Leonardo da Pisa

Leonardo da Pisa

Leonardo da Pisa

(1171 - 1249) Italian mathematician, nicknamed

Fibonacci

(“figlio di Bonacci”) is best remembered for the 1202 AD book

Liber Abaci

. The book reports practical and theoretical arithmetic,

algebra

,

geometry

, and weights and measures mathematics. Leonardo’s Arab sources had been gathered from Hellene and other sources, like Babylonian

square root

, and re-written into base 10 numerals after 800 AD. In the modern era the

Fibonacci sequence

recalls a possible aspect of Leonardo’s math life.

Leonardo’s father Gugliemo, better known as “Bonaccio,” was a merchant in Bugia, a port now part of Algeria. Bonaccio regularly used

Arabic numerals

and

Egyptian fraction

arithymetic in the course of his work. Leonardo was born in Pisa (then its own sovereign republic, now part of Italy) but spent his formative years in Bugia, helping his father. Leonardo described the theoretical side of Arabic-Hindu numeral arithmetic. The base 10 numeral aspect of the

Liber Abaci

had followed suggestions of Pope Sylvester. Pope Sylvester in 999 AD required Arabic numerals to be used in Latin mathematical documents.

Fibonacci’s 500 page book was fully translated from Latin to English by L.E. Sigler in 2002. Previously the book was translated by chapters, or by math topic. The first 1/4 of the book demonstrates practical, and theoretical examples written in three

subtraction

arithmetic styles. The subtraction styles were implemented by Arbs in 800 AD. Arabs had modified the Greek multication styles. Leonardo’s theoretical arithmetic otherwise followed Greek and

Egyptian math

methodologies. For example, Leonardo’s arithmetic reports 1500 BCE to 2000 BCE

2

n

arithmetic that had been translated into Hindu-Arabic numerals by Arabs. Arabs and Leonardo reported three distinct arithmetic notations. The first notation allowed vulgar fractions, and

aliquot parts

, to be linearly summed to larger vulgar fractions. Egyptian notations had not allowed vulgar fractions in final answers. The second, and third, notations condensed aliquot part information into circle patterns, showing Greeks, like Heron’s,

factoring

style. Sigler named the third notation after Euclid. All three notations reveal early uses of the

fundamental theorem of arithmetic

. All three notations were condensed, shortening Greek and Egyptian unit fraction answers.

It was in the

Liber Abaci

that Fibonacci indirectly mentioned a

sequence

for which he is famous to modern mathematicians. A form of the sequence may have been known to recreational mathematicians prior to Fibonacci.

James Joseph Sylvester

, in 1891, suggested that the sequence was used by Fibonacci. Sylvester connected a proposed greedy algorithm to

rational number

conversions within

n

-steps. Leonardo had written a two-step process based in a non-algorithmic method.

Fibonacci’s seventh method reported a sequence of unit fractions written in the first of three medieval notations, citing three alternatives, each using two-steps:

a.

4

9

-

1

13

=

3

13

×

49

=

(

1

319

+

0

637

+

1

617

+

1

319

+

1

13

), not elegant

b.

4

49

-

1

14

=

7

14

×

49

=

1

2

+

0

49

+

1

14

, elegant

c.

4

49

=

1

7

×

4

7

=

1

7

×

(

4

7

-

1

2

=

1

14

)

=

1

2

+

0

49

+

0

49

+

1

14

)

alternate elegant

A 2-step conversion method, using

multiples

26 and 6, was used by an EMLR student scribe to convert

1

8

. Ahmes also used a related two-step method to convert

28

97

, solving

2

97

, and

26

97

, by combining the unit fraction series. Hence, Fibonacci’s 2-step method was a restatement of earlier traditions.

Leonardo also wrote books on geometry and

Diophantine equations

, discussing these and other topics in the remaining 374 pages of Liber Abaci.3

Galileo was born before Stevin’s two books were approved by the Paris Academy, and thus learned square root from the tradition of Fibonacci:

Note that scribal shorthand notes suggested by academics prior to Dec. 2012 suggested incomplete square root steps even though major operational aspects of the same class of arithmetic and

algebraic

pesu steps have been found in the medieval era. In May 2013 the shorthand notes of Galileo reveal the same method was also used by Fibonacci and Archimedes.

In the square root of five step 3 was not needed.

B. square root of six (6) ,

step 1: estimated Q = 2, R = (6 -4)/4 = 1/2 such that

(2 + 1/2)^2 = 6 + (1/2)^2, error1 = 1/4

step 2:

reduced

1/4 error that divided by (2 + 1/2)

such that

1/4 x (2/10) = 1/20.

hence (2 + 1/4 - 1/400) =((2 + 99/400)*2, error2 = 1/400

Ahmes, Archimedes and Fibonacci may have stopped at this point and recorded

(2 + 99/400) as a unit fraction series

[2 + (80 + 10 + 8 + 1)/400] =

[2 + 1/5 + 1/40 + 1/50 + 1/400 ]

step 3 (as included Archimedes square root of three method) seemed optional

divided 1/400 by (400/1798) = 1/1798,

hence (2 + 99/400 - (1/1798)^2 = accurate (1/1798)^2

Archimedesâ actual

square root method

would have recorded

[2+ 1/5 + 1/40 + 1/50 + 1/400]

with a note that a longer series, with an error of (1/1798)^2 was easily found.

Looking forward from the life of Fibonacci,

Simon Stevin

formalized modern base 10 positional decimals, by using an algorithm, about 300 years after Leonardo’s death. John Napier improved the utility of Stevin’s decimal notation in ways that modern students appreciate, retaining base 10 numerals reported by Leonardo.

References

1

L.E. Sigler,

Fibonacci’s Liber Abaci, Leonardo Pisano’s Book of Calculations

, Springer, 2002.

2

Heinz LÃÂ¼neburg,

Leonardi Pisani Liber Abbaci oder LesevergnÃÂ¼gen eines Mathematikers

, Mannheim: B. I. Wissenschaftsverlag , 1993.

3

Oystein Ore

,

Number Theory

and its History

, McGraw-Hill, 1948.

.1

External links

•

http://liberabaci.blogspot.com

Sigler’s blog about his book

•

http://arxiv.org/PS_cache/math/pdf/0701/0701624v1.pdf

Heron and Fibonacci

addition

•

http://rmprectotable.blogspot.com/

2/n tables

•

http://emlr.blogspot.com

The blog of an EMLR student

Title

Leonardo da Pisa

Canonical name

LeonardoDaPisa

Date of creation

2014-03-22 7:22:34

Last modified on

2014-03-22 7:22:34

Owner

Mravinci (12996)

Last modified by

milogardner (13112)

Numerical id

41

Author

Mravinci (13112)

Entry type

Biography

Classification

msc 01A35

Synonym

Fibonacci

Synonym

Leonardo Pisano

Generated on Thu Feb 8 19:08:25 2018 by

LaTeXML