The Trachtenberg System

Chapter 0

The Story & the System

Jakow Trachtenberg was a Russian Jewish engineer and mathematician who, imprisoned in Nazi concentration camps between 1941–1945, kept his mind alive by inventing a new system of mental arithmetic. With no pencil or paper, he worked everything out in his head.

After the war he escaped to Switzerland, where he wrote up the system and opened the Hamburg Institute of Mathematics. Children who had been labelled "slow" were solving problems that astonished their teachers.

The core insight: every multiplication can be broken into a chain of simple operations on individual digits — no memorisation of large tables required. You only ever need to hold one or two digits in mind at a time.

The Trachtenberg System replaces hard multiplication with simple patterns: add neighbours, halve, double, subtract from 10 or 9. Each multiplier (2–12) has its own rule. The Direct Multiplication method handles any multiplier.

How to use this course: Each section teaches one method with step-by-step animated worked examples. Use the → Next Step button to walk through each digit. Then use the Practice block to test yourself with random problems.

Quick Reference

Multiply by	The Rule (one sentence)
2	Double each digit; carry if ≥ 10.
3	Double the digit + half the neighbour (use odd-flag). Process right→left.
4	Double the digit twice (or: double-of-double).
5	Half the neighbour; add 5 if the current digit is odd.
6	Digit + half neighbour; add 5 if digit is odd.
7	Double digit + half neighbour; add 5 if digit is odd.
8	Subtract digit from 10 (first digit); add 9 + neighbour's tens (rest). Keep units only + carry.
9	Subtract from 10 (last digit); subtract from 9 + add neighbour (middle); leftmost 1 less.
11	Add each digit to its right-hand neighbour.
12	Double each digit + its right-hand neighbour.
Any	Direct Multiplication: multiply digit pairs using a sliding "reference" column.

💡 Convention used throughout: "Neighbour" always means the digit immediately to the right of the current digit. We always work right to left (units place first), just like in long multiplication. A leading zero is imagined before the first digit.

Chapter 1

Multiplying by 2

The simplest rule — but it establishes the carry pattern used by every other method.

For each digit: double it. Write the units digit of the result. If the result is ≥ 10, carry 1 to the next position.

result_digit = (2 × d + carry_in) mod 10
carry_out = floor((2 × d + carry_in) / 10)

Worked Example — Step Through

Practice: × 2

Random problems · Type your answer · Press Enter

Chapter 2

Multiplying by 3

For each digit d with right-neighbour n:
Compute 2×d + half(n) + (5 if d is odd) + carry.
"Half" means floor(n/2). The "+5 if d is odd" is called the odd-flag.

📐 The odd-flag captures the 0.5 that would otherwise be lost when you halve an odd neighbour. It appears in methods ×3 through ×7.

Worked Example

Practice: × 3

Random problems · Type your answer · Press Enter

Chapter 3

Multiplying by 4

Apply the × 2 rule twice. Or equivalently: double each digit, then double again, propagating carries carefully.

💡 In practice, compute 4×d mentally (you only need 0–9 → 0–36), take the units digit, and carry the tens digit. Only digits 3–9 produce carries.

Worked Example

Practice: × 4

Random problems · Type your answer · Press Enter

Chapter 4

Multiplying by 5

For each digit d with right-neighbour n:
Result digit = half(n) + 5 if d is odd.
(The current digit d itself does not appear directly — only its parity.)

💡 This works because ×5 = ×10/2. You shift right (half the neighbour), and the "+5 if odd" compensates for the lost 0.5 from an odd digit × 5 = odd × 5 always ending in 5.

Worked Example

Practice: × 5

Random problems · Type your answer · Press Enter

Chapter 5

Multiplying by 6

For each digit d with right-neighbour n:
Result digit = d + half(n) + 5 if d is odd + carry.

Think of it as ×5 plus ×1: you keep the digit (×1) and add the ×5 contribution (half-neighbour plus odd-flag).

Worked Example

Practice: × 6

Random problems · Type your answer · Press Enter

Chapter 6

Multiplying by 7

For each digit d with right-neighbour n:
Result digit = 2d + half(n) + 5 if d is odd + carry.

Think ×7 = ×2 + ×5: double the digit, then add the ×5 part (half-neighbour and odd-flag).

Worked Example

Practice: × 7

Random problems · Type your answer · Press Enter

Chapter 7

Multiplying by 8

Rightmost digit: subtract from 10. (10 − d)
All other digits: 9 − d + neighbour's tens complement (i.e. nearest ten above neighbour) ??? — simpler form:
Use (−d − 1 + neighbour + 10) mod 10 with carry tracking.

Simplest practical form: Think of ×8 as ×10 − ×2. Compute ×2 first (mentally), then subtract from the shifted number.

⚠️ ×8 has two common formulations. We teach the complement approach: right digit is (10−d), middle digits are (9−d+n_right) mod 10 with borrow, leftmost gets reduced by 1. This is the authentic Trachtenberg form.

Worked Example

Practice: × 8

Random problems · Type your answer · Press Enter

Chapter 8

Multiplying by 9

Rightmost digit: 10 − d.
Middle digits: 9 − d + right-neighbour.
Leftmost digit: reduce by 1 (or write the tens digit of the 9×left calculation).

💡 ×9 = ×10 − ×1. The "10−d" at the end, "9−d+n" in the middle, and the left adjustment encode this idea digit by digit.

Worked Example

Practice: × 9

Random problems · Type your answer · Press Enter

Chapter 9

Multiplying by 11

For each digit d with right-neighbour n:
Result digit = d + n + carry.

The simplest non-trivial Trachtenberg rule. The rightmost digit is unchanged; then each subsequent position is the sum of the two adjacent digits in the original number.

Worked Example

Practice: × 11

Random problems · Type your answer · Press Enter

Chapter 10

Multiplying by 12

For each digit d with right-neighbour n:
Result digit = 2d + n + carry.

Think ×12 = ×2 + ×10. You double the current digit and add the next one to the right (which accounts for the ×10 shift).

Worked Example

Practice: × 12

Random problems · Type your answer · Press Enter

Chapter 11

Direct Multiplication

Direct Multiplication lets you multiply any two numbers digit by digit — no partial products, no long multiplication layout. You produce the answer one digit at a time, right to left.

The method uses a concept called cross-multiplication or the U-V pattern. For each output position, you sum all products of digit pairs whose positions add up to that column index.

To find output digit at position k (0 = units):
Sum all products a[i] × b[j] where i + j = k, plus any carry.
Write units digit, carry the tens.

📐 For a 2-digit × 2-digit multiplication AB × CD: units = B×D, tens = A×D + B×C + carry, hundreds = A×C + carry.

Worked Example: 3-digit × 1-digit

Extending to 2-digit × 2-digit: the same cross-product logic applies. For 43 × 27:

Worked Example: 2-digit × 2-digit

Practice: Direct ×

2-digit × 1-digit problems

Chapter 12

Trachtenberg Addition

Trachtenberg's addition method handles long columns of numbers without carrying as you go. Instead, you tally small groups then combine.

Two-finger method: Keep a running total in your head but only remember the units digit. When the total reaches 11 or higher, make a slash mark and subtract 10 (continue from the units of the total). Count your slash marks — that's the carry to the next column.

Work one column at a time, bottom to top.
Keep a mental tally. Each time it hits ≥ 10, subtract 10 and mark a tally (|).
The column's contribution = final units digit.
Number of tally marks = carry to next column.
Do NOT try to remember the running "tens" — just tally marks.

Worked Example

Practice: Column Addition

Sum the column · Type total · Press Enter

Final

Mixed Practice Quiz

A randomised set of problems across all methods. Use the Trachtenberg rules — no calculator, no long multiplication. Press Check when done.