Transcript FIBONACCI Presentation by: Angela Tersigni

FIBONACCI
Presentation by: Angela Tersigni
Known as : Leonardo Pisanus
Leonardus filius Bonacci
Leonardus Bigollus
“Bigollo” – Tuscan dialect – meaning “blockhead” or “traveller”
Never referred to himself as “Fibonacci”
Medieval biographies and portraits rare: All pictures and statues of
Leonardo of Pisa are only from artists’ imaginations
Exact date of birth and death not known
Background History
Leonardo was born 2 centuries after cultural
and economic slowdown in Europe known
as Dark Ages
Commercial Revolution was well underway
Both local and international trade occurred
Mediterranean Sea linked regions
representing different religions, political
entities and cultures
Three Italian cities dominated imports and
exports: Venice, Genoa and Pisa
Pisa: population of approx. 10,000 and was
a “commune” – independent republic
More History…
 His father, Giuliemo, held a diplomatic post in Bugia
(North Africa)
 Leonardo travelled extensively with his father:
Egypt, Syria, Constantinople, Sicily, France, Greece
 Leonardo acquired much knowledge of various
mathematical systems and texts during his travels
 Some argue he is not a true mathematician but only
an author of a very successful text (Liber Abaci)
 Yet, he did also compile his own techniques,
theorems and facts when he published his findings
 When we think of Fibonacci, we think of his
introduction on the Hindu-Arabic numerals (HAN) to
the Western world and the famous Fibonacci
sequence
Fibonacci’s Contributions
His writings provided us with more than HAN
and the Fibonacci sequence:
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Extraction of square and cube roots
Rule of 3 and Rule of 5 for problem solving
Averages
Compound interest
Diophantine equations; especially his unique method for finding
Pythagorean Triples
“res” term for unknown quantities
Chinese Remainder Theorem
Summing arithmetic and geometric series
Rule of False Position and Double False Position for problem solving
Working backwards for problem solving
Casting Out 9s to check for arithmetic accuracy
Roman Numerals
Not a positional system
Abacus provided place-value which was
missing in their notation
No need to memorize facts
Abacus made addition and subtraction quick
and easy but multiplication and division slow
and tedious
Disadvantage (abacus): work vanished once
done and it was impossible to explore
numbers and their relationships
Leonardo’s Contact with HAN
Arab businessmen travelled extensively
(China, Africa, Russia, India) and they
collected much scientific knowledge and
geography
Muslims well-known for their wealth of
knowledge. In the 9th century, Muslims were
reading Aristotle, Euclid, Hippocrates, Galen,
Ptolemy
They corrected and added their own findings
to these works
In India, Arabs acquired: algebra continued
from Greeks, place-value, numbers 1 to 9
and 0
By 7th century in India, the harmonious use
of 1-9, 0 and place-value was wellestablished
Advantage of HAN for business: work did not
vanish, easy to double-check work without
calculation from the beginning
Leonardo saw a better advantage: to explore
relationships between numbers
In 825 A.D., al-Khwarizmi wrote a book on
Hindu numerals which was translated by
Gerard of Cremona in 10th century
Translation was only known to small group
of scholars
Leonardo made these “Hindu numerals” (his
own words) well-known in Liber Abaci
Liber Abaci
Book of Abacus or Calculation
Ironically, it actually freed Western
mathematics from the abacus
Written in 1202 and revised in 1228 to
include mathematics for commercial
use
Fortunately, the 1228 version has
come down to us
Chapters 1-7:
-Hindu-Arabic numerals
-Arithmetic with HAN
Chapters 8-11: Commerce:
alloying, pricing,
conversions, interest,
wages, profit, barter,
negative numbers (debt)
Chapters 12-13: Recreational mathematics:
puzzles, “rabbit problem”
Chapter 14:
Extraction of Roots:
Approximation: (a^2+r)^1/2 = a + r/2a
Chapter 15:
Geometry:
Deals with geometry arithmetically
(unlike previous Greek works)
Use of “res” (unknown quantity) and
“census” (square of the unknown quantity
Lattice Multiplication
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02
17
24
26
15
13
18
17
13
09
00
______________
139676498390
Answer: 23,958,233 x 5,830 = 139,676,498,390
Liber Abaci Continued…
 Leonardo used many examples in describing
arithmetic using HAN
 Tables of facts included in his writing
 Memory of facts seen as a disadvantage of this
numeral system
 Tables for fractions reduced to unit fractions:
example: 3/8 reduced to 1/8 1/4 (1/8 +1/4)
Egyptian influence?
 Leonardo introduced horizontal bar for fractions
 He wrote mixed fractions from right to left: example:
¼ 4 instead of 4 ¼ (Arabic influence)
Problem Solving Methods
Double False Position
-Assumed tall tower was 10 ft. from the fountain
Where should the
Fountain be
Placed such that
2 birds flying at
The same rate
And leaving from
Each tower will
Arrive at the
Fountain at the
Same time?
-Using Pythagorean theorem: calculated first bird travelled 1700 ft. and
second bird 2500 ft.
-Added 5 ft. to distance of the fountain from the taller tower,
subtracting 5 ft. from the shorter tower
-This time the squares of the hypotenuses came out to 1,825 ft.and
2,125 ft.
-Original difference in squares of the hypotenuses was 800, and the
alteration by 5 ft. reduced it by 500 ft., another alteration by 3 ft. in the
same direction would reduce it by 300 ft., eliminating it completely.
-Thus, the fountain is 18 ft. from the taller tower and 32 ft. from the
shorter tower
Leonardo used many problem examples to
illustrate his problem-solving methods
His problem-solving methods as well as the
problems themselves were taken from
Muslim and Hindu texts
At times, he modified the problems to make
them more relevant and useful for
Europeans
Note: Leonardo always drew diagrams and
tables to better illustrate the problem and the
solution. He introduced the lines to show
which numbers were multiplied (example:
Rule of 3 and 5)
Leonardo would usually provide many
different strategies to solve one problem
Solving Backwards
Place-Value Game
The following game in Liber Abaci
demonstrates how comfortable Leonardo
was with the concept of HAN:
A group of men is seated in a row –one wearing a ring on a certain
joint of a certain finger of one hand. A leader counts the wearer’s
position in the row, doubles it, adds 5 to the product, multiplies the
sum by 5 and adds 10. To this figure, he adds a number indicating
the particular finger (little finger on left hand counts as 1 and the
thumb on the right hand is 10). Multiply this sum by 10. Add the
number indicating the joint of the finger on which the ring is placed:
using 1 for finger tip, 2 for middle and 3 for the lowest part.
The “guesser”, given the result, subtracts 350. He is then able to tell
the audience the ring wearer and on which hand and part of the finger
he is wearing the ring on.
Modern notation: 10[5(2x + 5) + 10 + y] + z where x is position in row of
Ring wearer, y is finger on which he wears it and z is the joint
Place-Value Game
The following game in Liber Abaci
demonstrates how comfortable Leonardo
was with the concept of HAN:
A group of men is seated in a row –one wearing a ring on a certain
joint of a certain finger of one hand. A leader counts the wearer’s
position in the row, doubles it, adds 5 to the product, multiplies the
sum by 5 and adds 10. To this figure, he adds a number indicating
the particular finger (little finger on left hand counts as 1 and the
thumb on the right hand is 10). Multiply this sum by 10. Add the
number indicating the joint of the finger on which the ring is placed:
using 1 for finger tip, 2 for middle and 3 for the lowest part.
The “guesser”, given the result, subtracts 350. He is then able to tell
the audience the ring wearer and on which hand and part of the finger
he is wearing the ring on.
Modern notation: 10[5(2x + 5) + 10 + y] + z where x is position in row of
Ring wearer, y is finger on which he wears it and z is the joint
Pratica Geometriae
Written in 1220
8 chapters of theorems mainly based
on Euclid’s “Elements” and “On
Divisions”
Practical measurement problems with
theoretical geometry
Problems treated algebraically
Trigonometry problems aimed at
surveyors
Prince Frederick ll
 Emperor and king of the 2 Sicilies
 Sicily was a meeting ground for Christian and
Muslim cultures of Europe and North Africa
 Frederick sent out questionnaires and scientific
problems to scholars in Egypt, Syria, Iraq, Asia and
Yemen in order to acquire knowledge and
understanding
 Indication of Leonardo’s fame at the time: Frederick
travelled to Pisa with his menagerie and entourage
to visit and question Leonardo
 Leonardo described the 3 problems posed to him in
Liber Quadratorum and Flos
Liber Quadratorum
“Book of Squares” written in 1225
Not as influential but best work
Major contribution to number theory between
Diophantus (4th century Alexandria) and
Pierre de Fermat (17th century)
Illustrated algebraic problems with lines or
geometric figures
He assigned letter labels to his lines which
represented the unknowns (start of algebraic
notation)
The following will show some work contained
in Liber Quadratorum:
 Leonardo’s method of finding Pythagorean triples:
most original thinking about numbers at the time
Notes square numbers can be constructed as sums of
Odd numbers:
1 = 1^2
1+3 = 2^2
1+3+5 = 3^2 etc..
The sequence and series of square numbers always
rise through regular addition of odd numbers.
Algebraically: n^2 + (2n+1) = (n+1)^2
Thus, when you wish to find 2 square numbers whose addition
Produces a square, take any odd square as one of the 2 square
Numbers and find the other square by the addition of all odd
Numbers up to but not including the odd square number.
 His general solution was worked out algebraically (using
words), not geometrically!
Some more of his original work include:
 x^2 + y^2 and x^2 – y^2 could not both
be squares ( Assertion: no right triangle
with rational sides could equal a square with a
rational side). The proof was incomplete; Fermat
sketched the proof
 x^4 – y^4 cannot be a square
 Expressed a number as 2, 3 or 4 squared numbers
or squared fractions
 Congruum: A number, K, is a congruum if:
K=ab(a+b)(a-b) ; if a+b is even or
K=4ab(a+b)(a-b) ; if a+b is odd where
a and b are integers
The following problem was presented to him during his meeting
with Prince Frederick ll in 1225:
Flos
“Flower”
Written in 1225
Contained the 2nd problem posed to him by
Prince Frederick ll
Although this problem was familiar to Arab
mathematicians, it was Leonardo who
recognized the fact that Euclid’s method for
solving such equations by square roots
would not work here. The solution required
cube roots which Euclid could not extract
using a ruler and compass
Problem:
x^3 +2x^2 +10x = 20 ; solve for x
Leonardo’s solution:
X= 1.3688081075 (modern
notation)
 He reasoned that 1<x<2 and narrowed down the
answer by trial and error
 He does not give any indication how he found it
 He preserved the sexagesimal fractions (Babylonian)
 This was a substitute for decimal fractions which did
not come to the West until late 16th century (probably
due to the fact that the monetary system did not have
a decimal relationship)
Conclusion
At first, HAN had opposition: business could
make due without it, difficult to learn tables,
records could be easily altered
Pure mathematics almost at a standstill 300
years after his death
By 15th century, the HAN system was
displacing the Roman numeral system
Printing press made his work better known;
he paved the way for algebraic notation and
aided scientific progress in the West
Many mathematicians borrowed from him
(Fermat, Pascal, Descartes etc..)
The well-known problem…
 Fibonacci Sequence: 1st known recursive sequence
in Western world
 Leonardo recognized it as such but attached no
other special importance to it
 He did not name this sequence after himself; Lucas
coined the phrase in 1877 after rediscovering it and
coming up with his own Lucas sequence
 German astronomer, Johannes Kepler in 1611: as n
increases, then the ratio of F(n)/F(n)+1 approaches
the “golden ratio”
 17th century formula: u(n+2) = u(n+1) +u(n)
 1830: A. Braun – bracts on pinecones
 1840s: Jacques Binet:
F(n)=1/(5)^1/2[[1+(5)^1/2]/2]^n - 1/(5)^1/2[[1-(5)^1/2]/2]^n
 1920: Church (botanist) – sunflower heads (spirals)
 1920s: Hambidge (botanist) – “dynamic symmetry”
 Dynamic symmetry: represented by a logarithmic /
equiangular spiral that does not change shape as it
is growing
Equiangular spiral can be drawn using the Fibonacci
squares and connecting their corners with a quarter
circle with the square side as the radius
Dynamic symmetry seen in shells,
animal horns, claws, beaks, ocean
waves, leaves on corn plants and trees
1963: Fibonacci Society was formed in
California – mathematicians share
ideas and stimulate research
surrounding the Fibonacci numbers
What would Leonardo think about this
peaked interest in his famous problem?
Summed arithmetic series:
2+4+6+8 = 4(8+2)/2 = 20
1+2+…+n = n(n+1)/2
Sum of odd numbers:
1+3+5+7+…+n = n^2 ; where n is the rank of the
series
Summed geometric series:
1+2+4+8 = 16-1 = 15 ; where 16 is the next number in
the series
Solved some Diophantine equations of
2nd degree
Are we looking too hard???
If we look hard enough, will we find
the pattern we are looking for all
around us?