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The Vedic multiplication algorithm is a very fast way of oral calculation. However, the basis of the algorithm is not available so far. The present paper demystifies the general Vedic algorithm for multiplication by establishment of foundation of the Vedic algorithm of product finding through end results of conventional multiplication. This novel approach, i.e., finding algorithm from the end results of conventional calculations may be useful in devising algorithms similar to Vedic in cases of other calculations. Though the availability of cheap calculators made the Vedic Method obsolete, the present trend resurrected Vedic algorithms by their use in the design of computer processors for enhancing speed and performance.

Vedic mathematics [

There are several Vedic algorithms for multiplication including a general one, which is universal, whereas each of other Vedic algorithms for multiplication suits a special situation.

Scope of this research is confined to finding the logic of Vedic Multiplication Algorithm for multiplication of general type as it has universal application and this finds product of a several-digit multiplicand and several-digit multiplier.

Interestingly, this research is purely based on end results of conventional multiplication procedure and this approach may be used in cases of other complicated mathematical procedures for finding shortcut-algorithms like the Vedic approach.

Results of this avant-garde research are very encouraging and verifiable. Besides, it may be path finder.

It may be interesting to note that the Vedic algorithm is based on the reorganization of conventional multiplication method only. In Vedic jargon, the algorithmic procedure is described in terms of urdhvak (vertical multiplication) and tiryak (crosswise multiplication) operations. The purpose of this work is to handover the very basis of the Vedic formulae to the users which can be a very valuable asset to them for clear understanding.

The ordinary multiplication method is used here as a vehicle in evolving the Vedic Sutra (Algorithm) starting with multiplication of 2-digit multiplicand and 2-digit multiplier. Also, alphabetical symbols shall be used to represent digits in doing so. This will be extended later for finding products of a several-digit multiplicand by another several-digit multiplier.

Let a_{1} and a_{2} be digits at units and tens of multiplicand. Similarly, let b_{1} and b_{2} 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

Critical observation of the result row of

Result at units b_{1}a_{1} prompts that this is the result of vertical multiplying the digits positioned at units for both multiplier and multiplicand (b_{1} and a_{1}). 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 a_{1} is vertically up or urdhvak to b_{1}. 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 b_{2}a_{1} + b_{1}a_{2} 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., b_{2}a_{2} 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

Tags for position of digits for place values Description, operations and result | Hundreds | Tens | Units |
---|---|---|---|

Multiplicand | a_{2} | a_{1} | |

Multiplier | b_{2} | b_{1} | |

First operation | b_{1}a_{2} | b_{1}a_{1} | |

Second operation | b_{2}a_{2} | b_{2}a_{1} | X |

Result of product | b_{2}a_{2} | b_{1}a_{2} + b_{2}a_{1} | b_{1}a_{1} |

Tags for position of digits for place values Description, operations and result | Hundreds | Tens | Units |
---|---|---|---|

Multiplicand | a_{2}_{ } | a_{1} | |

Multiplier | b_{2}_{ } | b_{1 } | |

Result of product | b_{2}a_{2} | b_{2}a_{1} + b_{1}a_{2} | b_{1}a_{1} |

Tags for position of digits for place values Description, operations and result | Hundreds | Tens | Units |
---|---|---|---|

Multiplicand | 3 | 4 | |

Multiplier | 6 | 7 | |

Vedic operations | 6 * 3 = 18 Add carry 4 to it: 18 + 4 = 22 | 7 * 3 + 6 * 4 = 45 Add carry 2, it is 47 | 7 * 4 = 28 |

Resulting Digit at place | 22 | 7 | 8 |

Carry | 4 | 2 |

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 a_{1}, a_{2} and a_{3} be digits at units, tens and hundreds respectively of multiplicand. Similarly, let b_{1}, b_{2} and b_{3} 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

From the result row in

Observations of results at hundreds:

1) b_{1}a_{3} 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

Tags for position of digits for place values Description, operations and result | Ten-thousands | Thousands | Hundreds | Tens | Units |
---|---|---|---|---|---|

Multiplicand | a_{3} | a_{2} | a_{1} | ||

Multiplier | b_{3} | b_{2} | b_{1} | ||

First operation | b_{1}a_{3} | b_{1}a_{2} | b_{1}a_{1} | ||

Second operation | b_{2}a_{3} | b_{2}a_{2} | b_{2}a_{1} | X | |

Third operation | b_{3}a_{2} | b_{3}a_{2} | b_{3}a_{1} | X | |

Result of product | b_{3}a_{2} | b_{2}a_{3} + b_{3}a_{2} | b_{1}a_{3} + b_{2}a_{2} + b_{3}a_{1} | b_{1}a_{2} + b_{2}a_{1} | b_{1}a_{1} |

Tags for position of digits for place values Description, operations and result | Ten-thousands | Thousands | Hundreds | Tens | Units |
---|---|---|---|---|---|

Multiplicand | a_{3} | a_{2} | a_{1} | ||

Multiplier | b_{3} | b_{2} | b_{1}_{ } | ||

Result of product | b_{1}a_{3} + b_{2}a_{2} + b_{3}a_{1} |

multiplication.

2) b_{3}a_{1} is also obtained by multiplying multiplier’s digit at hundreds by multiplicand’s digit at units crosswise or tiryak way.

3) b_{2}a_{2} is the result of finding product of digits at tens for both multiplicand and multiplier vertically or in urdhvak way.

Observations of result at thousands are in

1) b_{2}a_{3} is crosswise or tiryak product of multiplier’s digit at tens and multiplicand’s digit at hundred. This is shown by arrow in

2) Similarly, b_{3}a_{2} is also crosswise or tiryak product of multiplier’s digit at hundreds and multiplicand’s digit at tens. This is shown by arrow in

_{3}a_{2}, 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

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

Tags for position of digits for place values Description, operations and result | Ten-thousands | Thousands | Hundreds | Tens |
---|---|---|---|---|

Multiplicand | a_{3}_{ } | a_{2} | ||

Multiplier | b_{3}_{ } | b_{2} | ||

Result of product | b_{2}a_{3} + b_{3}a_{2} |

Tags for position of digits for place values Description, operations and result | Ten-thousands | Thousands | Hundreds | Tens | Units |
---|---|---|---|---|---|

Multiplicand | a_{3} | a_{2} | a_{1} | ||

Multiplier | b_{3} | b_{2} | b_{1} | ||

First operation | b_{1}a_{3} | b_{1}a_{2} | b_{1}a_{1} | ||

Second operation | b_{2}a_{3} | b_{2}a_{2} | b_{2}a_{1} | X | |

Third operation | b_{3}a_{2} | b_{3}a_{2} | b_{3}a_{1} | X | |

Result of product | b_{3}a_{2} | b_{2}a_{3} + b_{3}a_{2} | b_{1}a_{3} + b_{2}a_{2} + b_{3}a_{1} | b_{1}a_{2} + b_{2}a_{1} | b_{1}a_{1} |

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.

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

Tags for position of digits for place values Description, operations and result | Ten-thousands | Thousands | Hundreds | Tens | Units |
---|---|---|---|---|---|

Multiplicand | 4 | 6 | 7 | ||

Multiplier | 2 | 3 | 5 | ||

Operations | 4 * 2 = 8 Adding 2 carry it is 10 | 4 * 3 + 2 * 6 = 24 Adding 5 carry it is 29 | 5 * 4 + 2 * 7 + 6 * 3 = 52. Adding 5 carry it is 57 | 5 * 6 + 3 * 7 = 51 Adding 3 carry it is 54 | 7 * 5 = 35 |

Resulting Digit | 10 | 9 | 7 | 4 | 5 |

Carry | 2 | 5 | 5 | 3 |

Tags for position of digits for place values | Hundred-quadrillion | Ten- quadrillion | Quadrillion | Hundred-trillions | Ten-trillions | Trillions | Hundred-millions | Ten-millions | Millions | Hundred-thousands | Ten-thousands | Thousands | Hundreds | Tens | Units |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Description, operations and result | |||||||||||||||

Multiplicand | a_{8} | a_{7} | a_{6} | a_{5} | a_{4} | a_{3} | a_{2} | a_{1} | |||||||

Multiplier | b_{8} | b_{7} | b_{6} | b_{5} | b_{4} | b_{3} | b_{2} | b_{1} | |||||||

Operational calculations | b_{1}a_{8} | b_{1}a_{7} | b_{1}a_{6} | b_{1}a_{5} | b_{1}a_{4} | b_{1}a_{3} | b_{1}a_{2} | b_{1}a_{1} | |||||||

b_{2}a_{8} | b_{2}a_{7} | b_{2}a_{6} | b_{2}a_{5} | b_{2}a_{4} | b_{2}a_{3} | b_{2}a_{2} | b_{2}a_{1} | x | |||||||

b_{3}a_{8} | b_{3}a_{7} | b_{3}a_{6} | b_{3}a_{5} | b_{3}a_{4} | b_{3}a_{3} | b_{3}a_{2} | b_{3}a_{1} | x | |||||||

b_{4}a_{8} | b_{4}a_{7} | b_{4}a_{6} | b_{4}a_{5} | b_{4}a_{4} | b_{4}a_{3} | b_{4}a_{2} | b_{4}a_{1} | x | |||||||

b_{5}a_{8} | b_{5}a_{7} | b_{5}a_{6} | b_{5}a_{5} | b_{5}a_{4} | b_{5}a_{3} | b_{5}a_{2} | b_{5}a_{1} | x | |||||||

b_{6}a_{8} | b_{6}a_{7} | b_{6}a_{6} | b_{6}a_{5} | b_{6}a_{4} | b_{6}a_{3} | b_{6}a_{2} | b_{6}a_{1} | x | |||||||

b_{7}a_{8} | b_{7}a_{7} | b_{7}a_{6} | b_{7}a_{5} | b_{7}a_{4} | b_{7}a_{3} | b_{7}a_{2} | b_{7}a_{1} | x | |||||||

b_{8}a_{8} | b_{8}a_{7} | b_{8}a_{6} | b_{8}a_{5} | b_{8}a_{4} | b_{8}a_{3} | b_{8}a_{2} | b_{8}a_{1} | x | |||||||

Result | b_{8}a_{8} | b_{7}a_{8}+ b_{8}a_{7} | b_{6}a_{8}+ b_{7}a_{7}+ b_{8}a_{6} | b_{5}a_{8}+ b_{6}a_{7}+ b_{7}a_{6}+ b_{8}a_{5} | b_{4}a_{8}+ b_{5}a_{7}+ b_{6}a_{6}+ b_{7}a_{5}+ b_{8}a_{4} | b_{3}a_{8}+ b_{4}a_{7}+ b_{5}a_{6}+ b_{6}a_{5}+ b_{7}a_{4}+ b_{8}a_{3} | b_{2}a_{8}+ b_{3}a_{7}+ b_{4}a_{6}+ b_{5}a_{5}+ b_{6}a_{4}+ b_{7}a_{3}+ b_{8}a_{2} | b_{1}a_{8}+ b_{2}a_{7}+ b_{3}a_{6}+ b_{4}a_{5}+ b_{5}a_{4}+ b_{6}a_{3}+ b_{7}a_{2}+ b_{8}a_{1} | b_{1}a_{7}+ b_{2}a_{6}+ b_{3}a_{5}+ b_{4}a_{4}+ b_{5}a_{3}+ b_{6}a_{2}+ b_{7}a_{1} | b_{1}a_{6}+ b_{2}a_{5}+ b_{3}a_{4}+ b_{4}a_{3}+ b_{5}a_{2}+ b_{6}a_{1} | b_{1}a_{5}+ b_{2}a_{4}+ b_{3}a_{3}+ b_{4}a_{2}+ b_{5}a_{1}+ | b_{1}a_{4}+ b_{2}a_{3}+ b_{3}a_{2}+ b_{4}a_{1} | b_{1}a_{3}+ b_{2}a_{2}+ b_{3}a_{1} | b_{1}a_{2}+ b_{2}a_{1} | b_{1}a_{1} |

Tags for position of digits for place values | Hundred-quadrillion | Ten- quadrillion | Quadrillion | Hundred-trillions | Ten-trillions | Trillions | Hundred-millions | Ten-millions | Millions | Hundred-Thousands | Ten- thousands | Thousands | Hundreds | Tens | Units |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Description, operations and result | |||||||||||||||

Multiplicand | 1 | 0 | 2 | 3 | 1 | 0 | 2 | 1 | |||||||

Multiplier | 2 | 1 | 0 | 2 | 1 | 1 | 0 | 3 | |||||||

Operations | 2 * 1 | 1 * 1+ 2 * 0 | 0 * 1 2 * 2+ 0 * 1 | 2 * 1+ 2 * 3+ 0 * 0+ 2 * 1 | 1 * 1+ 2 * 1+ 0 * 2+ 1 * 3+ 2 * 0+ | 1 * 1+ 2 * 0+ 0 * 1+ 1 * 1+ 2 * 2+ 0 * 3 | 0 * 1+ 2 * 2+ 0 * 1+ 1 * 0+ 1 * 2+ 0 * 1+ 3 * 2 | 3 * 1+ 2 * 1+ 0 * 0+ 1 * 2+ 2 * 1+ 0 * 0+ 3 * 1+ 2 * 1+ | 3 * 0+ 1 * 1+ 2 * 0+ 0 * 2+ 3 * 1+ 2 * 0+ 1 * 1 | 3 * 2+ 0 * 1+ 0 * 3+ 2 * 2+ 1 * 1+ 1 * 0 | 3 * 3+ 2 * 1+ 1 * 0+ 1 * 2+ 1 * 0 | 3 * 1+ 1 * 1+ 0 * 0+ 1 * 2 | 3 * 0+ 1 * 1+ 2 * 0 | 3 * 2+ 0 * 1 | 3 * 1 |

Result | 2 | 1 | 5 | 0 | 6 | 7 | 3 | 4 | 6 | 2 | 3 | 6 | 1 | 6 | 3 |

Carry | 1 | 1 | 1 | 1 | 1 |

This example problem is tabulated in

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

1) The basis of the Vedic general multiplication algorithm can be found from the end results of ordinary multiplication. This demystifies the Vedic algorithm for general multiplication.

2) The approach of this paper may be used to find algorithm shortcuts for other types of arithmetical calculations.

3) A Vedic algorithm for the multiplication of a n-digit multiplicand by n-digit multiplier can be easily obtained by using alphabetic variables in lieu of numerals in the general conventional multiplication.

4) Vedic mathematics will continue in spite of the availability of electronic calculators or other calculating gadgets.

The authors express their sincere gratitude to the editor and his team for valuable guidance. Further, the authors are indebted to Mr. Prakash Atul, Director High Radius Corporation, Houston, Texas (US), Mrs. Apoorva Mathur, Mrs Richa, Mr. Nitish and Mrs. Reeta Mathur for encouraging them to conduct research in this field and for providing assistance in verification as well as computerization of this paper.

Mathur, M. and Aarnav (2017) Demystification of Vedic Multiplication Algorithm. American Journal of Computational Mathematics, 7, 94-101. https://doi.org/10.4236/ajcm.2017.71008