The Trachtenberg System: Fast Mental Multiplication
The Trachtenberg System is a method of rapid mental calculation, built for speed and accuracy and especially powerful for multiplication. Its trick: instead of memorizing times tables, you follow a simple rule for each multiplier. NumDojo breaks every rule into bite-sized lessons with instant feedback, worked examples, and XP rewards.
How the system works
Jakow Trachtenberg developed the system while imprisoned during World War II, working entirely in his head to keep his mind sharp. Rather than one giant multiplication algorithm, it gives each multiplier (2 through 12, and beyond) its own short, repeatable procedure. Because every step is small and mechanical, you make fewer errors, and it scales to numbers of any length.
Multiplying by 11 (the easiest rule)
The perfect starting point. Rule: write the last digit, then add each digit to its right-hand neighbor, and finally write the first digit.
Multiplying by 12
Rule: double each digit and add its right-hand neighbor.
Multiplying by 11 when digits carry
The first example had no carries. Real numbers usually do, so here is the same rule with carrying. When a sum passes 9, write the units digit and carry the ten into the next step.
Multiplying by 6
The larger multipliers use a “half the neighbor” idea. Rule for 6: to each digit add half of its right-hand neighbor (drop any remainder), and add 5 if the digit itself is odd. Pad the number with a leading zero so the first digit gets a turn.
Once the ×6 rule clicks, the rules for 5, 7, and the other multipliers follow the same shape.
Why learn Trachtenberg?
- No memorized times tables, just a simple rule per multiplier
- Systematic steps that work for large multiplications and reduce errors
- Builds number sense alongside raw speed
- Rules you can practise in timed drills and optional 1v1 battles
- Integrated with NumDojo's review queue for long-term retention
How to learn the whole system
- Start with ×11 until you can do it without writing anything down.
- Add ×12, then the “half the neighbor” rules (5, 6, 7).
- Drill each rule against a timer so the steps become automatic.
- Review with spaced repetition so the rules stick for months, not days.
NumDojo runs this whole path for you: one rule per lesson, timed practice, and a review queue that brings each rule back right before you would forget it. The full explainer walks through the history and every rule in order.
Frequently asked questions
What is the Trachtenberg System?
The Trachtenberg System is a set of rapid mental calculation rules, especially for multiplication. Instead of memorizing full times tables, you apply a short, repeatable rule for each multiplier (for example, rules for ×11 or ×12).
Who invented the Trachtenberg method?
Jakow Trachtenberg developed the system while imprisoned during World War II, refining rules he could practice entirely in his head. The published methods are widely taught for speed arithmetic.
Is Trachtenberg good for kids?
Yes. The rules are mechanical and step-by-step, which helps learners ages 8–18 build speed without heavy table memorization. NumDojo sequences the rules from easiest (×11) through more advanced multipliers.
How do I start learning Trachtenberg on NumDojo?
Create a free account, open the lessons curriculum, and begin with multiplying by 11. Worked examples, practice problems, and spaced-repetition review reinforce each rule.
Start the free Trachtenberg path
Create an account, then open Lesson 41, the free introduction to Jakow Trachtenberg and the system.