“Stand away from the door, Dr. Aarons,” came the voice of my host. I complied perforce; the madman entered, pushing before him a tea caddy bearing a really respectable meal, complete from bouillon to a bottle of wine. He propelled the cart with his withered left band; the right brandished the evil automatic.
“I trust you have used your time well,” he sneered.
“At least I have my first question,” I responded.
“Good, Dr. Aarons! Very good! Let us hear it.”
“Well,” I continued, “among numbers, expressions of quantity, mathematicians recognize two broad distinctions—two fields in which every possible numerical expression may be classified. These two classifications are known as real numbers on the one band, including every number both positive and negative, all fractions, decimals, and multiples of these numbers, and on the other hand the class of imaginary numbers, which include all products of operations on the quantity called 'c,' otherwise expressed as the square root of minus one.”
“Of course, Dr. Aarons. That is elementary!”
“Now then—is this quantity of yours real or imaginary?”
He beamed with a sinister satisfaction.
“A very fair question, sir! Very fair! And the answer—may it assist you—is that it is either!”
A light seemed to burst in my brain! Any student of numbers knows that only one figure is both real and imaginary, the one that marks the point of intersection between the real and imaginary numbergraphs. “I've got it!” The phrase kept running through my mind like a crazy drumbeat! With an effort I kept an appearance of calm.
“Mr. Strawn,” I said, “is the quantity you have in mind zero?”
He laughed—a nasty, superior laugh that rasped in my cars.
“It is not, Dr. Aarons! I know as well as you that zero is both a real and imaginary number! Let me call your attention to my answer: I did not say that my concept was both real and imaginary; I said it was either!” He was backing toward the door. “Let me further remind you that you have eight guesses remaining, since I am forced to consider this premature shot in the dark as one chance! Good evening!”
He was gone; I heard the bar outside the door settle into its socket with a thump. I stood in the throes of despair, and cast scarcely a glance at the rather sumptuous repast he had served me, but slumped back into my chair.
It seemed hours before my thoughts were coherent again; actually I never knew the interval, since I did not glance at my watch. However, sooner or later I recovered enough to pour a tumbler of wine and eat a bit of the roast beef; the bouillon was hopelessly cold. And then I settled down to the consideration of my third question.
From Strawn's several hints in the wording of his terms and the answers to my first and second queries, I tabulated what information I could glean. He had specifically designated a numerical expression; that eliminated the x's and y's of algebraic usage. The quantity was either real or imaginary and was not zero; well, the square of any imaginary is a real number. If the quantity contained more than one figure, or if an exponent was used, then I felt sure his expression was merely the square of an imaginary; one could consider such a quantity either real or imaginary. A means of determining this by a single question occurred to me. I scribbled a few symbols on a sheet of paper, and then, feeling a sudden and thorough exhaustion, I threw myself on the daybed and slept. I dreamed Strawn was pushing me into a nightmarish sea of grinning mathematical monsters.
The creaking of the door aroused me. Sunbeams illumined the skylight; I had slept out the night. Strawn entered balancing a tray on his left arm, holding the ever-present weapon in his free band. He placed a half dozen covered dishes on the tea-cart, removing the remains of the evening meal to his tray.
“A poor appetite, Dr. Aarons,” he commented. “You should not permit your anxiety to serve the ends of justice to upset you!” He chuckled with enjoyment of his sarcasm. “No questions yet? No matter; you have until four tomorrow for your next two.”
“I have a question,” I said, more thoroughly awakened. I rose and spread the sheet of paper on the desk.
“A numerical quantity, Mr. Strawn, can be expressed as an operation on numbers. Thus, instead of writing the numeral '4' one may prefer to express it as a product, such as '2 x 2,' or as a sum, as '3 + 1,' or as a quotient, as '8/2' or as a remainder, as '5—1.' Or even in other ways—as a square, such as 2(2)... All different methods of expressing the single quantity '4.' Now here I have written out the various mathematical symbols of operations; my question is this: Which if any of these symbols is used in the expression you have in mind?”
“Very neatly put, Dr. Aarons! You have succeeded in combining several questions in one.” He took the paper from me, spreading it on the desk before him. “This symbol, sir, is the one used.” He indicated the first one in my list—the subtraction sign, a simple dash!
And my hopes, to use the triviality of a pun, were dashed as well! For that sign eliminated my carefully thought-out theory of a product or square of imaginaries to form a real number. You can't change imaginary to real by addition or subtraction; it takes multiplication, squaring or division to perform that mathematical magic! Once more I was thoroughly at sea, and for a long time I was unable to marshal my thoughts.
And so the hours dragged into days with the tantalizing slow swiftness that tortures the condemned in a prison death house. I seemed checkmated at every turn; curious paradoxical answers defeated my questions.
My fourth query, “Are there any imaginaries in your quantity?” elicited a cool, definite “No.” My fifth, “How many digits are used in this expression?” brought forth an equally definite “Two.”
Now there you are! What two digits connected by a minus sign can you name whose remainder is either real and imaginary? “An impossibility,” I thought. “This maniac's merely torturing me!” And yet—somehow Strawn's madness seemed too ingenious, too clever, for such an answer. He was sincere in his perverted search for justice. I'd have sworn to that.
On my sixth question, I had an inspiration! By the terms of our game, Strawn was to answer any question save the direct one, “What is this expression?” I saw a way out! On his next appearance I met him with feverish excitement, barely waiting for his entrance to begin my query.
“Mr. Strawn! Here is a question you are bound by your own rules to answer. Suppose we place an equal sign after your quantity, what number or numbers will complete the equation: What is the quantity equal to?”
Why was the fiend laughing? Could he squirm out of that poser?
“Very clever, Dr. Aarons. A very clever question. And the answer is—anything!”
I suppose I shouted. “Anything! Anything! Then you're a fraud, and your game's a damnable trickery. There's no such expression!”
“But there is, Doctor! A good mathematician could find it!” And he departed, still laughing.
I spent a sleepless night. Hour after hour I sat at that hateful desk, checking my scraps of information, thinking, trying to remember fragments of all-but-forgotten theories. And I found solutions! Not one, but several. Lord, how I sweated over them! With four questions—two days—left to me, the solution of the problem began to loom very close. The things dinned in my brain; my judgment counseled me to proceed slowly, to check my progress with another question, but my nature was rebelling against the incessant strain. “Stake it all on your last four questions! Ask them all at once, and end this agony one way or the other!”
I thought I saw the answer. Oh, the fiendish, insane cleverness of the man! He had pointed to the minus sign on my list, deliberately misled me, for all the time the symbol had meant the bar of a fraction. Do you see? The two symbols are identical—just a simple dash—but one use means subtraction and the other division! “1—1” means zero, but “1/1” means one! And by division his problem could be solved. For there is a quantity that means literally anything, real number or imaginary, and that quantity is “0/0”! Yes, zero divided by zero. You'd think o
ffhand that the answer'd be zero, or perhaps one, but it isn't, not necessarily. Look at it like this: take the equation “2 x 3 = 6.” See? That's another way of saying that two goes into six three times. Now take “0 x 6 = 0.” Perfectly correct, isn't it? Well, in that equation zero goes into zero six times! Or “0/0 = 6”! And so on for any number, real or imaginary—zero divided by zero equals anything!
And that's what I figured the fiend had done. Pointed to the minus sign when he meant the bar of a fraction, or division!
He came in grinning at dawn.
“Are your questions ready, Dr. Aarons? I believe you have four remaining.”
I looked at him. “Mr. Strawn, is your concept zero divided by zero?”
He grinned. “No, sir, it is not!”
I wasn't disheartened. There was just one other symbol I had been thinking of that would meet the requirement—one other possibility. My eighth question followed. “Then is it infinity divided by infinity?”
The grin widened. “It is not, Dr. Aarons.”
I was a little panicky then! The end loomed awfully near! There was one way to find out if the thing was fraudulent or not; I used my ninth question:
“Mr. Strawn, when you designated the dash as the mathematical symbol used in your expression, did you mean it as the bar of a fraction or as the sign of subtraction?”
“As the subtraction sign, Dr. Aarons. You have one more question. Will you wait until tomorrow to ask it?”
The fiend was grinning in huge enjoyment. Thoroughly confident, he was, in the intricacies of his insane game. I hesitated in a torture of frenzied indecision. The appalling prospect of another agonized night of doubts decided me.
“I'll ask it now, Mr. Strawn!”
It had to be right! There weren't any other possibilities; I'd exhausted all of them in hour after hour of miserable conjecture!
“Is the expression—the one you're thinking of—infinity minus infinity?”
It was! I knew it by the madman's glare of amazed disappointment.
“The devil must have told you!” he shrieked. I think there were flecks of froth on his lips. He lowered the gun in his hand as I edged toward the door; he made no move to stop me, but stood in a sort of desolate silence, until I gained the top of the stairway. Then—
“Wait a minute!” he screamed. “You'll tell them! Wait just a minute, Dr. Aarons!”
I was down the stairs in two leaps, and tugging at the door. Strawn came after me, his gun leveled. I heard it crash as the door opened and I slipped out into a welcome daylight.
Yes, I reported him. The police got him as he was slipping away and dragged him before an alienist. Crazy, but his story was true; he had been mangled in an experimental laboratory explosion.
* * *
Oh, the problem? Don't you see? Infinity is the greatest expression of number possible—a number greater than any conceivable. Figure it out like this:
The mathematician's symbol for infinity is a tipsy eight—so: oo.
Well, take the question, oo + 6 = oo. That's true, because you can't add anything to infinity that will make it any greater than it is. See? It's the greatest possible number already. Well then, just by transposition, oo—oo = 6. And so on; the same system applies to any conceivable number, real or imaginary.
There you are! Infinity minus itself may equal any quantity, absolutely any number, real or imaginary, from zero to infinity. No, there was nothing wrong with Court Strawn's mathematics.
The End
THE CIRCLE OF ZERO
CHAPTER I
The Law of Chance
'IF THERE were a mountain a thousand miles high, and every thousand years a bird flew over it, just brushing the peak with the tip of its wing, in the course of inconceivable eons the mountain would be worn away. Yet all those ages would not be one second to the length of eternity...'
I don't know what philosophical mind penned the foregoing, but the words keep recurring to me since last I saw old Aurore de Neant, erstwhile professor of psychology at Tulane. When, back in '24, I took that course in Morbid Psychology from him, I think the only reason for taking it at all was that I needed an eleven o'clock on Tuesdays and Thursdays to round out a lazy program.
I was gay Jack Anders, twenty-two years old, and the reason seemed sufficient. At least, I'm sure that dark and lovely Yvonne de Neant had nothing to do with it; she was but a child of sixteen.
Old de Neant liked me, Lord knows why, for I was a poor enough student. Perhaps it was because I never, to his knowledge, punned on his name. Aurore de Neant translates to Dawn of Nothingness, you see; you can imagine what students did to such a name. 'Rising Zero'–'Empty Morning' — those were two of the milder sobriquets.
That was in '24. Five years later I was a bond salesman in New York, and Professor Aurore de Neant was fired. I learned about it when he called me up; I had drifted quite out of touch with University days.
He was a thrifty sort. He had saved a comfortable sum, and had moved to New York, and that's when I started seeing Yvonne again, now darkly beautiful as a Tanager figurine. I was doing pretty well, and was piling up a surplus against the day when Yvonne and I...
At least, that was the situation in August, 1929. In October of the same year, I was as clean as a gnawed bone and old de Neant had but little more meat. I was young, and could afford to laugh; be was old, and he turned bitter. And indeed, Yvonne and I did little enough laughing when we thought of our own future; but we didn't brood like the professor.
I remember the evening he broached the subject of the Circle of Zero. It was a rainy, blustering fall night, and his beard waggled in the dim lamplight like a wisp of gray mist. Yvonne and I had been staying in evenings of late; shows cost money, and I felt that she appreciated my talking to her father, and — after all — he retired early.
She was sitting on the davenport at his side when he suddenly stabbed a gnarled finger at me and snapped, 'Happiness depends on money!'
I was startled. 'Well, it helps,' I agreed.
His pale blue eyes glittered. 'We must recover ours!' he rasped.
'How?'
'I know how. Yes, I know how!' He grinned thinly. 'They think I'm mad. You think I'm mad; even Yvonne thinks so.'
The girl said softly, reproachfully, 'Father!'
'But I'm not,' he continued. 'You and Yvonne, and all the fools holding chairs at universities — yes! But not me.
'I will be, all right, if conditions don't get better soon,' I murmured.
I was used to the old man's outbursts.
'They will be better for us,' he said, calming. 'Money! We will do anything for money, won't we, Anders?'
'Anything honest.'
'Yes, anything honest. Time is honest, isn't it? An honest cheat, because it takes everything human and turns it into dust.' He peered at my puzzled face. 'I will explain,' he said, 'how we can cheat time.'
'Cheat?'
'Yes. Listen, Jack. Have you ever stood in a strange place and felt a sense of having been there before? Have you ever taken a trip and sensed that sometime, somehow, you had done exactly the same thing — when you know you hadn't?'
'Of course. Everyone has. A memory of the present, Bergson calls–'
'Bergson is a fool! Philosophy without science. Listen to me.' He leaned forward. 'Did you ever hear of the Law of Chance?'
I laughed. 'My business is stocks and bonds. I ought to know of it.'
'Ah,' he said, 'but not enough of it. Suppose I have a barrel with a million trillion white grains of sand in it, and one black grain. You stand and draw a single grain, one after the other, look at it, and throw it back into the barrel. What are the odds against drawing the black grain?'
'A million trillion to one, on each draw.'
'And if you draw half of the million trillion grains?'
'Then the odds are even.'
'So!' he said. 'In other words, if you draw long enough, even though you return each grain to the barrel and draw again, some day you will dr
aw the black one — if you try long enough!'
'Yes,' I said.
'Suppose now you tried for eternity?'
'Eh?'
'Don't you see Jack? In eternity the Law of Chance functions perfectly. In eternity, sooner or later, every possible combination of things and events must happen. Must happen, if it's a possible combination. I say, therefore, that in eternity, whatever can happen will happen!' His blue eyes blazed in pale fire.
I was a trifle dazed. 'I guess you're right,' I muttered.
'Right! Of course I'm right. Mathematics is infallible, Now do you see the conclusion?'
'Why — that sooner or later everything will happen.'
'Bah! It is true that there is eternity in the future; we cannot imagine ending. But Flammarion, before he died, pointed out that there is also an eternity in the past. Since in eternity everything possible must happen, it follows that everything must already have happened!'
I gasped. 'Wait a minute! I don't see–'
'Stupidity!' he hissed. 'It is but to say with Einstein that not only space is curved, but time, to say that after untold eons of millenniums, the same things repeat themselves because they must! The Law of Chance says they must, given time enough. The past and the future are the same thing, because everything that will happen must already have happened. Can't you follow so simple a chain of logic?'
'Why — yes. But where does it lead?'
'To our money! To our money!'
'What?'
'Listen. Do not interrupt. In the past, all possible combinations of atoms and circumstances must have occurred.' He paused, then stabbed that bony finger of his at me. 'Jack Anders, you are a possible combination of atoms and circumstances! Possible because you exist at this moment!'
'You mean — that I have happened before?'
The Lost Master - The Collected Works Page 59