Urdhva Tiryagbhyam

Meaning: "Vertically and crosswise". This general method for multiplication applies to any two numbers, breaking down the process into simpler vertical and crosswise steps.

How it works (e.g., for 2-digit by 2-digit numbers like AB x CD):

1. Step 1 (Rightmost Vertical): Multiply the rightmost digits vertically: B x D. This gives the rightmost digit(s) of the answer.
2. Step 2 (Crosswise): Multiply diagonally and add the products: (A x D) + (B x C). This gives the middle digit(s) of the answer. If the sum is a two-digit number, carry over the tens digit to the next step.
3. Step 3 (Leftmost Vertical): Multiply the leftmost digits vertically: A x C. Add any carry-over from Step 2 to this product. This gives the leftmost digit(s) of the answer.
4. Combine: Arrange the results from these steps to form the final product.

For larger numbers (e.g., 3-digit by 3-digit: ABC x DEF): The pattern extends. More crosswise products are involved.

• Rightmost: C x F
• Crosswise 1: (B x F) + (C x E)
• Crosswise 2 (Star product): (A x F) + (C x D) + (B x E)
• Crosswise 3: (A x E) + (B x D)
• Leftmost: A x D

Each step's result contributes to a part of the final answer, managing carries as you go from right to left.

Advantages:

• Universal: Works for all multiplication.
• Mental Math: With practice, it can be performed mentally.
• Efficient: Reduces the number of intermediate steps compared to traditional long multiplication.
• Foundation for algebraic multiplication: The same principle applies to multiplying algebraic expressions.