Welcome to the Trachtenberg System! We begin with the easiest and most famous rule: multiplying by 11. This method is incredibly fast and requires only simple addition, making it a perfect introduction to the power of this system.
To multiply any number by 11, you work from right to left. For each digit, you simply add it to its "neighbor" (the digit on its immediate right) and any carry from the previous step.
First, add a leading zero to your number: 0678.
Reading the digits gives the final answer: 7,458.
Enter any number to see the step-by-step "Add the Neighbor" method.
Multiplying by 11 is the same as multiplying by 10 and then adding the number once more (11N = 10N + 1N). Adding a zero to the right of a number is the same as multiplying by 10. The 'Add the Neighbor' rule is a step-by-step method of adding the original number to this 10-times-larger version.
This is where carrying comes in, just like in regular addition. If you calculate 7 + 8 = 15, you write down the 5 and carry the 1 over to the next calculation on the left.
Adding a leading zero makes the rule consistent for every digit. Without it, you would need a special final step for the leftmost digit. With the zero, the last step is simply '0 + leftmost digit + carry,' which gives you the final, correct digit for the answer automatically.