One of my teachers -- I think it was Mr Smith in "College Algebra" during my freshman year at Westminster -- gave me these words to remember:
When confronted
with a statement
that seems true
for all positive integers
the wise student
uses mathematical induction
as her proof technique.
That advice has served me well. From it I move to a pair of mathematical statements (the aforementioned Principle of Mathematical Induction and the Pigeonhole Principle) that are among my favorite mathematics poems.
Principle of Mathematical Induction
Bilingual pronunciation note: The expression S1 is read "ess-sub-one"
and other similar expressions are read similarly,
including the final Sk+1 as "ess-sub-kay-plus-one."
Suppose we have an infinite list of statements,
S1 S2, S3, . . . -- one for each positive integer.
Then all of these statements are valid
if these two conditions hold:
S1 is valid;
For any positive integer k,
if Sk is valid then so is Sk+1.
Pigeonhole Principle
Dear reader: despite its informal nature, the statement below
is a canonical instance of a very general idea -- it can, for example, be applied
to real pigeons as they nest in the compartments of a pigeon house
or to a collection of lucky numbers to be distributed to fortune-seekers.
If the apartment-house manager
has more letters to distribute
than there are pigeonholes
on the reception room wall,
then at least one pigeonhole
will get more than one envelope.
Friday, July 29, 2011
Mathematical Induction -- principle, perhaps poem
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