Read Online The Unimaginable Mathematics of Borges' Library of Babel by William Goldbloom Bloch - Free Book Online
Authors: William Goldbloom Bloch
Tags: Non-Fiction
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the 25 symbols in a minimum of four places to
distinguish 24,000 separate numbers, because
25 4 =
390,625
while
25 3 =
15,625
which doesn't provide enough
distinct signifiers to take us up to base 24,000. Anyway, not only wouldn't
this convention save much space, it also leads back to the previous dilemma:
writing out the names of the numbers will result in waterfalls of gibberish.
Finally, a
potential catalogue entry might take a different tack. It might give
coordinates, such as, "Go up ninety-seven floors, move diagonally left
four thousand hexagons, and then move diagonally right another two hundred and
twenty." Although this might, at first blush, seem appealing, the same
sorts of problems arise, for most hexagons are unimaginably far away. The
example provided above works simply because the numbers involved—97, 4,000, and
220—are so miniscule, so accessible. The Library is neither.
The
Library is its own catalogue. Any other catalogue is unthinkable.
Math
Aftermath: Numb and Number (Theory)
A metaphysician is one who,
when you remark that twice two makes four, demands to know what you mean by
twice, what by two, what by makes, and what by four. For asking such questions
metaphysicians are supported in oriental luxury in the universities, and
respected as educated and intelligent men.
—H.
L. Mencken, A Mencken Chrestomathy
Below are two outgrowths from
the sprawling yet spare field of number theory; together they form a pair of
relatively straightforward mathematical confections. Both revolve around using
prime numbers decisively to reach interesting conclusions.
Consider the
25 1,312,000 distinct volumes in the Library: a simple rethinking of
this number will produce a result surely unimagined by Borges. Now, as we all
know, the number 25 factors into 5 5, so
A prime number is a
positive integer greater than one that is divisible only by itself and by one.
The unique factorization theorem, proved by Euclid in The Elements, says
that every positive integer is decomposable into exactly one product of
primes, each of which is raised to a power greater than or equal to one. For
example, we all know that 100 = 10 10, and it's also true that 100 = 4 25. So, what is 100 equal to, 10 10 or 4 25? Of course you're laughing at us, because 100 is obviously
equal to both products. Neither of these answers, though, is written exactly as
a product of primes, in which each prime is raised to a power greater than or
equal to one. Based on the two factorizations—10 10 and 4 25—it's easy to see that
100 = 10 10 = (2 5) (2 5) = (2 2) (5 5) = (2 2 ) (5 2 )
and
100 = 4 25 = (2 2) (5 5) = (2 2 ) (5 2 ).
Because 100 is so familiar,
it's probably not surprising to you that both of the initial factorizations
lead to the unique one. And perhaps it is equally intuitive that no matter how
large an integer we begin with, no matter how we might try, there will be only
one way to factor it into powers of primes. Still, it's nice to know that
Euclid showed that it must always be true.
By the work
above, 5 2,624,000 is a unique factorization of 25 1,312,000 into
primes, each raised to a power greater than or equal to one. In this case,
plainly the number of distinct books uniquely decomposes to one prime (5)
raised to a power greater than one (2,624,000). It follows that the only
numbers that can divide 25 1,312,000 are powers of five. Now, as is
easily inferred from the story, each hexagon in the Library contains 640 books.
The number 640 uniquely factors into 2 7 5, and so the number 640 does not divide 25 1,312,000 ,
for
and none of the seven 2s in
the denominator may divide any of the millions of 5s in the numerator. This
means that the books do not exactly fill out all the hexagons, which entails
that either the Library is not complete (!!!), or that there is a
special hexagon that is not full, or that at least one
distinguish 24,000 separate numbers, because
25 4 =
390,625
while
25 3 =
15,625
which doesn't provide enough
distinct signifiers to take us up to base 24,000. Anyway, not only wouldn't
this convention save much space, it also leads back to the previous dilemma:
writing out the names of the numbers will result in waterfalls of gibberish.
Finally, a
potential catalogue entry might take a different tack. It might give
coordinates, such as, "Go up ninety-seven floors, move diagonally left
four thousand hexagons, and then move diagonally right another two hundred and
twenty." Although this might, at first blush, seem appealing, the same
sorts of problems arise, for most hexagons are unimaginably far away. The
example provided above works simply because the numbers involved—97, 4,000, and
220—are so miniscule, so accessible. The Library is neither.
The
Library is its own catalogue. Any other catalogue is unthinkable.
Math
Aftermath: Numb and Number (Theory)
A metaphysician is one who,
when you remark that twice two makes four, demands to know what you mean by
twice, what by two, what by makes, and what by four. For asking such questions
metaphysicians are supported in oriental luxury in the universities, and
respected as educated and intelligent men.
—H.
L. Mencken, A Mencken Chrestomathy
Below are two outgrowths from
the sprawling yet spare field of number theory; together they form a pair of
relatively straightforward mathematical confections. Both revolve around using
prime numbers decisively to reach interesting conclusions.
Consider the
25 1,312,000 distinct volumes in the Library: a simple rethinking of
this number will produce a result surely unimagined by Borges. Now, as we all
know, the number 25 factors into 5 5, so
A prime number is a
positive integer greater than one that is divisible only by itself and by one.
The unique factorization theorem, proved by Euclid in The Elements, says
that every positive integer is decomposable into exactly one product of
primes, each of which is raised to a power greater than or equal to one. For
example, we all know that 100 = 10 10, and it's also true that 100 = 4 25. So, what is 100 equal to, 10 10 or 4 25? Of course you're laughing at us, because 100 is obviously
equal to both products. Neither of these answers, though, is written exactly as
a product of primes, in which each prime is raised to a power greater than or
equal to one. Based on the two factorizations—10 10 and 4 25—it's easy to see that
100 = 10 10 = (2 5) (2 5) = (2 2) (5 5) = (2 2 ) (5 2 )
and
100 = 4 25 = (2 2) (5 5) = (2 2 ) (5 2 ).
Because 100 is so familiar,
it's probably not surprising to you that both of the initial factorizations
lead to the unique one. And perhaps it is equally intuitive that no matter how
large an integer we begin with, no matter how we might try, there will be only
one way to factor it into powers of primes. Still, it's nice to know that
Euclid showed that it must always be true.
By the work
above, 5 2,624,000 is a unique factorization of 25 1,312,000 into
primes, each raised to a power greater than or equal to one. In this case,
plainly the number of distinct books uniquely decomposes to one prime (5)
raised to a power greater than one (2,624,000). It follows that the only
numbers that can divide 25 1,312,000 are powers of five. Now, as is
easily inferred from the story, each hexagon in the Library contains 640 books.
The number 640 uniquely factors into 2 7 5, and so the number 640 does not divide 25 1,312,000 ,
for
and none of the seven 2s in
the denominator may divide any of the millions of 5s in the numerator. This
means that the books do not exactly fill out all the hexagons, which entails
that either the Library is not complete (!!!), or that there is a
special hexagon that is not full, or that at least one
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