The Rule for 4: Subtract from 10

The Trachtenberg rule for 4 is a fascinating technique that relies on subtraction rather than direct multiplication. It showcases the system's creative approach to simplifying complex arithmetic.

The Rule: A 3-Case Process

Working from right to left (with a leading zero):

For the rightmost digit: Subtract it from 10.
For all middle digits: Subtract the digit from 9, then add half of its right-hand neighbor.
For the leading zero: Take half of its neighbor and subtract 1.
Special Rule: At each step, if the digit itself is odd, add an extra 5 to the sum.

Example: Multiply 324 by 4

Add a leading zero: 0324.

Digit 4: $10 - 4 = 6$ . (4 is even). Write 6.
Digit 2: $(9 - 2) + \text{half of } 4 = 7 + 2 = 9$ . (2 is even). Write 9.
Digit 3: $(9 - 3) + \text{half of } 2 = 6 + 1 = 7$ . (3 is odd, so add 5). $7 + 5 = 12$ . Write 2, carry 1.
Digit 0: $(\text{half of } 3) - 1 = 1 - 1 = 0$ . Add carry 1. $0 + 1 = 1$ . Write 1.

The final answer is 1,296.

🔢Multiplication by 4 Lab

Enter any number to see the step-by-step Trachtenberg process for multiplying by 4.

🧠Quick-Fire Quiz!

Frequently Asked Questions

Why subtract from 10 for the first digit, but 9 for the others?

This is a clever way to handle the 'borrowing' that happens in traditional subtraction. Subtracting the first digit from 10 accounts for the units place, while subtracting subsequent digits from 9 accounts for the fact that we have already 'borrowed' from them.

What is the 'add 5 for odd digits' part for?

This rule, like in the rule for 3, is a correction factor. It compensates for the decimal part that is dropped when taking 'half of the neighbor,' ensuring the calculation remains accurate without needing to work with fractions.

Why does the leading zero rule involve subtracting 1?

The final step of taking half the neighbor and subtracting 1 is the last part of the subtraction algorithm. It finalizes the calculation by accounting for the last 'borrow' from the leftmost position of the number.