Transcript geometric method for solving equations

geometric method for
solving equations
Mesopotamia , Persia , Ancient
Greece, China , European
mathematicians
Generally…
Nowadays , Geometric method for
solving equations concerns solving
equations and systems of equations
using their graph…
►The
Greeks created a geometric algebra
where terms were represented by sides of
geometric objects.Mathematicians of
ancient China created calculating
methods for solving equations and they
applied them to geometric problems. The
Arab mathematicians were able to
interpret the results of certain cubic
equations geometrically.
Persian – Arabian
mathematicians…
Omar Khayyam , Ibn al-Haytham ,
Sharaf al-Din al-Tusi ,
Ghiyath Al-Din
Jamshid Al-Kashi…
Omar Khayyám (1048–1122)
► He is well known for inventing the general
method of solving cubic equations by
intersecting a parabola with a circle. He also
combined the use of trigonometry and
approximation theory to provide methods of
solving algebraic equations by geometrical
means. He also solved the cubic equation
x3 + 200x = 20x2 + 2000 by considering the
intersection of a rectangular hyperbola and a
circle.
Ibn al-Haytham (Alhazen) (945-1040)
This mathematician was able to solve by purely
algebraic means certain cubic equations, and
then to interpret the results geometrically.
Sharaf al-Din al-Tusi (1135-1213)
He used an algebraic method similar to
Newton’s method.
Ghiyath Al-Din Jamshid Al-Kashi
(1380-1429)
► In
The Treatise on the Chord and Sine, al-Kashi
computed sin 1° to nearly as much accuracy as his
value for π.He also improved the method that his
ancestor Sharaf al-Dīn al-Tūsī used. In algebra and
numerical analysis, he developed an iterative method
for solving cubic equations In order to determine sin
1°, al-Kashi discovered the following formula often
attributed to François Viète in the 16th century:
Chinese mathematicians…
Qin Jioushao ,Yang Hui , Liu Hui…
Qin Jiushao (1202 - 1261)
► . He wrote the mathematical treatise Shushu
Jiuzhang (Mathematical Treatise in Nine
Sections). In the treatise gives an equation
whose coefficients are variables and Heron's
formula for the area of a triangle.
Yang Hui (about 1238 - about 1298).
► In his works he described multiplication,
division, root-extraction, quadratic and
simultaneous equations, series, computations
of areas of a rectangle, a trapezium, a circle,
and other figures. He also gave a wonderful
account of magic squares and magic circles.
Liu Hui (3rd century)
► He found approximations to π using regular
polygons with 3 × 2n sides drawn in a circle..
Liu also wrote Haidao suanjing or Sea Island
Mathematical Manual article. In it Liu uses
Pythagoras's theorem to calculate heights of
objects and distances to objects which cannot
be measured directly.
A proof of the
pythagorean theorem
by liu hui
Greek mathematicians…
Methods Euclides used to solve equations…
1. x2– 5x - 36 =0
► ρ1ρ2
= 36 ρ1+ ρ2 = 13
► ● we construct a circle
with diameter 13 and a square
with side 6.it can be easily
proved that ρ1=AΓ=4
And p2=ΓΒ=9.
2. x2 – 5x - 36 =0
►
the one radical x1 is negative
ρ1 =- Ιρ1Ι
►
the other radical x2 is positive
ρ2= Ιρ2Ι
►
we construct a circle of semi
diameter 5.in a coincidental
point
we construct a geometric
tangent ΑΓ=6. We adject point
Γ with
► the centre of the circle. The
half-line cuts the circle at
points Α & Δ.
►
For the two radicals it will be:
Ιρ1Ι= ΓΒ = 4
Ιρ2Ι = ΓΔ = 9
ρ1= - 4
ρ2 = 9
►
European mathematicians
►
Techniques of applying geometrical constructions to
algebraic problems were also adopted by a number of
Renaissance mathematicians such as Gerolamo Cardano
and Niccolò Fontana "Tartaglia" on their studies of the
cubic equation. The geometrical approach to construction
problems, rather than the algebraic one, was favoured by
most 16th and 17th century mathematicians, notably Blaise
Pascal .The French mathematicians Francisco Vieta and
later René Descartes and Pierre de Fermat started the
conventional way of thinking about construction problems
through the introduction of coordinate geometry. They
were interested primarily in the properties of algebraic
curves, such as those defined by Diophantine equations (in
the case of Fermat), and the algebraic reformulation of the
classical Greek works on conics and cubics (in the case of
Descartes).
Nowadays…
solve the quadratic equation :x² – ax + b sing a
ruler and compasses only.
first of all, we cannot solve the
equation if the numbers a & b
are not >1.
► Secondly, we make the points
Y&Z have coordinates (0,1)and
(a , b) respectively.
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Method
1. Find the centre of the circle
by finding the centre of YZ.
2.Using the compasses, draw
the circle with diameter YZ.
3.the circle at the x-axis,
creates the points x1[with
coordinates(x1,0)] and
thepointx2[with
coordinates(x2,0)].These are
the roots of the equation x ² ax + b = 0
►Although
this method is really
simple ,we have to be really accurate
when constructing the circle.
Thank you all..
► Maria
Chantzopoulou
3rd general lyceum of karditsa