At midnight, how many balls are in the vase? Zero
ountless times I have posed this thought experiment to friends and associates, to university audiences and management seminars. The answers I invariably hear are of the form: The mathematician places net nine balls into the vase at each step; after an infinite number of steps, there must be an infinite number of balls in the vase.
Not so, I chortle. At the stroke of midnight, the vase would be empty.
Name any ball and I will tell you it is not in the vase. Not only that, but I will tell you the exact moment at which it was removed from the vase:
• Ball number one is not in the vase; the mathematician removed it at one minute to midnight.
• Ball number two is not in the vase; the mathematician removed it at one-half of a minute to midnight.
• Ball number three is not in the vase; the mathematician removed it at one-third of a minute to midnight.
• Ball number four is not in the vase;...
How can infinity and zero both be the right answer?

Off on a Tangent

onsider the trigonometric tangent of a 90-degree angle.

As a transcendental function, the tangent does something mighty strange in the vicinity of 90 degrees.

• At 89.99999o, the tangent has a value of plus 5,729,578, which is not infinity or anything, but it sure is heading in that general direction.
• At 90.00001o, the tangent has a value of minus 5,729,578, which is heading toward a negative infinity.
Half-way in between -- at exactly 90.000...0o, the tangent is supposed to be infinite; however, it must also be passing through zero at the same exact place.

Mathematicians shrug. "What do you expect from a discontinuous function?" they explain.

 Where two parallel lines meet, so too do the infinite and the infinitesimal. -- Paul Niquette, 1966

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