30 automaton

An automaton in action

1

as asymptote: values of the function become infinitely

large as xapproaches the value 3 from the right, and

infinitely large and negative as xapproaches the value

3 from the left.

A function y= f(x) has a horizontal asymptote y= L

if limx→∞ f(x)= Lor limx→–∞f(x)= L. For example,

the function has horizontal asymp-

tote y= 3, since

.

An asymptote need not be horizontal or vertical,

however. For example, the function

approaches the line y= x+ 1 as x

becomes large, thus y= x+ 1 is a “slant asymptote” for

the curve.

If a

HYPERBOLA

is given by the equation ,

then manipulating yields the equation .

The right hand side tends to zero as xbecomes large,

showing that the curve has slant asymptotes given by

, that is, by the lines and

.

Extending the definition, we could say that the

curve has the parabola y= x2as

an asymptote.

automaton (plural, automata) An abstract machine

used to analyze or model mathematical problems is

called an automaton. One simple example of an

automaton is a “number-base machine,” which consists

of a row of boxes extending infinitely to the left. One

places in this machine a finite number of pennies in the

rightmost box. The machine then redistributes the pen-

nies according to a preset rule.

A “1 ←2” machine, for example, replaces a pair

of pennies in one box with a single penny in the box

one place to the left. Thus, for instance, six pennies

placed into the 1 ←2 machine “fire” four times to

yield a final distribution that can be read as “1 1 0.”

This result is the number six written as a

BINARY NUM

-

BER

, and this machine converts all numbers to their

base-two representations.

A 1 ←3 machine yields base-three representations,

and a 1 ←10 machine yields the ordinary base-ten rep-

resentations. The process of

LONG DIVISION

can be

explained with the aid of this machine.

Variations on this idea can lead to some interesting

mathematical studies. Consider, for example, a 2 ←3

machine. This machine replaces three pennies in one

box with two pennies in the box one place to the left.

In some sense, this is a “base one and a half machine.”

For instance, placing 10 pennies in this machine yields