Trachtenberg System Basics

Mental Math · Axiom Academy

LESSON Trachtenberg System Basics One recipe per multiplier, run digit by digit from the right, using each digit s neighbour. 1. One Rule, One Digit at a Time Take 234 × 12 . The recipe for 12 is five words long: double the digit, add the neighbour . Your neighbour is always the digit immediately to your right — never the one to the left, and never a digit from the other number. Start at the far right and walk left. Put your finger on the rightmost digit. It has no neighbour, so treat its neighbour as a 0 . Apply the recipe to those two digits, then add whatever carry is waiting in this column. Write the last digit of that total. Anything above it becomes the carry, and it waits in the next column to the left . Step left and repeat — including at least one step past the front of the number, where the digit is a phantom 0 but the neighbour is real. Four columns, right to left. Read the written digits back the other way. When the front of the number is where it gets interesting In 934 × 12 the third column, 2 × 9 + 3 + 1, comes to 22 — a carry of 2 , not 1. Two more steps are needed past the front of the number before there is nothing left to write: The cards for 5 , 6 and 7 share one strange ingredient: you take half of your neighbour , and if that half does not come out whole you simply throw the leftover half away . On its own that would lose you something. It does not, because of a second ingredient that quietly puts it back: if the digit under your finger is odd, add 5 .

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