Alex's Adventures in Numberland - Alex Bellos [52]
Again, the first digit can be derived in any one of the four ways: 8 + 7 – 10 = 5, or –2 – 3 + 10 = 5, or 8 – 3 = 5 or 7 – 2 = 5. The second digit is the product of the digits in the second column, –2×–3 = 6.
Tirthaji’s tactic reduces the multiplication of two single-digit numbers to an addition and the multiplication of the differences of the original numbers from ten. In other words, it reduces the multiplication of two single digit numbers larger than five to an addition and the multiplication of two numbers less than five. Which means that it is possible to multiply by six, seven, eight and nine without going higher than our five-times table. This is useful to people who find learning their times tables difficult.
In fact, the technique explained by Tirthaji is the same as a method of finger calculation used at least since the Renaissance in Europe, and still used by farmhands in parts of France and Russia as late as the 1950s. On each hand the fingers are assigned the numbers from 6 to 10. To multiply two numbers together, say 8 and 7, touch the 8 finger to Weinger. The number of digits above the linking fingers on one side is subtracted from the linking finger on the other side (either 7 – 2 or 8 – 3) to give 5. The number of digits above the linking fingers on each side, 2 and 3, are multiplied to make 6. The answer, as above, is 56.
How to calculate 8 ×7 with ‘peasant’ finger multiplication.
Tirthaji continued his talk by demonstrating that the method also works when multiplying two-digit numbers, this time using the example 77×97. He wrote on the board:
77
97
Then, instead of writing the difference of 77 from 10, he wrote the difference of each number from 100. (This is where the second sutra comes in. When subtracting a number from 100, or any larger power of 10, all the digits of the number are subtracted from 9 apart from the last one, which is subtracted from 10, as I showed on chapter 3):
77
–23
97
–3
As before, in order to get the first part of the answer there are four options. He chose to show the two diagonal additions: 77 – 3 = 97 – 23 = 74.
The second part is derived by multiplying both digits in the right-hand column: –23×–3 = 69.
The answer is 7469.
Tirthaji then proceeded to an example with three figures: 888×997. This time the difference is calculated from 1000.
Diagonal addition gives 885 for the first part, and multiplication of the right column gives 336 for the second, for an answer of 885,336.
‘Equations are rendered much easier by these formulae,’ Tirthaji commented. The students reacted with spontaneous hearty laughter. Perhaps the chuckles came from the absurdity of an 82-year-old guru in a robe teaching basic arithmetic to some of the smartest maths students in the US. Or perhaps it was in appreciation of the playfulness of Tirthaji’s arithmetical tricks. Arabic numerals are a mine of hidden patterns, even at such a simple level as multiplying two single digits together. Tirthaji then tinued his talk with techniques for squaring, dividing and algebra. The response seems to have been enthusiastic, judging by a transcription made of the lecture notes: ‘Immediately following the end of the demonstration, one student was heard to ask his friend beside him, “What do you think?” His friend’s reply, “Fantastic!”’
On his return to India, Tirthaji was summoned to the holy city of Varanasi, where a special council of Hindu elders discussed his breach of protocol in leaving the country. It was decided that his trip was to be the first and last time that a Shankaracharya was allowed to travel abroad, and Tirthaji undertook a purification ritual just in case he had consumed unHindu food while on his travels. Two years later,