The Sanskrit sutra "Urdhva Tiryagbhyam", meaning "Vertically and Crosswise", gives a general technique for reducing long multiplication to a single-line shortcut. It works like this.
To multiply two numbers (of two or more digits), split each number into two parts. If the first number is a1 + b1 and the second number is a2 + b2, then the product of the two numbers is:
(a1 x a2) + (a1 x b2 + b1 x a2) + (b1 x b2)
The solution comprises three parts (as shown by the boxes and arrows above): the head, the middle, and the tail.
- The digits on the right are multiplied vertically to get the tail part: b1 x b2 (excess carried over)
- All digits are multipled crosswise and added together to get the middle part: a1 x b2 + b1 x a2 (excess carried over)
- The digits on the left are multiplied vertically to get the head part: a1 x a2
Here is a simple example to illustrate this technique.
23 x 41 = 943
The steps are:
- 3 x 1 = 3
- 2 x 1 + 3 x 4 = 14, put down 4 and carry over 1
- 2 x 4 = 8, plus the 1 carried over, is 9
The speed gain using this technique (over the conventional method of multi-line long multiplication) becomes more apparent when handling larger numbers. Here is another example involving excess carryover at each stage.
108 x 64 = 6912
The steps are:
- 8 x 4 = 32, put down 2 and carry over 3
- 10 x 4 + 8 x 6 = 88, plus the 3 carried over, is 91; put down 1 and carry over 9
- 10 x 6 = 60, plus the 9 carried over, is 69
This powerful technique can be expanded upon to cover all cases of multiplication, not just two or three-digit numbers.
