Alex's Adventures in Numberland - Alex Bellos [54]
During the interview the room filled up with about 20 people, who sat silently as the Shankaracharya spoke. As the session drew to a close, a middle-aged software consultant from Bangalore asked a question about the significance of the number 1062. The number was in the Vedas, he said, so it had to mean something. The Shankaracharya agreed. It was in the Vedas and, yes, it had to mean something. This prompted a discussion about how the Indian government is neglecting the country’s heritage, and the Shankaracharya lamented that he spent most of his energy in trying to protect traditional culture so he could not devote more time to maths. This year he had managed only 15 days.
Over breakfast the next day I asked the computer consultant about his interest in the number 1062, and he answered with a lecture on the scientific achievements of ancient India. Thousands of years ago, he said, Indians understood more about the world than what is known today. He mentioned that they flew aeroplanes. When I asked if there was any proof of this, he replied that stone engravings of millennia-old aircraft have been found. Did these planes use the jet engine? No, he said, they were powered using the Earth’s magnetic field. They were made from a composite material and flew at a low speed, between 100 and 150kmph. He then became increasingly annoyed by my questions, interpreting my desire for proper scientific explanations as an affront to Indian heritage. Eventually, he refused to speak to me.
While Vedic science is fantastical, occultist and barely credible, Vedic Mathematics stands up to scrutiny, even though the sutras are mostly so vague as to be meaningless and to accept the story of their origin in the Vedas requires the suspension of disbelief. Some of the techniques are so specific as to be nothing more than curiosities – such as a tip for calculating the fraction in decimal. But some are very neat indeed.
Consider the example of 57× 43 from earlier. The standard method of multiplying these numbers is to write down two intermediary lines, and then add them:
Using the third sutra, Vertically and Cross-wise, we can find the answer quite handily as follows.
Step 1: Write the numbers on top of each other:
5
7
4
3
Step 2: Multiply the digits in the right-hand column: 7×3 = 21. The final digit of this number is the final digit of the answer. Write it below the right-hand column, and carry the 2.
Step 3: Find the sum of the cross-wise products: (5 × 3) + (7 × 4) = 15 + 28 = 43. Add the 2 that is carried from the previous step to get 45. The final digit of this number, the 5, is written underneath the left-hand column, with 4 carried.
Step 4: Multiply the digits in the left-hand column, 5×4 = 20. Add the 4 that has been carried to make 24, to give the final answer:
The numbers have been multiplied vertically and cross-wise, as the sutra said on the tin. This method can be generalized to multiplications of numbers of any size. All that changes is that more numbers need to be vertically and cross-multiplied. For example, 376×852:
3
7
6
8
5
2
Step 1: We start with the right column: 6×2 = 12
Step 2: Then the sum of the cross-products between the units and the tens column, (7 × 2) + (6 × 5) = 44, plus the 1 carried above. This is 45.
Step 3: Now we move to the cross-products between the units and the hundreds column, and add them to the vertical product of the tens column, (3 × 2) + (8 × 6) + (7 × 5) = 89, plus the 4 carried above. This is 93.
Step 4: Moving leftwards, we now cross-multiply the first two columns: (3 × 5) + (7 × 8) = 71, plus the 9 carried above. This is 80.
Step 5: Finally, we find the vertical product of the left column, 3 × 8 = 24, plus 8 carried above. This is 32. The final answer: 320,352.
Vertically and Cross-wise, or ‘cross-multiplication