Let a₁ and a₂ be digits at units and tens of multiplicand. Similarly, let b₁ and b₂ represent the digits at units and tens of multiplier. With these assumptions, the conventional multiplication is performed to evolve the algorithm for such product finding tasks. A conventional multiplication table is given below in Table 1.

Critical observation of the result row of Table 1 is the basis of evolution of Vedic Sutras or Vedic Formulae for multiplication shortcut. For this reason, Table 2 is made without showing the operations of multiplication.

Result at units b₁a₁ prompts that this is the result of vertical multiplying the digits positioned at units for both multiplier and multiplicand (b₁ and a₁). In Vedic jargon, this is known as urdhvak. It is a Sanskrit word and when it is translated, it means upward or rising up or vertically up. Here a₁ is vertically up or urdhvak to b₁. This is the Vedic formula to get quickly the digit of product at units. This is marked by arrows here for better understanding.

Observation of result at Tens b₂a₁ + b₁a₂ points out that the sum of crosswise multiplication 1) of digit at Units of multiplier with digit at Tens of multiplicand and crosswise multiplication 2) of digit at Tens of multiplier with digit at Units of multiplicand. The result at Hundreds, i.e., b₂a₂ indicates that this is vertical or urdhvak multiplication of digits at the tens for both multiplier and multiplicand. This is similar to getting result at units.

Lemma: In all product finding cases, the result of first place, i.e., units and last place (depending on number of digits) are vertical (urdhvak) products.

It may be noted that like conventional multiplication, carry, if any, is to be carried forward for addition to next higher place value.

Example 1: find product 34 and 67.

The above problem is tabulated in Table 3.

So, 34 * 67 = 2.278.

Example 2: Find 78 * 59.

Orally 9 * 8 = 72. 2 is kept as result at units whereas 7 is carry. 9 * 7 + 5 * 8 = 63 + 40 = 103. Adding carry 7 to it, it is 110. So, 0 is retained as answer digit at tens and 11 is carry. Lastly 7 * 5 = 35. Adding carry 11 to it, it is now 46. This goes to hundreds. Therefore the final answer is 4602.

Let a₁, a₂ and a₃ be digits at units, tens and hundreds respectively of multiplicand. Similarly, let b₁, b₂ and b₃ be digits at units, tens and hundreds respectively of multiplier. With these assumptions, the conventional multiplication is performed to evolve the Vedic algorithm for product finding tasks. Conventional multiplication table is given in Table 4.

From the result row in Table 4, it can be visualized that the Vedic formulae for finding product’s digits at units and ten-thousands would be same as in case of multiplication of 2-digit variable number by another 2-digit variable number. Here ten-thousandth place is the last position and hence the last digits of multiplicand and multiplier will be multiplied vertically (urdhvak). As in previous case, Table 5 is made after deleting the rows for operations and keeping result at only hundreds.

Observations of results at hundreds:

1) b₁a₃ is found after multiplying multiplier’s digit at tens with multiplicand’s digit at hundred. This is shown by slant arrow i.e. crosswise or tiryak of Vedic

multiplication.

2) b₃a₁ is also obtained by multiplying multiplier’s digit at hundreds by multiplicand’s digit at units crosswise or tiryak way.

3) b₂a₂ is the result of finding product of digits at tens for both multiplicand and multiplier vertically or in urdhvak way.

Table 5 is now made after deleting all results save result at thousands to find Vedic algorithm to get product’s digit at this place. Also, forget the digits at units now and onwards for the sake of calculations. Therefore, the column for units has been deleted.

Observations of result at thousands are in Table 4.

1) b₂a₃ is crosswise or tiryak product of multiplier’s digit at tens and multiplicand’s digit at hundred. This is shown by arrow in Table 6.

2) Similarly, b₃a₂ is also crosswise or tiryak product of multiplier’s digit at hundreds and multiplicand’s digit at tens. This is shown by arrow in Table 6.

Table 3 is reproduced below as Table 7 to observe result at ten-thousands. Digit at ten-thousands is b₃a₂, which is nothing but vertical multiplication of digits at hundreds for both multiplier and multiplicand. It is noteworthy to keep in mind that results at extreme places (here units and ten-thousands) are always vertical multiplications. This rule is universally applicable to multiplication of any-digit number by other any-digit number.

Example 1: Find product of 467 and 235.

The above problem is tabulated below in Table 8.

Therefore, 467 * 235 = 109,745.

Proceeding this way, one can find formulae for getting the result of multiplication of an n-digit multiplicand by n-digit multiplier. As an example, an 8-digit

multiplicand is multiplied by 8-digit multiplier in order to show the efficacy of the findings of this research. The Vedic formulae are worked out on the basis of conventional multiplication output. Table 9 shows the conventional multiplication of an 8-digit multiplicand by 8-digit multiplier. From this conventional multiplication result, the product of any two 8-digit numbers can be orally found. The end results of conventional product finding in Table 9 is used to get the Vedic formulae for such product finding. The next table, Table 10, uses this algorithm to find the product of 8-digit multiplicand and 8-digit multiplier orally.

Example: Find the product of 10,231,021 and 21,021,103.

This example problem is tabulated in Table 10.

Therefore, 10,231,021 * 21,021,103 = 21,506,734,623.