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Fibonacci Numbers & Counting
Reprinted w/permission of Asimov On Numbers
Consider Leonardo Fibonacci, for instance, the most accomplished mathematician
of the Middle Ages. (He was born in Pisa, Italy, so he is often called
Leonardo of Pisa.) About 1200, when Fibonacci was in his prime, Pisa was
a great commercial city, engaged in commerce with the Moors in North Africa.
Leonardo had a chance to visit that region and profit from a Moorish education.
The Moslem world by then learned of a new system of numeration from Hindus.
Fibonacci picked it up and in a book, Libber Abaci, published in
1202, introduced these “Arabic numerals” and passed them on
to a Europe still suffering under the barbarism of the Roman numerals.
Since Arabic numerals are only about a trillion times as useful as Roman
numerals, it took a mere couple of centuries to convince European merchants
to make the change.
In this same book, Fibonacci introduces the following problem: “How
many rabbits can be produced from a single pair in a year if every month
each pair begets a new pair, which from the second month on become productive,
and no deaths occur?” (It is also assumed that each pair consists
of a male and female and that rabbits have no objection to incest.)
In the first month, we begin with a pair of immature rabbits, and in
the second month, we still have one pair, but now they are mature. By
the third month, they have produced a new pair, so there are two pairs,
one mature, one immature. By the fourth month the immature pair has become
mature and the first pair has produced another immature pair, so there
are three pairs, two mature and one immature.
You can go on if you wish, reasoning out how many pairs of rabbits there
will be each month, but I will give you the series of numbers right now
and save you the trouble. It is:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144
At the end of the year, you see, there would be 144 pairs of rabbits
and that is the answer to Fibonacci’s problem.
The series of numbers evolved out of the problem is the “Fibonacci
Number Series” and the individual numbers of the series are
the “Fibonacci numbers.” Looking at the series shows each
number (from the third member on) is the sum of the two preceding numbers.
This means we needn’t stop the series at the twelfth Fibonacci number
(F12). We can construct F13 easily enough by adding F11 and F12. Since
89 and 144 are 233, that is F13. Adding 144 and 233 gives us 377 or F14.
We can continue with F15 equal to 610, F16 equal to 987, and so
on for as far as we care to go. Simple arithmetic, nothing more than addition,
will give us all the Fibonacci numbers we want.
To be sure, the process gets tedious after a while as the Fibonacci numbers
stretch into more and more digits and the chances of arithmetical error
increase. One arithmetical error anywhere in the series, if uncorrected,
throws off all the later members of the series.
But why should anyone want to carry the Fibonacci sequence on and on
and on into large numbers? Well, the series has its applications. It is
connected with cumulative growth, as the rabbit problem shows, and, as
a matter of fact, the distribution of leaves spirally about a lengthening
stem, the scales distributed about a pine cone, the seeds distributed
in the sunflower center, all have an arrangement related to the Fibonacci
series. The series is also related to the “golden section,”
which is important to art and aesthetics as well as to mathematics.
But beyond all that, there are always people who are fascinated by large
numbers. (I can’t explain the fascination but believe me it exists.)
And if fascination falls short of working away night after night with
pen and ink, it’s now possible to program a computer to do the work,
and get large numbers it would be impractical to try to work out the old-fashioned
way.
Recreational Mathematics Magazine lists the first 571 Fibonacci numbers
as worked out on an IBM 7090 computer. The fifty-fifth Fibonacci number
passes the trillion mark, so we can say F55 is greater than T-1.
From that point on, every interval of fifty-five or so Fibonacci numbers
(the interval slowly lengthens) passes another T-number. Indeed, F481
is larger than a googol. It is equal to almost one and a half googols,
in fact.
Those multiplying rabbits, in other words, will quickly surpass any conceivable
device to encourage their multiplication. They will outrun any food supply
that can be dreamed up, any room that can be imagined. There might be
only 144 at the end of a year, but there would be nearly 50,000 at the
end of two years, 15,000,000 at the end of three years, and so on. In
30-years there would be more rabbits than there are subatomic particles
in the known universe, and in 40-years there would be more than a googol
of rabbits.
To be sure, human beings do not multiply as quickly as Fibonacci’s
rabbits, and old human beings do die. Nevertheless, the principle
remains. What those rabbits can do in a few years, we can do in a few
centuries or millenniums. Soon enough. Think of that when you tend to
minimize the population explosion.
For the fun of it, I would like to write F571, which is the largest number.
Leonardo Fibonacci was born in Pisa Italy about 1170 and he died about
1230. His greatest achievement was in popularizing the Arabic numerals
in his book Liber Abaci. In this, he had been anticipated by the
English scholar Adelard of Bath (tutor of Henry II before that prince
had succeeded to the throne) a century earlier. It was Fibonacci’s
book, however, that made the necessary impression.
But why did he call it Liber Abaci, or Book of the Abacus? Because, oddly
enough, the use of Arabic numerals was implicit in the “abacus,”
a computing device that dates back to Babylonia and the earliest days
of history.
The abacus, in its simplest form, is most easily visualized as a series
of wires on each of which ten counters are strung. There is room on the
wire to move one or more of the counters some distance to the right or
left.
If you want to add 5 and 4r, for instance, you move five counters
leftward, then four more, and count all you have moved-nine. If you want
to add five and eight, you move five counters, but only have five more,
not eight more, to move. You move the five, convert the ten counters
into one counter in the wire above, then move the remaining three. The
counters in the wire above are “tens,” so you have one ten and
three ones for a total of thirteen.
The wires represent, successively, units, tens, hundreds, thousands,
and so on, and Arabic numerals, in essence, give the number of counters
moved in each of the wires. The manipulations required in the abacus are
those required in Arabic numerals. What was needed was a special symbol
for a wire in which no counters were moved. This was zero, 0, and Arabic
numerals were in business.
Editor’s Note: It’s recommended traders study
Fibonacci Numbers, as the numbers do seem to be reflected in the markets.
In particular, .382%, 50% and .618% seem very significant.
Many traders watch for resistance and support areas at these numbers
by calculating these levels between important highs and lows. They may
also be used as far as time is concerned, not just prices.
Some traders say time is more important than price. Many times Gann said
in his writings “when time is up watch for a change of trend.”
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