Ted Chiang - Division by Zero

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Fantastic Metropolis

Division by Zero

By Ted Chiang

(Reprint, added 13 December 2003.)

Stories of Your Life and Others

Ted Chiang

Dividing a number by zero doesn't produce an infinitely large number as an answer. The reason is that

division is defined as the inverse of multiplication; if you divide by zero, and then multiply by zero, you

should regain the number you started with. However, multiplying infinity by zero produces only zero, not

any other number. There is nothing which can be multiplied by zero to produce a nonzero result;

therefore, the result of a division by zero is literally "undefined."

Renee was looking out the window when Mrs. Rivas approached.

"Leaving after only a week? Hardly a real stay at all. Lord knows I won't be leaving for a long time."

Renee forced a polite smile. "I'm sure it won't be long for you." Mrs. Rivas was the manipulator in the

ward; everyone knew that her attempts were merely gestures, but the aides wearily paid attention to her

lest she succeed accidentally.

"Ha. They wish I'd leave. You know what kind of liability they face if you die while you're on status?"

"Yes, I know."

"That's all they're worried about, you can tell. Always their liability--"

Renee tuned out and returned her attention to the window, watching a contrail extrude itself across the

sky.

"Mrs. Norwood?" a nurse called. "Your husband's here."

Renee gave Mrs. Rivas another polite smile and left.

Carl signed his name yet another time, and finally the nurses took away the forms for processing.

He remembered when he had brought Renee in to be admitted, and thought of all the stock questions at

the first interview. He had answered them all stoically.

"Yes, she's a professor of mathematics. You can find her in Who's Who."

"No, I'm in biology."

And:

"I had left behind a box of slides that I needed."

"No, she couldn't have known."

And, just as expected:

"Yes, I have. It was about twenty years ago, when I was a grad student."

"No, I tried jumping."

"No, Renee and I didn't know each other then."

And on and on.

Now they were convinced that he was competent and supportive, and were ready to release Renee into

an outpatient treatment program.

Looking back, Carl was surprised in an abstracted way. Except for one moment, there hadn't been any

sense of deja vu at any time during the entire ordeal. All the time he was dealing with the hospital, the

doctors, the nurses: the only accompanying sensation was one of numbness, of sheer tedious rote.

There is a well-known "proof" that demonstrates that one equals two. It begins with some definitions:

"Let a = 1; let b = 1." It ends with the conclusion "a = 2a," that is, one equals two. Hidden

inconspicuously in the middle is a division by zero, and at that point the proof has stepped off the brink,

making all rules null and void. Permitting division by zero allows one to prove not only that one and two

are equal, but that any two numbers at all -- real or imaginary, rational or irrational -- are equal.

As soon as she and Carl got home, Renee went to the desk in her study and began turning all the papers

facedown, blindly sweeping them together into a pile; she winced whenever a corner of a page faced up

during her shuffling. She considered burning the pages, but that would be merely symbolic now. She'd

accomplish as much by simply never glancing at them.

The doctors would probably describe it as obsessive behavior. Renee frowned, reminded of the indignity

of being a patient under such fools. She remembered being on suicide status, in the locked ward, under

the supposedly round-the-clock observation of the aides. And the interviews with the doctors, who were

so condescending, so obvious. She was no manipulator like Mrs. Rivas, but it really was easy. Simply

say "I realize I'm not well yet, but I do feel better," and you'd be considered almost ready for release.

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Ted Chiang - Division by Zero

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