Contribution of Ghiyath Al'Din Jamshid Mas'Ud Al'Kashi in the Field of Mathematics

5/22/2018 Contribution of Ghiyath Al'Din Jamshid Mas'Ud Al'Kashi in the Field of Mathematics

1/33

Well Come to thepresentation of Math

term paper(Group-6)Prepared By:


2/33

Contribution of GhiyathalDin Jamshid MasudalKashi in the field of

mathematics


3/33

IntroductionAl-Kashi came from early fifteenth

century. He is a part of great momentwhen Islamic culture was welldeveloped. That made his inventory

was greatly affected so much by thisculture.


4/33

IntroductionAs he lived, Al-Kashi made a great invention to

mathematics field. He was the man whodeveloped and put decimal fraction into aconvenient notation. He introduced the way weare recognizing by now the notation of

fraction which is presented by horizontal line

separating two integer parts, namely a and bfrom decimal part. He was also the man whocompleted Hindu-Arabic place value system.


5/33

He was also paid for his contribution by anastronomer who ruled a major domain incentral asia that moment. The astronomer

was called Ulugh Beg, the one who wasengaged in ambitious project of completingsinus and tangent table.Al-Kashi made

sin table based on a circle of radius 60. Hewas also the man who came with an idea oftriple angle formula as we have already

known at this moment.

Introduction


6/33

Al-Kashi (1380 - 1429) was a renownedmathematician and astronomer in early fifteenth-

century Persia and Central Asia. In 1424 Al-Kashi published a treatise on circumference, inwhich he calculated "pi", the ratio of a circle's

circumference to its diameter, to nine decimalplaces. Nearly two hundred years would passbefore another mathematician surpassed thisachievement.

A BRIEF HISTORY OF GHIYATHAL-DIN JAMSHID MASUD AL-KASHI


7/33

A BRIEF HISTORY OF GHIYATHAL-DIN JAMSHID MASUD AL-KASHIThere are various spellings given for Al-Kashi s full name, but the Standard English

transliteration seems to be Ghiyath al-DinJamshid Mas ud al-Kashi. He was born in1380 in Kashan, a desert town in Iran locatednear the Central Iranian Range


8/33

Books of Al - KashiMiftal isb(TheKey of Arithmetic)or Miftal ussb fi ilm al isb(TheKey of Reckoners in the Science ofArithmetic).

Risal muityya(Treatiseon the Circumference)Talkhis al Miftah(Compendiumof the Key)Risla dar shar i lt i rasd (Treatise on the Explanation of ObservationalInstruments) ukhtasar dar lim i hayat(Compendiumon the Science ofAstronomy)orRisla dar hayat ( TreatiseonAstronomy).Zij i Khaqni f takml i Zij i lkhn(Khaqni Zijperfection of lkhnZij)

Risla al watar wal jaib(Treatiseon the Chord and Sine)Ilkat an Nuzha(Supplementto the Excursion)


9/33

Books of Al - KashiSullam al-sam fi all ishkl waqaa lil-muqaddim fil-abd

wal-ajrm( The Stairway of Heven, on Resolution of Difficulties Met byPredecessors in the Determination of Distances and Sizes)

Nuzha al-adiq fi kayfiyya ansa al-la al-musamm bi tabaq al-mantiq(The Garden Excursion,; on the Method of Construction of theInstrument Called Plate of Heavens)


10/33

In The Treatise on the Chord and Sine, al-Kashicomputed sin 1 to nearly as much accuracy as his

value for .He also improved the method that hisancestor Sharaf al-Dn al-Ts used. In algebraandnumerical analysis, he developed an iterative method

for solving cubic equations In order to determine sin

1, al-Kashi discovered the following formula oftenattributed to Franois Vitein the 16th century:

Ghiyath Al-Din Jamshid Al-Kashi

(1380-1429)
http://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Algebrahttp://en.wikipedia.org/wiki/Numerical_analysishttp://en.wikipedia.org/wiki/Iterative_methodhttp://en.wikipedia.org/wiki/Cubic_equationhttp://en.wikipedia.org/wiki/Vi%C3%A8te%27s_formulahttp://en.wikipedia.org/wiki/Fran%C3%A7ois_Vi%C3%A8tehttp://en.wikipedia.org/wiki/Fran%C3%A7ois_Vi%C3%A8tehttp://en.wikipedia.org/wiki/Fran%C3%A7ois_Vi%C3%A8tehttp://en.wikipedia.org/wiki/Fran%C3%A7ois_Vi%C3%A8tehttp://en.wikipedia.org/wiki/Vi%C3%A8te%27s_formulahttp://en.wikipedia.org/wiki/Cubic_equationhttp://en.wikipedia.org/wiki/Cubic_equationhttp://en.wikipedia.org/wiki/Cubic_equationhttp://en.wikipedia.org/wiki/Iterative_methodhttp://en.wikipedia.org/wiki/Iterative_methodhttp://en.wikipedia.org/wiki/Iterative_methodhttp://en.wikipedia.org/wiki/Numerical_analysishttp://en.wikipedia.org/wiki/Numerical_analysishttp://en.wikipedia.org/wiki/Numerical_analysishttp://en.wikipedia.org/wiki/Algebrahttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%AB


11/33

i i d


12/33

Multiple Algorithms andMultiple Solution

Five different multiplication algorithms are givenin Treatise

The first one using a lattice contains as many rows

as the number of multiplicand digits, and as manycolumns as the number of multiplier digits; then wedivide each cell diagonally into two triangles. Forexample, the multiplication of 7806 by 175 isrepresented by


13/33

Multiple Algorithms and MultipleSolution

Figure 1: multiplication process represented by

lattice from al-Nabulsi (1977, p.57)


14/33


we start by multiplying the first digit from the multiplier byevery digit from the multiplicand. The multiplication of 358by 624 is represented by the following:

Figure 2: multiplication process representation

from al-Nabulsi (1977, p. 59).


15/33


Another way: If the number of digits in themultiplicand is large. The multiplication of2783 by 456 is represented as follows:

Figure 3 : Multiplication processre resentation from al-Nabulsi 1977 . 61 .

l i l l i h d


16/33

Multiple Algorithms and

Multiple Solution

The sixth problem in Section I is:A piece of jewellery is made from gold and pearl. Its weight is three

methqals3

The Key to Arithmetic introduces four different methods to solve

this problem. In modern symbols these become and its price istwenty-four dinars. The price of one methqal of gold is five dinars,and of pearl is fifteen dinars. We want to know the weight of eachkind

By algebra: let

be the gold weight. Then the pearl weightbecomes 3.The pearl price =15(3)=4515The jewellery price =4515+5=4510=24We get =2.1 methqals, the gold weight. Then the pearl weight is

0.9 methqal.

lti l l ith d lti l


17/33


One of al-Kashismethods to solve word problems

is using these rules; he refers to this as maftoohat (.(

By maftoohat: the gold weight is (translating theoriginal Arabic words to mathematical symbols)


18/33


By geometry: We represent the problem as in the following picture.

Figure 6: Geometric illustration from al-Nabulsi (1977, p. 500).

This last approach presents a way to think geometricallyabout this problem, by considering weight and pricemagnitudes represented by sides of a rectangle, and the

product of these magnitudes is the rectangle area.


19/33

Law of Cosines and Jamshid al-Kashi

Figure: Triangle.

The angles , , and are respectively opposite the sides a,b, and c.In trigonometry, the law of cosines(also known as thecosine formulaor cosine rule) is a statement about ageneral triagnle that relates the lengths of its sides to thecosine if one of its angles. Using notation as in Fig., thelaw of cosines states that


20/33


The law of cosines may be used forcalculating the length of one side of a triangle

when the angle of the opposite this side, andthe length of the other two sides, are known.The law may be expressed as c2= a2+ b 2 -2abcos(C), where a, b, and c are triangle sidelengths and C is an angle between sides a andb.


21/33


The law of cosines is useful for computing the thirdside of a triangle when two sides and their enclosedangle are known, and in computing the angles of a

triangle if all three sides are known.By changing which sides of the triangle play theroles of a, b, and c in the original formula, one

discovers that the following two formulas also statethe law of cosines:

L C J


22/33

Law o Cos nes an Jams a -Kashi

Application of Law of Cosines: An ExampleTwo cars leave a city at the same time and travel along straight highwaysthat differ in direction by 800. One car averages 60 miles per hour and the

other averages 50 miles per hour. How far apart will the cars be after 90minutes?

Solution:Determine how far each car has traveled during the 90 minutes

d1 = distance of car 1

d2 = distance of car 2

r1 = rate of car 1 = 60 mphr2 = rate of car 2 = 50 mph

t = time traveled = 90 minutes = 1.5 hours

d1 = r1 t , d2 = r2 t

d1 = (60 mph) (1.5 hrs), d2 = (50 mph) (1.5 hrs)

d1 = 90 mi, d2 = 75 mi


23/33


Draw diagram of situation

Calculation: c2

= a2

+b2

-2abcoscc2= (75)2+ (90)2-27590cos800c2=11380.6224c=106.68 107 mi


24/33

Fixed Point Iteration Method

He wrote The Reckoners Key which summarizes arithmeticand contains work on algebra and geometry. In anotherwork, alKashiapplied the method now known as fixed-point

iteration to solve a cubic equation having as a root.Generally, for an equation of the formx = f(x).

We define the iteration.

Another more famous fixed point iteration coming much later

is Newtons Method-

He also worked on solutions of systems of equations anddeveloped methods for finding the nth root of a number

Horners method today.


25/33



26/33



27/33

Calculation of PI

Al-Kshwrote his most important works inSamarkand. In July 1424 he completed Risla al-muitiyya (The Treatise on the Circumference),

masterpiece of computational technique resulting inthe determination of 2 to sixteen decimal places.

Concretely, al-Kshposed the problem of calculatingthe said ratio with such precision that the error in the

circumference whose diameter is equal to 600,000diameters of the earth will be smaller than the

thickness of a horses hair. Al-Kshused thefollowing Old Iranian units of measurement:


28/33

Calculation of PI

1 parasang (about 6 kilometers) = 12,000 cubits,

1 cubit = 24 inches (or fingers),

1 inch = 6 widths of a medium-size grain of barley,

I width of a barley grain = 6thicknesses of a horseshair.

The great-circle circumference of the earth is considered

to be about 8,000 parasangs, so al-Kshsrequirement is equivalent to the computation of with an error no greater than the width of a horseshair.


29/33

Calculation of PI

Al-Kshsmeasurement of the circumference is basedon a computation of the perimeters of regularinscribed and circumscribed polygons, as had been

done by Archimedes, but it follows a somewhatdifferent procedure. Al-Kashi applied hisfundamental theorem to calculate successively thevalue of the chord cnof the arc n0= 180o

360o/3(2n) (where n 0). From his fundamentaltheorem al-Kashi obtained the identity


30/33

Calculation of PI

From here he found the lengths of the sides of inscribedand circumscribed regular polygons each with 3(2n)sides (n 1), in a given circle. Then he determined

the number of sides of the inscribed regular polygonin a circle whose radius is six hundred times theradius of the Earth in a such a way that the differencebetween the circumference of the circle and the

perimeter of the inscribed regular polygon in thiscircle will become less than the width of a horses hair.


31/33

Calculation of PI

Al Kashi continued to use his fundamental theorem to calculatethe value of , correct to 16 decimal places, using inscribedand circumscribed polygons, each with 3(228) = 805,306,368sides.

In the third section of his treatise al-Kshascertains that therequired accuracy will be attained in the case of the regularpolygon with 3228= 805, 306, 368 sides.

He resumes the computation of the chords in twenty-eightextensive tables; he verifies the extraction of the roots by

squaring and also by checking by 59 (analogous to thechecking by 9 in decimal numeration); and he establishes thenumber of sexagesimal places to which the values used must betaken.


32/33

Calculation of PI

Finally the best approximation for 2 r isaccepted as the arithmetic mean whosesexagesimal value for r = 1 is 6 16I59II28III

1IV34V 51VI46VII14VIII50IX, where all places arecorrect. In the eighth section al-Ksh

translates this value into the decimal fraction

2= 6.2831853071795865, correct to sixteendecimal places. This superb result farsurpassed all previous determinations of .


33/33

Conclusion

Finally we can say that Al Kashi was aremarkable scientist, one of the most famousin the world, who had a perfect command of the

science of the ancients, who contributed to itsdevelopment, and who could solve the mostdifficult problems.

He made a platform to do research onmathematics by expressing mathematics in asimple way.

Contribution of Ghiyath Al'Din Jamshid Mas'Ud Al'Kashi in the Field of Mathematics

Documents

Transcript of Contribution of Ghiyath Al'Din Jamshid Mas'Ud Al'Kashi in the Field of Mathematics