The rule for multiplying by 8 is: Subtract the last digit from 10 and double it. For middle digits, subtract from 9, double it, and add the neighbor. For the first digit, subtract 2.
The final answer is 1,872.
Enter any number to see the step-by-step Trachtenberg process for multiplying by 8.
Both rules are based on subtraction. The rule for 9 is 'subtract from 9/10 and add the neighbor.' The rule for 8 takes this a step further: you subtract from 9/10, *double* the result, and then add the neighbor. The doubling accounts for the difference between 9 and 8.
This is a standard technique in these subtraction-based methods to handle 'borrowing' implicitly. Subtracting the first (units) digit from 10 is the standard operation. Subtracting all subsequent digits from 9 accounts for the 'borrow' that would have occurred from that position.
This is a simplified final step. The full rule would be to add two leading zeros and apply the 'subtract from 9, double, add neighbor' rule to the first leading zero. That calculation simplifies to just taking the neighbor of that zero (which is the first digit of the original number) and subtracting 2.
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Both rules are based on subtraction. The rule for 9 is 'subtract from 9/10 and add the neighbor.' The rule for 8 takes this a step further: you subtract from 9/10, *double* the result, and then add the neighbor. The doubling accounts for the difference between 9 and 8.
This is a standard technique in these subtraction-based methods to handle 'borrowing' implicitly. Subtracting the first (units) digit from 10 is the standard operation. Subtracting all subsequent digits from 9 accounts for the 'borrow' that would have occurred from that position.
This is a simplified final step. The full rule would be to add two leading zeros and apply the 'subtract from 9, double, add neighbor' rule to the first leading zero. That calculation simplifies to just taking the neighbor of that zero (which is the first digit of the original number) and subtracting 2.