Man and Mystery v10 Math Wonders by Pablo C Agsalud Jr Rev 06.pdf

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A collection of intriguing topics and fascinating stories
about the rare, the paranormal, and the strange
Math Wonders
Volume 10
Discover what’s intriguing about this most hated subject
Pablo C. Agsalud Jr.
Revision 6
Foreword
In the past, things like
television,
and words and
ideas like
advertising, capitalism, microwave
and
cancer
all seemed too strange for the ordinary
man.
As man walks towards the future, overloaded with
information, more mysteries have been solved
through the wonders of science. Although some
things remained too odd for science to reproduce
or disprove, man had placed them in the gray
areas between
truth
and
skepticism
and labeled
them with terminologies fit for the modern age.
But the truth is, as long as the strange and
unexplainable cases keep piling up, the more likely
it would seem normal or natural. Answers are
always elusive and far too fewer than questions.
And yet, behind all the wonderful and frightening
phenomena around us, it is possible that what we
call
mysterious
today won’t be too strange
tomorrow.
This book might encourage you to believe or refute
what lies beyond your own understanding.
Nonetheless, I hope it will keep you entertained
and astonished.
The content of this book remains believable for as
long as the sources and/or the references from the
specified sources exist and that the validity of the
information remains unchallenged.
Intriguing Numbers
Wikipedia.org
Explore the world of mathematics and discover the most baffling
number theories.
Fibonacci number
Wikipedia.org
In mathematics, the Fibonacci numbers are the numbers in the following integer sequence:
0,1,1,2,3,5,8,13,21,34,55,89,144,…
By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each
subsequent number is the sum of the previous two.
In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence
relation
with seed values
The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci.
Fibonacci's 1202 book Liber Abaci introduced the sequence to Western European mathematics,
although the sequence had been described earlier in Indian mathematics. (By modern
convention, the sequence begins with F0 = 0. The Liber Abaci began the sequence with F1 =
1, omitting the initial 0, and the sequence is still written this way by some.)
A tiling with squares whose sides are successive Fibonacci numbers in length
A Fibonacci spiral created by drawing circular arcs connecting the opposite corners of squares
in the Fibonacci tiling; this one uses squares of sizes 1, 1, 2, 3, 5, 8, 13, 21, and 34.
Fibonacci numbers are closely related to Lucas numbers in that they are a complementary pair
of Lucas sequences. They are intimately connected with the golden ratio, for example the
closest rational approximations to the ratio are 2/1, 3/2, 5/3, 8/5, ... . Applications include
computer algorithms such as the Fibonacci search technique and the Fibonacci heap data
structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed
systems. They also appear in biological settings, such as
branching in trees,
arrangement of
leaves on a stem,
the fruit spouts of a
pineapple,
the flowering of
artichoke,
an uncurling
fern
and the arrangement of a
pine cone.
Origins
The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody.
In the Sanskrit oral tradition, there was much emphasis on how long (L) syllables mix with the
short (S), and counting the different patterns of L and S within a given fixed length results in
the Fibonacci numbers; the number of patterns that are m short syllables long is the Fibonacci
number F
m + 1
.
Susantha Goonatilake writes that the development of the Fibonacci sequence "is attributed in
part to Pingala (200 BC), later being associated with Virahanka (c. 700 AD), Gopāla (c.1135
AD), and Hemachandra (c.1150)". Parmanand Singh cites Pingala's cryptic formula misrau cha
("the two are mixed") and cites scholars who interpret it in context as saying that the cases
for m beats (Fm+1) is obtained by adding a [S] to F
m
cases and [L] to the F
m−1
cases. He
dates Pingala before 450 BCE.
However, the clearest exposition of the series arises in the work of Virahanka (c. 700AD),
whose own work is lost, but is available in a quotation by Gopala (c.1135):
Variations of two earlier meters [is the variation]... For example, for [a meter of
length] four, variations of meters of two [and] three being mixed, five happens.
[works out examples 8, 13, 21]... In this way, the process should be followed in all
mAtrA-vr.ttas (prosodic combinations).
The series is also discussed by Gopala (before 1135AD) and by the Jain scholar Hemachandra
(c. 1150AD).
In the West, the Fibonacci sequence first appears in the book Liber Abaci (1202) by Leonardo
of Pisa, known as Fibonacci. Fibonacci considers the growth of an idealized (biologically
unrealistic) rabbit population, assuming that: a newly born pair of rabbits, one male, one
female, are put in a field; rabbits are able to mate at the age of one month so that at the end
of its second month a female can produce another pair of rabbits; rabbits never die and a
mating pair always produces one new pair (one male, one female) every month from the
second month on. The puzzle that Fibonacci posed was: how many pairs will there be in one
year?
At the end of the first month, they mate, but there is still only 1 pair.
At the end of the second month the female produces a new pair, so now there are 2
pairs of rabbits in the field.
At the end of the third month, the original female produces a second pair, making 3
pairs in all in the field.
At the end of the fourth month, the original female has produced yet another new pair,
the female born two months ago produces her first pair also, making 5 pairs.
At the end of the nth month, the number of pairs of rabbits is equal to the number of new
pairs (which is the number of pairs in month n − 2) plus the number of pairs alive last month
(n − 1). This is the nth Fibonacci number.
The name "Fibonacci sequence" was first used by the 19th-century number theorist Édouard
Lucas.
The first 21 Fibonacci numbers F
n
for n = 0, 1, 2, ..., 20 are:

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