The Trachtenberg Speed System — Complete Interactive Guide

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Core Principles

Work right to left. Start with the units digit and move left.
Neighbor = the digit immediately to the right of the one you're working on.
“Halve” means floor division: halve(7)=3, halve(8)=4. Ignore remainders.
“Add 5 if odd” — if the current digit is odd, add 5 to your running total for that step.
Leading zero: Most rules start by writing an extra 0 in front of your number.

Multiplication Rules (Interactive)

×0

Multiply by 0

Rule: result is always 0.

Try:

×1

Multiply by 1

Rule: copy the number.

Try:

×2

Multiply by 2

Rule: multiply each digit by 2 with carrying.

Try:

×3

Multiply by 3

Rule: subtract rightmost from 10, others from 9, double, add half neighbor +5 if odd, leading subtract 2

Try (example 492):

×4

Multiply by 4

Rule: subtract rightmost from 10, others from 9, add half neighbor +5 if odd, leading subtract 1

Try (example 346):

×5

Multiply by 5

Rule: half neighbor, add 5 if digit odd

Try (example 43):

×6

Multiply by 6

Rule: add digit + half neighbor, add 5 if digit odd

Try (example 357):

×7

Multiply by 7

Rule: double digit + half neighbor, add 5 if digit odd

Try (example 693):

×8

Multiply by 8

Rule: subtract rightmost from 10, others from 9, double, add neighbor, leading subtract 2

Try (example 456):

×9

Multiply by 9

Rule: subtract rightmost from 10, others from 9, add neighbor, leading subtract 1

Try (example 2130):

×10

Multiply by 10

Rule: add zero

Try:

×11

Multiply by 11

Rule: add digit to neighbor

Try (example 3425):

×12

Multiply by 12

Rule: double digit and add neighbor

Try (example 316):

×13

Bonus: Multiply by 13

Rule: triple digit and add neighbor (brief mention). The system extends: ×13 = 3×digit + neighbor, with carrying. Learn 11 and 12 first, then 13 follows the same pattern.

General Multiplication — Two-Finger Method

For any multiplier, Trachtenberg taught a “two-finger” method using UT pairs (Units and Tens). You compute each answer digit by combining products of pairs of digits from the multiplicand and multiplier. It's like long multiplication but done left-to-right in your head.

Worked example from the book: 123456 × 789 = 97,406,784

The rule: for each answer position, sum the products of digit-pairs where the place values add to that position, plus carries. Practice with the simplified demo below (shows standard partial products):

Multiplicand

Multiplier

Addition & Subtraction (Brief)

Addition: Trachtenberg adds left-to-right, keeping a running total and correcting with “nines complements.” For each column, say the provisional sum, then adjust when the next column causes a carry. This lets you speak the answer as you go.

Subtraction: Use complements. To subtract, add the nines-complement of the subtrahend and drop the leading 1. Think: 753 - 286 → complement of 286 is 713, add: 753+713=1466, drop leading 1 → 466+1 = 467.

Division Overview

Trachtenberg division uses a “ready-made” partial dividend table for the divisor. You work left-to-right estimating one digit at a time, using the same halve/double ideas to avoid trial-and-error. It’s the most advanced part of the system — master multiplication first.