त्रिभुजस्य फलशरीरं समदल•ôôटी भुजार्घ संवर्गः ।
Area = 1/2 AB x CP (Aryabhatta)चतुरघि•¢ शतमष्टगुणं द्वाषष्टिस्तथा सहस्नाणां ।
अयुतद्वय विष्•म्भस्य आसन्नो वृत्तपरिणाहः ॥
AB2 = AC2 + BC2 (Baudhayana)In addition to Mt. Rushmore, one of Gutzon Borglum's great works as a sculptor is the face of Lincoln which sits in Washington, D.C. He cut it from a large square block of stone in his studio. One day, when the face of Lincon was just becoming recognisable out of the stone, a young girl was visiting the studio with her parents. She looked at the half-done face of Lincoln, her eyes registering wonder and astonishment. She stared at the piece for a moment and then ran to the sculptor.
"Is that Abraham Lincoln?" she asked.
"Well," said the little girl, "how in the world did you know he was inside there?'
The Vedas are the storehouse of all the wisdom of the universe. They represent an inexhaustible mine of the profoundest wisdom in matters both spiritual and temporal. However, for most of us, the message of our ancient Indian Vedic lore and the knowledge therein is a cryptic mystery masked in a mystical language. Yet, the great sages of the past were able to see the portrait of wisdom concealed within our Vedic scriptures.
They were not sculpted by the chisel and hammer of the ordinary systematic enquiry methods of modern science. But, they are the eternal truths revealed by God to the ancient rishis, who through their spiritual endeavours, were competent enough to receive them from a source which was perfect and all-knowing. Through their spiritual purity the great sages of the past had acquired an immense knowledge in all reailms of life.
That is why, time and again, the great men of antiquity - the sculptors of wisdom - have carved out from these sacred books, and made recognisable, the face of knowledge. As a result, the common man's mind has registered wonder and astonishment at their profound discoveries. The ones that surprise and interest most are those along scientific lines.
It was Aryabhatta who sculpted the notion that the earth revolves around the sun, 1000 years before Copernicus.
It was the ancient Indian astronomer, Bhaskaracharya, who discovered the law of gravity 1200 years before Sir Isaac Newton.
It was the Indian mathematician, Baudhayana, who carved out the so called Pythagoras' Theorem long before Pythagoras.
It was India that moulded geometry 1200 years before it was introduced to Europe in the 16th century.
These may seem abstract statements, but they become more plausible when we consider the great extent to which mathematics was developed in the Vedas.
The maths was not developed for entertainment, but for a purpose. The foundation-stone of science is mathematics. To advance scientifically one has to advance mathematically. The fact that such advanced mathematical concepts and their amazingly simple application existed in the Vedas indicates the extent to which science was developed.
'Vedic Mathematics', as it is called, remained hidden in the Vedas for many centuries. Then the late Shankaracharya, Bharati Krishna Tirthaji Maharaj, extracted the 'Sixteen Simple Mathematical Formulae' from the Vedas. He reconstructed them from the 'Atharvaveda' after assiduous research and penance for about eight years in the forests surrounding Sringiri.
Initially, he wrote 16 volumes - one for each formula. The manuscripts were placed in the house of one of his disciples, from where they were misplaced and lost forever. It was a colossal loss. But the Shankaracharya was not perturbed, saying that everything was stored in his memory and that he could re-write the 16 volumes.
In 1957, during his foreign tour, he wrote one volume - an introductory account of the 16 formulae. A month later his health deteriorated and a short while later he passed away, leaving only one introductory volume.
Vedic Mathematics presents a strikingly new theory and method, now almost unknown. The method is obviously radically different from the one adopted by the modern mind.
The following are introductory examples which will help demonstrate this point.
- Vulgar Fractions
- The conversion of vulgar fractions whose numerator is 1 and whose denominator ends with a 9 i.e. 1/19, 2/29, 1/39 etc. can be easily done using the 'one more than previous' formula. To demonstrate let us take 1/19. Converting 1/19 using conventional methods requires dividing 19 by 100, guessing how many times 19 will fit into 100, finding the remainder, etc. But the Vedic method is quite simple. (Follow along on a separate sheet of paper).
- Step 1) Start by placing 1 as the last digit (i.e. the right-hand-most digit) of the answer.
- Step 2) Proceed to multiply leftward, continually multiplying by the 'factor'. The factor = 1 + the penultimate (second-to-last) number of the denominator (19). In this case the penultimate number is 1. So the factor = 1 + 1 = 2.
- Step 3) So our first multiplication should be 1x2 which gives 2. The result should look as follows: 21
- Step 4) Next multiply the 2 by the factor (2) to get 4. The result should look as follows: 421
- Step 5) The 4 is multiplied by the factor to get 8. Then when 8 is multiplied by the factor the result is 16. Here the 6 is placed in the answer while the 1 is carried. The result should be as follows: 168421.
- Step 6) Then, when 6 is multiplied by the factor the result is 12, but the 1 which was carried in the previous step has to be added, giving 13. The 3 is placed in the answer and again the 1 is carried. At this point your answer should look as follows: 1368421.
- Step 7) This process of multiplying and carrying should be continued until the multiplication results in a figure equivalent to the difference between the denominator (19) and the numerator (1) which is 18 (i.e. 19-1 = 18). So in this case we stop at 947368421. Since 9 x 2 (the factor) = 18 and at this point, for this example, our answer should look as follows: 947368421
- Step 8) Now take the 9's compliment of this figure to arrive at the first half of the answer as follows:
- Step 9) Now put the two halves together to get the complete answer. You should arrive at the following answer:
This can be verified on your calculator. (If it is capable of being this accurate!)
To arrive at the answer from left to right we use the division process:
- Step 1) Dividing 1 (the first digit of the dividend) by 2, (the factor) we see the quotient is zero and the remainder is 1. We therefore set 0 down as the first digit of the quotient and prefix the remainder (1) to that very digit of the quotient and thus obtain 10 as our next dividend.
- Step 2) Dividing this 10 by 2, we get 5 as the second digit of the quotient; and as there is no remainder (to be prefixed thereto), we take up that digit 5 itself as our next dividend.
- Step 3) So, the next quotient digit is 2; and the remainder is 1. We therefore, put 2 down as the third digit of the quotient and prefix the remainder (1) to that quotient digit (2) and thus have 12 as our next dividend:
- Step 4) This is continued until we reach the figure equivalent to the difference between numerator and denomiator. And then take the 9's compliment as in the first method. Combining the two figures we get the answer.
Multiplying 12 x 13 (two-digit numbers)
- Step 1) Multiply the left-hand-most digit (1) of the multiplicand vertically by the left-hand-most digit (1) of the multiplier. Their product (1) is set down as the left-hand-most part of the answer
- Step 2) Then cross multiply 1 and 3, and 1 and 2, add the two, get 5 as the sum and set it down as the middle past of the answer.
- Step 3) Now multiply 2 and 3 vertically, get 6 as their product and put it down as the last (the right-hand-most) part of the answer:
1 : 3+2 : 6 = 156
Other examples: 23x21
4 : 2+6 : 3 = 483
94 x 81
72 : 9+32 : 4 = 72:41:4 = 7614
Multiplying 9 by 7
- Step 1) We should take, as the base for our calculations, that power of 10 which is the nearest to the numbers to be multiplied. In this case 10 itself is that power.
- Step 2) Put the two numbers 9 and 7 above and below on the left-hand side
- Step 3) Subtract each of them from the base (10) and write down the remainders (1 and 3) on the right-hand side with a connecting minus sign between them, to show that the numbers to be multiplied are both less than 10:
- Step 4) The product will have two parts, one on the left side and one on the right.
- Step 5) Now, the left-hand-side digit of the answer can be arrived at in one of four ways, of which only two will be shown here:
(a) Cross-subtract the deficiency (3) in the second row from the original number (9) in the first row to get 6.
(b) Cross-subtract the deficiency (1) in the first row from the original number (7) in the second row to get 6.
- Step 6) To arrive at the right-hand-side digit multiply the numbers in the right column vertically (i.e. 1x3) to give 3. So our answer is 7x9 = 63.
Note: This is a common feature of the Vedic system and is of great advantage as it enables us to test and verify the correctness of our answer at each step.
56 x 99
For multiplicands and multipliers above a certain power all rules remain the same, except that, instead of cross-subtracting you cross-add. Therefore for 1005 x 1009 (remember, the closest power of 10 here is 1000):
If one number is above and the other is below a power of 10 then the cross-adding or substracting will be done according to the sign, and vertical multiplication will produce a negative product which will be subtracted from the left-hand side. For example: 107 x 93
107 + 7
93 - 7
100/-49 = 99/51
Squaring of numbers ending in 5.
For example, 152. Here the last digit is 5 and the preceeding digit is 1. So, one more than that is 2. Now, the formula in this context tells us to multiply the previous digit (1) by one more than itself (i.e. by 2), so the left-hand side digit is 1x2; and the right hand side is the vertical - multiplication product (i.e. 5x5 = 25) as usual.
Thus: 152 = 1x2/25 = 2/25
252 = 2x3/25 = 6/25
352 = 3x4/25 = 12/25 etc.
In this way the sixteen formulae of Vedic Mathematics which have been found apply to everything from simple arithmetical computations (as we have briefly seen here) to subjects such as analytical conics and everything in between such as differential calculus, integration, geometry, cubic equations, and others. Although the introductory volume gives an overview of all these branches, the remaining 15 volumes have been lost.
The truths that the Vedic seers intuitively discovered are today being intellectually established through the discoveries of modern science. Our ancestors did not rest content with their intuitive realisations. They wanted to rub it on the touch-stone of reason and establish the validity of their knowledge. Thus was science born in ancient India.