×9 Technique

In the Trachtenberg system multiplying by 9 reduces to three steps: subtract the last digit from 10; subtract each middle digit from 9 and add its right-hand neighbour; subtract 1 from the first digit. For 87×9: ones 10−7=3; next step 9−8+7=8; leading digit 8−1=7 → 783. Single-digit numbers show the same rule: 6×9 gives 10−6=4 and 6−1=5 → 54.

The rule rests on the identity 9 = 10 − 1: the number is multiplied by ten and then subtracted from the result, and the procedure hands you the final digits directly. That is the beauty of the system — even a child who has not memorised the times table can multiply three- and four-digit numbers by 9 without error, using only addition and subtraction.

• Age: 9-10, once the concept of multiplication is familiar.
• Common mistake: subtracting the middle digits from 10 — only the last digit uses 10, the rest use 9.
• Practice tip: verify answers with the digit sum: multiples of 9 always have a digit sum divisible by 9 (783 → 7+8+3=18).