# What is the Trachtenberg Method: All You Need to Know

9 min read · Aug 12, 2024

## Table of contents

Hi there!

Welcome to the thirteenth chapter in my mental math series.

Today we'll answer what is the Trachtenberg Method? (along with any other question related to it)

### Introduction

The Trachtenberg Method is a system of rapid mental calculations, without the need for pen and paper.

Specifically, it focuses on multiplication, division, and addition.

It has an interesting way of breaking down big calculations into baby steps that you can do quickly. For example, when you're multiplying by 11, you just add up the numbers next to each other, starting from the right (I'll explain this later).

One of the cool tricks in this system is what they call the "neighbor" concept. You look at each number and think about how it relates to the number next to it on the right. They also talk about "leading zeros" for some multiplications.

All this stuff helps you keep track of what you're doing in your head.

### Why Learn the Trachtenberg Method?

It's got some pretty sweet benefits.

First of all, you'll be able to do all sorts of complicated calculations in your head, fast. According to people who studied its effectiveness, it's designed to make you about 20% faster at doing math.

Moreover, people who get good at this often find they can concentrate better, remember more stuff, and solve problems more easily.

## Historical Background

### Origin and Development

Alright, let's talk about where this Trachtenberg Method came from. It's quite a story.

Jakow Trachtenberg, a Russian Jewish engineer and mathematician, came up with this system while he was stuck in Nazi concentration camps during World War II.

Now, in these camps, you couldn't exactly whip out a calculator or even a pencil and paper. So Trachtenberg figured out how to do all these complex calculations in his head. He took his engineering smarts and broke down all the tricky math into smaller, easier steps.

Why? It helped him keep his mind sharp and hold onto hope in a really dark time.

Over time, he kept practicing and tweaking his method, making it better and better. He came up with tricks for adding, multiplying, and dividing- all designed for you to do quickly in your head without any help.

### Jakow Trachtenberg's Life and Contributions

Since he's a fascinating character, let me tell you a bit more about the man himself.

He was born in Odessa back in 1888, and right from the get-go, he was a smart guy. He graduated top of his class from from the Mining Engineering Institute in St. Petersburg. Then, before he was even 25, he was running the show at the Obukhov arms factory, bossing around 11,000 men.

Even though he was working in an arms factory, Trachtenberg was all about peace. During World War I, he set up this group called the Society of Good Samaritans. They taught Russian students how to take care of wounded people.

This made him earn recognition even from the Tsar (Nicholas II).

After escaping from a labor camp in Trieste in 1945, Trachtenberg and his wife and ended up in Switzerland. That's where he set up his Mathematical Institute in Zurich. This place was all about teaching his method to other people.

By the way, Trachtenberg wasn't just about numbers. He also came up with a way to teach foreign languages that a bunch of German schools started using.

## Core Techniques: Overview of Addition, Multiplication, and Division

We'll go into more details in the next chapter. For now this is just a quick overview.

### Addition Techniques

The big idea here is something called the "**rule of 11**." Basically, when you're adding up numbers in columns, you never let the total go over 11.

Here's how it works: You line up all your numbers in columns. If some numbers are shorter, you just slap some zeros on the front to make them all the same length.

Now, here's the cool part. You start counting up or down the column, and if you hit 11 or more, you immediately knock off 11 and put a little tick mark there. This helps manage larger sums without the need for complex carrying operations.

This is way different from how you normally add numbers:

- You don't have to carry big numbers, which is a real brain-saver.
- That "rule of 11" keeps things nice and tidy.
- Those tick marks help you keep track of what you're doing.
- You can just keep rolling through the addition without stopping to carry numbers all the time.

### Multiplication Techniques

These techniques are divided into two main categories: the general multiplication method (for big numbers) and specialized methods (for small numbers).

The general multiplication method involves adding zeros to the front of the first number, as many as there are digits in the second number.

So if you're multiplying 972 by 18, you'd write it as 00972 x 18. You then proceed using a system with outer and inner pairs of digits.

For multiplying by small numbers from 2 to 12, Trachtenberg's got some special shortcuts:

- Multiplying by 11? Just add up pairs of digits, starting from the right.
- Multiplying by 12? Double the number and add it to ten times the number.
- Multiplying by 5? Cut the number in half and multiply by 10.
- Multiplying by 9? Subtract 1 from the number, then subtract that from 10 times the number.

These tricks are easy to remember and quick to use (once you get the hang of them).

### Division Techniques

Instead of writing everything out like in long division, Trachtenberg sets it up with three rows: two for dividends and one for a working figure. This setup lets you tackle division step by step in your head.

One of the big ideas here is using partial dividends. Instead of trying to chew on the whole big number at once, you break it down into smaller bites. This makes it easier to do the math in your head and helps you avoid making mistakes.

The coolest part? Once you get good at this, you can do it all in your head. No need to write anything down!

## Advantages of the Trachtenberg Method

### Speed and Efficiency

According to the book "The Trachtenberg Speed System of Basic Mathematics," (which I recommended before) a student using the Trachtenberg Method was able to multiply two 10-digit numbers (5132437201 times 452736502783) in just 70 seconds.

Now, I'm not saying you'll be doing that too... But it shows you what's possible when you get good at this stuff!

Like I said before, you can expect to do math about 20% faster than the old-fashioned way.

### Mental Math Improvement

The more you practice this Trachtenberg stuff, the better you get at mental math overall. Here's how it helps:

- You start seeing patterns in numbers, enhancing general numerical fluency.
- You learn how to break big problems into smaller ones.
- You get better at picturing numbers in your head (which I know from personal experience how hard this is!!!!)
- Your working memory (that's the part of your brain that juggles information while you're using it) gets stronger.

### Error Reduction

Here's another cool thing about Trachtenberg's method - it helps you make fewer mistakes.

The way it works is pretty slick:

- It breaks everything down into simple steps. That way, you're less likely to mess up because you're not trying to do too much at once.
- Every operation follows the same rules every time. It's like having a map - you always know where you're going.
- You do everything step-by-step. This means if you do make a mistake, you'll catch it early before it snowballs into something bigger.

But wait, there's more! Trachtenberg even built in ways to check your work. There's this trick using nines and elevens that can tell you if your answer is right 99% of the time. And you can do it quick, without having to redo the whole problem.

## Applications of the Trachtenberg Method

### Academic Settings

In math class, using Trachtenberg's tricks can make you look like a regular Einstein. It helps you crunch numbers faster, and it makes you really understand how numbers work together.

That's not just good for arithmetic - it'll help you when you get to the tougher stuff like algebra too.

But here's where it gets really exciting - tests. You know those big, scary standardized tests like the SAT or ACT? Well, Trachtenberg's method can give you a real edge there. These tests are all about speed and accuracy, and that's exactly what this method gives you. You'll be zooming through the math sections, leaving more time for the tricky questions. And you know what that means? Higher scores, my friend.

### Professional Applications

Now, you might be thinking, "Okay, but what about when I grow up?" Well, let me tell you, this Trachtenberg stuff isn't just for school - it's got real-world applications too.

Take finance, for example. Imagine being able to calculate percentages or interest rates in your head faster than the person next to you.

Or how about engineering? Those folks are always needing to do quick checks on their calculations. With Trachtenberg's method, they can do mental math to verify what their computers are telling them.

### Everyday Life

But here's the best part - this stuff is useful in everyday life too.

Think about when you're out shopping. You can figure out discounts in your head, compare prices between different sized packages, all that stuff.

And when it comes to managing your own money? Forget about it. You'll be budgeting like a pro, figuring out how much to save, splitting bills with your friends - all in your head.

## Learning Curve and Accessibility

### How Long Does it Take to Learn the Trachtenberg Method?

Like I said before, don't expect to be a math wizard overnight, but you'd be surprised how quickly you can pick up some of these tricks!

Those easy ones, like multiplying by 11 or doing simple addition with that "rule of 11"? Most folks can get the hang of those in **just a few days**.

Now, the trickier stuff, like the fancy multiplication techniques or the advanced division methods, that's going to take a bit more elbow grease. But don't get discouraged! Most people start seeing real improvements after **a month or two of regular practice**.

And if you stick with it, you'll just keep getting better and better 🙂

Of course, how fast you learn depends on a few things:

- If you're already pretty good at math, you might pick it up quicker.
- Practicing a little bit every day works better than cramming once in a while.
- Some people learn better by seeing things, others by doing. Figure out what works for you.

### Who Can Benefit from Learning this Method?

Now, here's the best part - this Trachtenberg Method isn't just for math geniuses or kids in school. It's for everybody!

If you're in a job where you need to crunch numbers fast, like in finance or engineering, this could be your secret weapon. And even if you're not a "math person," these tricks can help you with everyday stuff like figuring out tips or managing your money.

And get this - it's great for older folks too! It's like a workout for your brain.

## Comparison with Other Mental Math Techniques

We've talked about traditional mental math, vedic math, and the abacus method.

But here's why Trachtenberg's method is special:

- It's like a recipe - you follow the same steps every time.
- You don't have to memorize a bunch of stuff.
- It's got built-in ways to check your work.
- You can start using it pretty quick - most folks see improvements in just a few weeks.
- It's all in your head - no tools needed.
- It works for big numbers too, which is great if you're into engineering or money stuff.
- It gives you different ways to solve problems, which makes your brain more flexible.

But again, all methods have their good points.

## Criticisms and Limitations

Now, don't go thinking this Trachtenberg stuff is perfect. It's got its problems too, and we should talk about them.

Some folks say it **focuses too much on being fast and not enough on really understanding math** (a bit like learning to play a song really quickly without knowing how to read music).

Another thing is that it's mostly for basic math - adding, multiplying, dividing. It doesn't help much with the harder stuff like algebra or geometry.

Learning all of Trachtenberg's tricks can take a while too. Some people might not have the time or need for it, especially if they're not doing a lot of math in their daily life.

But don't let all this scare you off! Even with these drawbacks, a lot of people say Trachtenberg's method is great for making your brain quicker, helping you calculate faster, and making you feel more confident about math. It's not meant to replace regular math learning, but it can be a great add-on.

Remember, no method is perfect. The key is to use what works for you and helps you understand better. If Trachtenberg's method does that for you, then go for it!

## Conclusion

And that's all for today!

I hope you enjoyed this overview. In the final chapter, we'll take a deeper look at how beginners can approach the Trachtenberg Method.

See you in the next chapter 🙂 Be well.

### About the author

#### Richard Reis

Founder of Mental Math Pro! Also currently doing a fun little "25 AI Startups in 50 Weeks" challenge.