cried with relief. He was fourteen.
Now, as they walk though the door, Orel says, “I’ve got it. Your argument would make sense if the time needed to cross each fraction of the distance was constant. In that case velocity would be constantly decreasing and time versus distance would be an asymptotic curve, never reaching its goal. But the time it takes to cross each fraction of the distance is getting smaller at the same rate that the distances are decreasing. The sum of all those times will still be finite, even if you divide the distance infinitely.”
They walk down the long, narrow tunnel that divides the power station from Hydroponics. A single, fluorescent light runs along the ceiling. Mold grows on the walls.
“But I am dividing it infinitely,” Bernie says. “Don’t you see? No matter how much distance I cross, there must still be a tiny distance left keeping me from my goal.”
“Not really. Because the distance left to you and the time it takes to cross that distance quickly become not just very small, but infinitely small.”
“But still greater than zero.”
“No. Because an infinitely small number is equal to zero.”
“What?”
“I can prove it.” They pass through another thick door and enter the brightly lit corridors of Hydroponics. A tiny maintenance robot rolls past them. “Let’s take an infinitely small number: a decimal point followed by an infinity of zeros and then a one. We’ll call it ‘Almost Zero.’ If we’re dividing a finite distance infinitely, we must reach this number sometime, right?”
“Okay . . . for argument’s sake, I’ll agree.”
“What’s one minus Almost Zero?”
“Hmm.” Bernie turns and looks at him. Between his ugly black metal jaw and his ugly black bristling hair, his two handsome blue eyes are crinkled in amusement. “It’s a decimal point followed by an infinite number of nines. Although, in this case, the infinity involved is actually greater by one than the infinite number of zeros in Almost Zero.”
“Good boy,” Orel says dryly. “Let’s call this number ‘Almost One.’ Now, Almost One can be expressed as the sum of a decimal point followed by an infinity of threes and a decimal point followed by an infinity of sixes — the decimal equivalents of one-third and two-thirds. So Almost One is actually equal to one. So if we subtract Almost One from both sides of the equation, we see that Almost Zero is, in fact, equal to zero. So if you keep dividing the distance infinitely, you will quickly reach a point where you have zero distance left to travel.
“And we’re here.” With a dramatic flourish, Orel punches the lock plate for the entrance to Gimmel Eight. Nothing happens. “Door’s stuck,” he says, hitting it a few more times. He smiles weakly. Reality has, once again, failed to conform to his rhetoric.
Bernie examines the control panel. “Just like you to change the subject when I’m winning an argument. There’s been a pneumatic breach. That’s why the computer won’t let you open the door. That’s fumatory in there. You’d suffocate in a minichron.”
“Damn.” Orel peers through the tiny window in the door. It is completely dark on the other side. “Lights are out, too. What do you suppose happened?”
“Let’s find out.” Bernie unsnaps the cover to the manual override and throws a switch. A thick, translucent web of plastic whips out from the doorjamb, stretching across the door in an iris pattern. Bernie and Orel unclip their respirators from their belts and attach them to their faces. “Ready?” Bernie asks, his voice muffled by the filter.
Orel nods, and Bernie hits one last button. The door behind the webbing slides open. The webbing bulges toward them, pushed by the pressure of the air behind it, then flexes as the strands react to the tension. When the web appears stable, Orel sticks his hand into the center of the opaque swirl of filaments. The webbing gives, then tightens. Cool, oiled
Heinrich Fraenkel, Roger Manvell