Will Royson
Listen, numbers are hard. But the Trachtenberg Method makes ‘em simpler.
Ever feel like you forgot everything you learned in grade school thanks to your handy-dandy calculator? Like… complex multiplication. Who can still do that without whipping out a smartphone?
People who can use the Trachtenberg Method can.
Jakow Trachtenberg developed the Trachtenberg Method while passing time when he was imprisoned in a Nazi concentration camp during World War II. After he escaped and fled to Switzerland, he founded the Institute of Mathematics.
Here’s how you can use the method to make math (and life) easier.
What Is the Trachtenberg Method?
While there’s a whole book about it (The Trachtenberg Speed System of Basic Mathematics) the Trachtenberg method is basically a mathematical order of operations.
It allows you to solve math problems more easily by following a simple set of rules.
Benefits of the Trachtenberg Method
The Trachtenberg method has tons of benefits. The most obvious benefit to the Trachtenberg method is that it makes solving problems super simple. Even children with basic addition and subtraction skills use it, so you can, too.
Here are some other benefits to the Trachtenberg method:
- Applying the Trachtenberg method takes no time at all. The rules stay the same, so it’s just a matter of applying them to different problems.
- If you can memorize the rules to the Trachtenberg method, you can quickly solve even complex problems. This saves you time and effort.
- You don’t need to remember how to perform mental calculations with the Trachtenberg method. Instead, you only need to understand the rules, which you can apply in various situations.
Limitations of the Trachtenberg Method
There’s no doubt that the Trachtenberg Method makes math simpler. But there are various rules you need to follow. And if you have a difficult time remembering those rules, you’ll have a difficult time applying the Trachtenberg method.
Luckily, with memorization tools like Dorothy, you can better remember the Trachtenberg method’s rules so that they are easy to apply on the spot.
How to Learn the Trachtenberg Method
Here are some steps to learn the Trachtenberg method to the best of your abilities.
1. Master the terminology.
Before you try to tackle the Trachtenberg method, you first need to understand some basic mathematical terminology.
Multiplicand: This refers to the number that is being multiplied (generally written on the left side of the “x” symbol).
Multiplier: This refers to the number that the other is being multiplied by (generally on the right side of the “x” symbol).
See the example problem given: 712 x 11. In this case, 712 is the multiplicand and 11 is the multiplier.
Number: The number is the specific digit with which you are working.
Neighbor: A neighbor is the digit to the right of the “number.”
Leading Zero: A leading zero is the zero that’s added to the front of the multiplicand as a reminder that the answer may have a digit in its place.
Dot: This may also be identified as a remainder.
Half: When dealing with even numbers, half simply means half. However, the Trachtenberg Method does not deal with fractions. For this reason, “half” of an odd number will round down. (For example: half of 7 is 3; half of 5 is 2.)
2. Work through a practice problem.
There are a few rules to understand, but they may be easier to learn by doing. So, let’s start by walking through an example problem, step by step. This way, you can get the basic idea behind the method.
Once you have chosen a problem, place a leading zero in front of the multiplicand to ensure you have a placeholder in the event the answer contains an extra digit.
For this example, both the number and its neighbor will be underlined to help you identify each. Remember, the neighbor is always to the right of the number with which you are working.
Let’s use this equation again: 0712 x 11
Step one: Find the rule below for multiplying by 11″
Step two: Take the last number of the multiplicand and place it under the equation.
0712 x 11
————
2
Step three: Add each successive number to its neighbor.
In this case, (1 + 2 = 3). So, we place a “3” to the left of the “2” below the equation.
0712 x 11
————
32
Next, (7 + 1 = 8). So, we place an ‘8’ to the left of the ‘3’ below the equation.
0712 x 11
————
832
Step four: The first number (leftmost digit) of the multiplicand becomes the leftmost digit of the answer.
0712 x 11
————
7832
And that’s it. There’s your solution: 7832.
3. Memorize the rules.
Now that you’ve seen how easy the Trachtenberg method can be, you’ll need to take a closer look at the rules, as they vary depending on the multiplier.
While the rules may seem daunting at first, rest assured that this method is quite simple once you’ve worked through a few problems. Continue practicing the method, and the rules will eventually become like second nature to you.
Below is a list of rules for multipliers 1 through 12.
Rules of the Trachtenberg Method
Here are the rules, listed in the same order as the book.
Multiply by 1:
- Multiplying by 1 does not change any number, so just copy down each digit of the given number
Multiply by 2:
- Simply multiply each digit by two.
Multiply by 11:
- Add the neighbor.
Multiply by 12:
- Double the number, then add the neighbor.
Multiply by 6:
- Add “half” of the neighbor. If the number is odd, add five.
Multiply by 7:
- Double each digit, and then add “half” of the neighbor. If the number is odd, add five.
Multiply by 5:
- Take “half” of the neighbor. If the number is odd, add five.
Multiply by 8:
- Righthand digit: Subtract the righthand digit from 10, and then double it.
- Subtract middle digits from nine, and then double it. Add “half” of the neighbor.
- Subtract two from the left-hand digit.
Multiply by 9:
- Subtract the righthand digit from 10.
- Subtract middle digits from nine, and then add the neighbor.
- Subtract one from the left-hand digit.
Multiply by 4
- Righthand digit: Subtract the righthand number from 10. If the number is odd, add five.
- Subtract middle digits from nine. If the number is odd, add five. Now, add “half” of the neighbor.
- Take “half” of the left-hand digit, and then subtract one.
Multiply by 3:
- Subtract the righthand digit from 10, and then double what you get. If the number is odd, add five.
- Subtract middle digits from nine, and then double the answer. Again, if the number is odd, add five. Now, add “half” of the neighbor.
- Take half of the left-hand digit, and then subtract two.
“Multiplying by 3 is similar to multiplying by 8 but with a few exceptions,” according to the book. “Instead of adding the neighbor, as we did for 8, we now add only ‘half’ the neighbor. It is understood, of course, that we also add the extra 5 if the number is odd. Adding half the neighbor always carries with it the extra 5 for odd numbers.”
How to Find the Rest of the Rules
The Trachtenberg Method is just one part of the larger Trachtenberg System. The system extends beyond multiplication to division, addition and subtraction.
If you’re looking for the rest of the rules, the original textbook, published in 1960, is a great resource. It contains seven chapters, each of which explore a different method within the Trachtenberg System.
But, at the end of the day, the Trachtenberg System is really all about memorization. Once you have spent enough time practicing this system, the rules become secondhand. That’s where Dorothy can help. Dorothy uses spaced repetition to increase your ability to memorize, remember and recall anything. It’ll quiz you to the rules every so often so you won’t forget them.