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Impure Mathematics
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To prove once and for all that math can be fun, we
present: Wherein it is related how that paragon of womanly
virtue, young Polly Nomial (our heroine) is accosted by that
notorious villain Curly Pi, and factored (oh horror!!!)
Once upon a time (1/t) pretty little Polly Nomial was
strolling across a field of vectors when she came to the boundary
of a singularly large matrix. Now Polly was convergent, and her
mother had made it an absolute condition that she must never
enter such an array without her brackets on. Polly, however,
who had changed her variables that morning and was feeling
particularly badly behaved, ignored this condition on the basis
that it was insufficient and made her way in amongst the complex
elements. Rows and columns closed in on her from all sides.
Tangents approached her surface. She became tensor and tensor.
Quite suddendly two branches of a hyperbola touched her at a
single point. She oscillated violently, lost all sense of
directrix, and went completely divergent. As she tripped over a
square root that was protruding from the erf and plunged
headlong down a steep gradient. When she rounded off once more,
she found herself inverted, apparently alone, in a non-Euclidean
space.
She was being watched, however. That smooth operator,
Curly Pi, was lurking inner product. As his eyes devoured her
curvilinear coordinates, a singular expression crossed his face.
He wondered, "Was she still convergent?" He decided to
integrate properly at once.
Hearing a common fraction behind her, Polly rotated and
saw Curly Pi approaching with his power series extrapolated.
She could see at once by his degenerate conic and dissipative
that he was bent on no good.
"Arcsinh," she gasped.
"Ho, ho," he said, "What a symmetric little asymptote
you have I can see you angles have lots of secs."
"Oh sir," she protested, "keep away from me I haven't
got my brackets on."
"Calm yourself, my dear," said our suave operator, "your
fears are purely imaginary."
"I, I," she thought, "perhaps he's not normal but
homologous."
"What order are you?" the brute demanded.
"Seventeen," replied Polly.
Curly leered "I suppose you've never been operated on."
"Of course not," Polly replied quite properly, "I'm
absolutely convergent."
"Come, come," said Curly, "let's off to a decimal place
I know and I'll take you to the limit."
"Never," gasped Polly.
"Abscissa," he swore, using the vilest oath he knew.
His patience was gone. Coshing her over the coefficient with a
log until she was powerless, Curly removed her discontinuities.
He stared at her significant places, and began smoothing out her
points of inflection. Poor Polly. The algorithmic method was
now her only hope. She felt his digits tending to her asymptotic
limit. Her convergence would soon be gone forever.
There was no mercy, for Curly was a heavyside operator.
Curly's radius squared itself; Polly's loci quivered. He
integrated by parts. He integrated by partial fractions. After
he cofactored, he performed runge - kutta on her. The complex
beast even went all the way around and did a contour
integration. What an indignity - to be multiply connected on
her first integration. Curly went on operating until he
completely satisfied her hypothesis, then he exponentiated and
became completely orthogonal.
When Polly got home that night, her mother noticed that
she was no longer piecewise continuous, but had been truncated
in several places But it was to late to differentiate now. As
the months went by, Polly's denominator increased monotonically.
Finally she went to L'Hopital and generated a small but
pathological function which left surds all over the place and
drove Polly to deviation.
The moral of our sad story is this: "If you want to
keep your expressions convergent, never allow them a single
degree of freedom."