Monday, June 17, 2013

Source of the Cipher 6: Much Ado

            6. Much Ado

            Namagiri wrung her hands. When will Ramanujan return? And how did he do? Did they listen? Ever since he set off to the meeting, she had been waiting here, standing at the doorway of their room, standing right on the threshold all this time, waiting.
            “There you are!” she cried. They rushed into an embrace. Then she broke off, held him at arm’s length, and said, “And?”
            “It went just as you planned, dearest,” he said. “I am so sorry...”
            “You told them everything?”
            “And did they listen?”
            Ramanujan sighed. “They listened enough. They knew the counting board; they understood the abacus; and they accepted sunyata - ”
            “- thank goodness!”
            “ - and the tables. Base two they almost liked.”
            “And the mill?”
            Ramanujan shook his head.
            “Never mind,” Namagiri consoled him. “The mill can bide its time... Did you mention me?”
            “Not as originator, as you requested. Not until the very end.” He sighed. “And it was just as you predicted; the moment I gave you credit, they stopped listening.” He lowered his head. “I am so sorry.”
            “Credit? Of course there’s sunyata credit for me,” Namagiri replied. “And I’m so sorry about the mill. I know you wanted it...”
            Ramanujan shrugged. “Somebody else will build it, someday.”
            “Look on the bright side, dear; until then, they’ll still need you.” Namagiri took her husband by the hand. “Now come in and tell me what they said. Tell me every word.” She  led him into their room.
                        *                      *                      *

            Ramanujan bowed deeply. “Sire and esteemed guest, I come to report success!”
            Prince and Sheik exhaled ropy smoke. Prince Rahni said, “In what wise?”
            Ramanujan said, “I have found a system of computation that is easy, accurate, and teachable. It can be imparted to the young and the old, the wise and the simple. I know this because I was able to teach it to my wife, an illiterate washerwoman.”
            “Impressive,” said the Sheik. “Tell us more...”
            Ramanujan felt a pang of shame for deceiving his superiors. After all, it was not he who taught her but she who taught him. (But Namamgiri had ordered him not to credit her, and so he did not.) He said, “Recall positional numeration...”
            Sheik Kahmunni said, “You mean the abacus numbers?” Ramanujan nodded, and the Sheik said, “The counting-board scheme?” Ramanujan nodded, and the Sheik said, “I recall that it failed! For what of two hundred and three versus two hundred and thirty?”
            Ramaujan said, “Those numbers are distinguished on the counting-board by having different void columns, is that not so?” Sheik and Prince nodded. “Then clearly what we need is a sign to denote which columns are empty. That is, we need a digit for the void.” Ramanujan wrote a sunyata. “Sunyata; the number of nothingness.”
            “What?!” cried Prince and Sheik in unison.
            Ramanujan continued, “Thus we distinguish two hundred and thirty from two hundred and three,” and he wrote on his slate:
                        230                  203
            “Call it a placeholder,” he assured the skeptical Sheik and the baffled Prince. “Numerical punctuation. A book-keeping device, for ease of reading.”
            “Ha. Hm,” said the Sheik. “Well, I’ll play along with your little game.”
            “It’s all a mystery to me,” the Prince complained.
            “Go ahead, and faith will follow you,” Ramaujan assured him. “Sunyata is the crowning digit, the missing link; it completes the system. By using all ten digits we can write down numbers of any size and form.”
            “I see,” said the Sheik. “Any number fits onto a counting board, so you can write it in positional notation - if you allow void digits for void columns.”
            “Just so!”
            The Sheik said, “How strange, that counting boards should teach a new number!”
            The Prince said, “Why not? Don’t they know numbers better than we do?”
            Ramanujan said, “But now you can know just as much as counting boards do! Consider these tables!” He gave them a leaf of papyrus inscribed with these figures:

        +       0       1       2       3       4       5       6       7       8       9

        0       0       1       2       3       4       5       6       7       8       9
        1       1       2       3       4       5       6       7       8       9       10
        2       2       3       4       5       6       7       8       9       10      11
        3       3       4       5       6       7       8       9       10      11      12
        4       4       5      6       7       8       9       10      11      12      13
        5       5       6      7       8       9     10       11      12      13      14
        6       6       7      8       9       10    11      12       13      14      15
        7       7       8      9       10      11    12       13      14      15      16
        8       8       9     10       11      12    13       14      15      16      17
        9       9     10     11        12      13    14       15      16      17      18

        ×       0       1       2       3       4       5       6       7       8       9
        0       0       0       0       0       0       0       0       0       0       0
        1       0       1       2       3       4       5       6       7       8       9
        2       0       2       4       6       8       10     12      14      16      18
        3       0       3       6       9       12      15     18      21      24      27
        4       0       4       8       12      16      20     24      28      32      36
        5       0       5       10      15      20      25     30      35      40      45
        6       0       6       12      18      24      30     36      42      48      54
        7       0       7       14      21      28     35     42       49     56      63
        8       0       8       16      24      32      40      48      56      64      72
        9       0       9       18      27      36      45      54      63      72      81

            Ramanujan said, “The first is a table of sums of the digits, the second a table of products of  digits. Know these and you know all computation!” While Sheik and Prince wordlessly inspected the plus-times tables, Ramanujan added, “These tables are so simple that I could teach them to my wife, an illiterate washerwoman. Indeed, it was she who wrote these tables down, as I called out each entry.” (He did not add: at her urging.)
            “Very interesting,” said the Sheik. “May I have this copy?” Ramaujan nodded, and the Sheik said, “Thank you; I shall show this to ben-Tennon at the earliest opportunity. Just one thing puzzles me; this new digit...” He pointed to the table, at the entry 5×2 = 10. “That sunyata number. It is a number, is it not?”
            “It is a placeholder digit,” Ramanujan explained.
            “And digits are numbers,” Sheik Kahmunni insisted.
            “And numbers are for counting things, yes? But this sunyata number... it’s for counting an empty column?”
            “But if the column is empty, then what is there to count?”
            “Nothing; and that is sunyata.”
            “But how much is sunyata?”
            Ramanjuan said, “Sunyata is as much as there is water in the desert.”
            The Sheik said, “But there is no water in the desert!”
            “And that’s sunyata water. Sunyata is as many as there are elephants in this room.”
            The Sheik said, “But there are no elephants in this room!”
            “And that's sunyata elephants. Sunyata is the number of things that are not.”
            The Sheik said, “But there are no things that are not!”
            “And that’s sunyata,” Ramanujan explained.
            “Aha!” Prince Rahni exclaimed. “It’s a mystical number!”
            “A transcendent number,” Ramanjuan agreed. “A keystone number. A central number. Perhaps the central number.”
            “And you have found a practical use for this mysticism?” the Sheik asked.
            “It'll balance your accounts,” Ramanujan said.
            Prince Rahni laughed. “How rare and delightful to see something transcendent busy itself in financial matters! At last speculation pays off!”
            The Sheik said, “You believe this bizarre numerology of the nonexistent?”
            Prince Rahni replied, “What does it matter if it is strange, so long as it is useful? His ideas are absurd, but then again, so are many other profitable ideas; why pick on that one? The question is not, is it crazy, but rather, is it crazy enough to work?”
            Sheik Kahmunni nodded. “I agree. Your mad computer has a fine idea, one which I shall be honored to steal.”
            Ramanujan said, “Sire and esteemed guest... may I interject?”
            Prince Rahni nodded.
            Ramanujan said, “I have taken these new methods to heart, woven them into my mind. As a result my arithmetical powers have expanded. I can now tackle problems that even I did not dare attempt before. Do you recall the anise-seed number? The smallest number that is a sum of two cubes in two different ways?”
            “One thousand, seven hundred and twenty nine?” Sheik Kahmunni asked.
            “That very number! Well, what is the smallest number that is a sum of two fourth powers in two different ways?”
            Sheik and Prince just looked at each other. They took a puff from the water pipe. Prince Rahni said, “I have no idea.”
            Sheik Kahmunni hazarded, “Perhaps it exists... but if so, it must be very large.”
            Ramanujan said, “Just so! In fact it’s six three five, three one eight, six five seven!”
            Prince Rhani said, “What?”
            Ramanujan bowed. “In other words... six hundred and thirty-five million, three hundred and eighteen thousand, six hundred and fifty seven. For it equals one hundred and thirty-four, raised to the fourth power, plus one hundred and thirty-three, raised to the fourth power; and it also equals one hundred and fifty-eight, raised to the fourth power, plus fifty-nine, raised to the fourth power.”
            Sheik Kahmunni said, “If you say so...”
            Ramanujan said, “Now I can do even the Sessa problem!”
            Prince Rahni said, “You can? Good!” 
            Sheki Kahmunni said, “What Sessa problem?”
            Prince Rahni said, “It is a story-problem. Once upon a time Prince Minister Sessa invented the game of chess; and the King offered him a reward for his ingenuity. Sessa replied, ‘Take a chessboard and put one grain of wheat on the first square; then two upon the second square; then four upon the third square; then eight upon the fourth square; and so on, doubling the number of wheat-grains per square, until you cover all sixty-four squares of the chessboard.’ The King granted this wish; but soon discovered that this was too much wheat for him to give. How many grains of wheat did Sessa demand?”
            Ramanujan said, “To cover the first square took one grain; to cover the first two squares took three grains; to cover the first three squares took seven grains; the first four squares took fifteen grains; and all these numbers equal one subtracted from a power of two. In like wise, the number of grains it would take to cover sixty-four squares equals one subtracted from the sixty-fourth power of two.”
            The Sheik said, “Clear enough; but what is that sixty-fourth power?”
            Ramanujan said, “I asked my wife to calculate the first sixty-four powers of two; and she did so in a very short time, using only this new method. Behold!” He produced a leaf of papyrus inscribed with these figures, in Namagiri’s handwriting:


            Ramanujan said, “Therefore Sessa wanted this number;  one eight, four four six, seven four four, sunyata seven three, seven sunyata nine, five five one, six one five.”
            “What?” said Prince Rahni.
            Ramanujan bowed. “Eighteen million million million, four hundred and forty-six thousand million million, seven hundred and forty-four million million, seventy-three thousand million, seven hundred and nine million, five hundred and fifty-one thousand, six hundred and fifteen.”
            The Sheik scanned Namagiri’s list of powers of two. His scowl slowly softened to a smile. “This seems to be in order... May I keep this copy?” Ramanujan nodded, and the Sheik said, “I will let my clerk check these figures in full.”
            “He will find them entirely correct.”
            “No doubt he will,” said Sheik Kahmunni. “And that is what I find truly impressive. Far more than your previous tricks, O computer; for now that you have explained your magic, it is all the more amazing for being knowable by ordinary mortals.”
            Prince Rahni said, “Most magic suffers upon exposure.”
            “Not this,” said the Sheik, “for it is real magic. May I make a small confession?”
            Prince Rahni smiled. “By all means.”
            “For many, many years I have been a devoted reader of fantastic narratives. Far and wide across the land, east, west, north and south, I have sought mystic fables, cosmic epics, and magical romances. I could tell you a thousand and one tales!”
            Prince Rahni said, “As if your real life were not adventurous enough?”       
            Sheik Kahmunni shrugged. “It’s a vice, I admit it! And like all vices, insatiable; for every fantasy I have ever read lacks a certain something.”
            Prince Rahni asked, “That being?”
            “Reliable information!” Sheik Kahmunni exclaimed. “Always, I have wanted to know how the wizard's spells work! I wanted to know his methods; how he made his miracles. Not when, why, nor for whom! Curse the plot; I wanted practical instruction!”
            “Aha!” said Prince Rahni. “You wanted the wizard’s secrets. His powers.”
            “Exactly! And so for decades I have nursed a secret grudge against all swindling fabulists, who promise marvels that they have neither intention nor ability to reveal!”
            Prince Rahni shook his head sadly. “Fraudulent practices indeed.”
            “But consider these plus-times tables,” Sheik Kahmunni urged. “What are they but the Secret of Arithmetic? They deliver mastery over number! And what’s more, they use a mystic symbol; a new number, a name for nothing, an invisible digit that unlocks infinity! What secret teaching, what occult glyph, what magical spell could top that?”
            Prince Rahni added, “And this magic is not merely for slaying a dragon or defeating an evil empire. No, it serves a higher purpose. It’ll make us rich!”
            Sheik Kahmunni nodded. He said, “To get rich is glorious.” He puffed on the water-pipe and said, “You know, this reminds me of the time I was in Mesopotamia. I was exploring an ancient Babylonian tomb - ”
            “ - and why were you, a poor merchant, exploring an ancient Babylonian tomb?”
            “I was taking inventory,” the Sheik said cooly. “And there on the wall, we found inscriptions; number signs! Most were useless astronomical records, but a good fifth was a tally of the tomb’s contents, compiled for us by its sponsors and occupants!”
            Prince Rahni puffed on the water-pipe and said, “How thoughtful of them.”
            “Ben-Tennon was able to match the numbers to the merchandise - and a rich haul it was, too. It was a positional number system like the one your computer is proposing, but with a base of sixty rather than ten. Yes, each place worth sixty times its successor. What’s more, they too used a symbol much like this sunyata. It looked like this.” He drew a picture of a diagonal wedge. “We decided that it was a kind of punctuation mark, to space out the missing powers of sixty. A placeholder; but now I see it as sunyata, their sixtieth digit.”
            “Brilliant!” Prince Rahni said. “But why did you not adopt their system?”
            “It was too crude, too cumbersome. Sixty digits are hard to remember; and their plus-times tables would have thirty-six hundred entries each.”
            Prince Rahni laughed. “How impractical!”
            “Indeed,” said the Sheik. He puffed on the water-pipe. “Such methods are fit for astrologers, priests, and tomb-decorators; but a businessman such as myself needs a system that is quick, simple, efficient, and friendly to the user. Such as base ten.”
            Prince Rahni puffed on the water-pipe and said, “You know, this reminds me of what I told my astrologer once. Nine-tenths of life is just showing up; and nine-tenths of science is neither invention nor discovery, but publicity!”
            The Sheik said, “Publish or perish. That too I learned in the tombs of Babylon.”
            Ramanujan said, “Sire and esteemed guest... may I interject?”
            Prince Rahni nodded.
            Ramanujan said, “It is true that base ten is simpler and easier than base sixty; but would you be interested in a system that is even easier?”
            Prince and Sheik nodded.
            “Then consider base two! Each digit is worth twice its predecessor, and there are only two digits; one and sunyata.”
            “Being and nothing, lingham and yoni,” Prince Rahni said, smiling.
            “Active and passive, male and female,” Sheik Kahmunni replied.
            “Yes and no, on and off,” Ramanujan concluded. “Base two arithmetic is easy and clear. Behold!” And he wrote down the following:

            +  |  0   1                 ×  |  0   1
            ---|---------               ---|----------
            0  |  0   1                 0  |  0   0
               |                           |
            1  |  1  10                 1  |  0   1

            Sheik and Prince marvelled over these tables.
            The Prince said, “How simple!”        
            The Sheik said, “How direct!”
            Ramanujan said, “This too I was able to teach to my wife.” Another lie; for it was she who taught him. But Namagiri had ordered him not to give her credit, until all else had been accomplished.
            “His wife!” the Prince muttered. “Then anyone can learn it!”
            “And not just women and children,” Ramanujan added. “For this system is so simple, that I believe that we can use it to create a kind of mill, or loom, for numbers. By manipulating cords, levers, and weights, we can emulate the rules of base-two arithmetic.”
            “A mechanical computer!” the Prince exclaimed.
            “You could call it that,” Ramanujan affirmed.
            “It’ll be tireless, swift, inerrant...” the Prince breathed.
            “Perhaps only two out of those three,” said the Sheik, more skeptical.
            “Mechanical computation,” Prince Rahni said. “A whole new world.”
            Sheik Kahmunni said, “Is that so? Tell me, computer; what is the base-two name for... say... fourteen hundred and ninety-two?”
            Ramanujan instantly replied, “One sunyata one one one sunyata one sunyata one sunyata sunyata!”
            Sheik added, “Now try two thousand and one.”
            “One one one one one sunyata one sunyata sunyata sunyata one!”
            Prince and Sheik looked at each other. They smirked. Then they burst out laughing. After rollicking awhile, both took long drags on the water-pipe.
            The Sheik said, “What slave or servant could remember such gibberish?”
            The Prince chimed in, “Numbers that long are for demons!”
            The Sheik replied, “Or djinns!”
            “Or Naugas!”
            “In any case, not humans!” And they laughed. Suddenly they stopped laughing, and looked sideways at Ramanujan.
            Prince Rahni said kindly, “Present company excepted, of course.”
            Sheik Kahmunni said, “There’s no need for mechanical computers; is not Ramanujan himself a perfectly fine computer? And isn’t his base-ten method, plus abacus, enough to make a computer out of any sufficiently diligent servant?”
            Ramanujan said, “Actually, it wasn’t really my idea...”
            “Then I needn’t give you credit,” the Sheik retorted. “So whose idea is it?”
            “Well... actually, it was my wife who supplied the key concept...”
            “Your wife?” Prince Rahni exclaimed.
            “What concept?” Sheik Kahmunni asked.
            “My wife,” Ramanujan said. “An illiterate washerwoman. And also the most brilliant mind in millennia; for she saw what none have seen before. It was Namagiri who gave name and sign to sunyata, the number of nothingness.”
            Once again Sheik and Prince looked at each other. Once again they smirked. And once again they burst into gusts of laughter. After awhile they calmed down.
            Prince Rahni said, “This sunyata is a woman's number?”
            Sheik Kahmunni said, “Ridiculous!”
            The Prince said, “Absurd!”
            Sheik Kahmunni said, “Let us speak of other matters now.”
            Prince Rahni said kindly, “You are dismissed, computer...”

No comments:

Post a Comment