rules to be able to remember all of them, would you believe me?

You should, and I will explain why.

The first thing to remember is:

Whenever you are using "half" the neighbor, you **always** add 5 if the number is odd.

To show what I mean we will look at the rule for 11 and 5:

11 | Add the neighbor |

5 | Use “half” the neighbor, plus 5 if the number is odd |

If you look carefully at the rules you will see this is true in every case where "half" the neighbor is used.

### Grouping the Rules

We can put the rules for the numbers above into five groups, those groups are:

Notice anything about the groups? How much lower is each number from the one on its right? That's right! they are five apart.

Lets have a look at the rules for the first group of numbers:

12 | Double the number and add the neighbor |

7 | Double the number and add 5 if the number is odd, and add “half” the neighbor |

2 | Double the number. |

Do you see a pattern? Well for a start they all double the number. The other pattern is as you go down from 12 to 7 then to 2 you start by adding the neighbor, then you add "half" the neighbor and finally you ignore the neighbor.

This pattern of using the neighbor, use "half" the neighbor, and finally, for the groups of three numbers, ignore the neighbor holds for all the groups of numbers above.

Lets look at another group:

8 | First step: subtract from 10 and double. Middle step: subtract from 9, double and add the neighbor. Last step: reduce left hand digit of multiplicand by 2 |

3 | First step: subtract from 10 and double, and add 5 if the number is odd. Middle step: subtract from 9, double and add 5 if the number is odd, and add “half” the neighbor. Last step: take “half” the left-hand digit of multiplicand and reduce by 2. |

For 8 you use the neighbor and for 3 use use "half" the neighbor. As we have mentioned above, every time you use "half" the neighbor you also add five if the number is odd.

For 8, in step 3, we reduce the left-hand digit by 2 but for 3 we take "half" the left hand digit and reduce it by 2.

## The 5 number rules to learn

With what we have learnt above we can see that if we learn the rules for the numbers: 8, 9, 10, 11, and 12 we can then work out the rules for the remaining numbers.

12 | Double the number and add the neighbor |

11 | Add the neighbor |

10 | Use the neighbor |

9 | First step: subtract from 10 Middle steps: subtract from 9 and add the neighbor. Last step: reduce left hand digit of multiplicand by 1 |

8 | First step: subtract from 10 and double. Middle step: subtract from 9, double and add the neighbor. Last step: reduce left hand digit of multiplicand by 2 |

Now that is starting to look a whole lot easier. However, there is a way that I use to make learning the rules super easy.

## Dan Ate Mine!

I remember all the rules above by simply remembering the following:

**DAN ATE MINE**

That is it! I can extrapolate all the rules from that.

Let me explain.

DAN |
stands for: Double the number, Add the Neighbor |

ATE |
is 8 |

MINE |
is 9 |

Okay, so how does that help you wonder. Well the next step of the extrapolation is this:

12 | DAN |

11 | AN |

10 | N |

8 | is 10 - 2 DAN |

9 | is 10 - 1 AN |

I start with twelve and use DAN, then as I go down to 11 then 10 I remove the first letter each time from DAN, then AN so for 10 we just have N. Then at 8 I begin with DAN again and remove the first letter as I go to 9.

I also note how much I have to subtract from 10 to get 8 and 9, which is 2 and 1.

So now we have all five of our numbers 8, 9, 10, 11, and 12 and taking the extrapolation one more step:

12 | Double the number and Add the Neighbor |

11 | Add the Neighbor |

10 | Use the Neighbor |

8 | First step: subtract from 10 and Double.Middle step: subtract from 9, Double and Add the Neighbor.Last step: reduce left hand digit of multiplicand by 2 |

9 | First step: subtract from 10Middle steps: subtract from 9 and Add the Neighbor.Last step: reduce left hand digit of multiplicand by 1 |

For the 8 and 9 the subtracting from 10 reminds me they are the three step rules where the first step is subtracting from ten. This first step can be thought of as including adding the neighbor like the middle steps, but since there is no neighbor nothing is added. The difference from 10 also tells me how much to subtract in the third rule, subtract 2 for 8 and subtract 1 for 9.

Now I have my first 5 rules back and I know that as I go down by 5 I find the next number that follows the same rule and each time I go down 5 if I use the neighbor, the next step down is "half" the neighbor and the final step is ignore the neighbor.

12 | Double the number and Add the Neighbor |

7 | Double the number, add “half” the Neighbor and add 5 if the number is odd |

2 | Double the number. |

11 | Add the Neighbor |

6 | Add “half” the neighbor, Add 5 to the number only if it is odd |

1 | Copy down the multiplicand unchanged |

10 | Use the Neighbor |

5 | Use “half” the neighbor, plus 5 if the number is odd |

0 | Zero times any number at all is zero |

8 | First step: subtract from 10 and Double. Middle step: subtract from 9, Double and Add the Neighbor. Last step: reduce left hand digit of multiplicand by 2 |

3 | First step: subtract from 10 and double, and add 5 if the number is odd. Middle step: subtract from 9, double and add 5 if the number is odd, and add “half” the neighbor. Last step: take “half” the left-hand digit of multiplicand and reduce by 2. |

9 | First step: subtract from 10 Middle steps: subtract from 9 and add the neighbor. Last step: reduce left hand digit of multiplicand by 1 |

4 | First step: subtract from 10, and add 5 if the number is odd. Middle step: subtract from 9 and add “half” the neighbor, plus 5 if the number is odd. Last step: take “half” the left-hand of the multiplicand and reduce by 1. |

Now it should be easy to see the relationship between the rules and see how easy it can be to remember the rules.

## Practice, Practice

Once I got my mnemonics sorted out I practiced by picking a number between 0 and 12 then from "DAN ATE MINE" mentally extrapolating the rule(s) for that number.

Once you have done that several times for each number you will find it hard to forget.

After some practice the rules become automatic and you don't really have to think about them.

If you have any questions or comments about how To Remember The Trachtenberg Basic Multiplication System Rules or anything else to do with the Trachtenberg System please leave them below and I will respond asap.

tsm com says

July 10, 2017 at 2:17 amWish you have a youtube video that covers this… like the others.

The memory solution with a visual will make all the difference.

🙂

Tony says

July 10, 2017 at 7:48 amI have more videos planned, and this is one of them. It is just a matter of finding the time to get everything I would like to do done for this site.

Steven says

June 26, 2019 at 10:29 amexcellent work. thank you for the mnemonics DAN ATE MINE. It surely is logical and easy to remember