A report on AlgebraGeometry and Mathematics

The word algebra comes from the title of a book by Muhammad ibn Musa al-Khwarizmi.
An illustration of Desargues' theorem, a result in Euclidean and projective geometry
3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)
A page from Al-Khwārizmī's al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala
A European and an Arab practicing geometry in the 15th century
The distribution of prime numbers is a central point of study in number theory. This Ulam spiral serves to illustrate it, hinting, in particular, at the conditional independence between being prime and being a value of certain quadratic polynomials.
Italian mathematician Girolamo Cardano published the solutions to the cubic and quartic equations in his 1545 book Ars magna.
Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c. 1310).
The quadratic formula expresses concisely the solutions of all quadratic equations
Linear algebra lecture at the Aalto University
An illustration of Euclid's parallel postulate
Rubik's cube: the study of its possible moves is a concrete application of group theory
Algebraic expression notation:
 1 – power (exponent)
 2 – coefficient
 3 – term
 4 – operator
 5 – constant term
 x y c – variables/constants
Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique angles.
The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
The graph of a polynomial function of degree 3
A sphere is a surface that can be defined parametrically (by  or implicitly (by x2 + y2 + z2 − r2 = 0.)
Archimedes used the method of exhaustion, depicted here, to approximate the value of pi.
Visual checking of the Pythagorean theorem for the (3, 4, 5) triangle as in the Zhoubi Suanjing 500–200 BC. The Pythagorean theorem is a consequence of the Euclidean metric.
The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.
The Koch snowflake, with fractal dimension=log4/log3 and topological dimension=1
A page from al-Khwārizmī's Algebra
A tiling of the hyperbolic plane
Leonardo Fibonacci, the Italian mathematician who introduced the Hindu–Arabic numeral system invented between the 1st and 4th centuries by Indian mathematicians, to the Western World.
Differential geometry uses tools from calculus to study problems involving curvature.
Leonhard Euler created and popularized much of the mathematical notation used today.
A thickening of the trefoil knot
Carl Friedrich Gauss, known as the prince of mathematicians
Quintic Calabi–Yau threefold
The front side of the Fields Medal
Discrete geometry includes the study of various sphere packings.
The Cayley graph of the free group on two generators a and b
Bou Inania Madrasa, Fes, Morocco, zellige mosaic tiles forming elaborate geometric tessellations
The Pythagoreans discovered that the sides of a triangle could have incommensurable lengths.
Euler's identity, which American physicist Richard Feynman once called "the most remarkable formula in mathematics".

Geometry is, with arithmetic, one of the oldest branches of mathematics.

- Geometry

Mathematics is an area of knowledge that includes such topics as numbers (arithmetic, number theory), formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

- Mathematics

Algebra is one of the broad areas of mathematics.

- Algebra

Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example).

- Algebra

For instance, the introduction of coordinates by René Descartes and the concurrent developments of algebra marked a new stage for geometry, since geometric figures such as plane curves could now be represented analytically in the form of functions and equations.

- Geometry
The word algebra comes from the title of a book by Muhammad ibn Musa al-Khwarizmi.

5 related topics with Alpha

Overall

Archimedes used the method of exhaustion to calculate the area under a parabola.

Calculus

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Archimedes used the method of exhaustion to calculate the area under a parabola.
Alhazen, 11th-century Arab mathematician and physicist
Isaac Newton developed the use of calculus in his laws of motion and gravitation.
Gottfried Wilhelm Leibniz was the first to state clearly the rules of calculus.
Maria Gaetana Agnesi
The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus.

Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.

A page from The Compendious Book on Calculation by Completion and Balancing by Al-Khwarizmi

Mathematics in medieval Islam

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A page from The Compendious Book on Calculation by Completion and Balancing by Al-Khwarizmi
Omar Khayyám's "Cubic equations and intersections of conic sections" the first page of the two-chaptered manuscript kept in Tehran University
To solve the third-degree equation x3 + a2x = b Khayyám constructed the parabola x2 = ay, a circle with diameter b/a2, and a vertical line through the intersection point. The solution is given by the length of the horizontal line segment from the origin to the intersection of the vertical line and the x-axis.
Engraving of Abū Sahl al-Qūhī's perfect compass to draw conic sections.
The theorem of Ibn Haytham.

Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta).

Important progress was made, such as full development of the decimal place-value system to include decimal fractions, the first systematised study of algebra, and advances in geometry and trigonometry.

The first use of an equals sign, equivalent to 14x + 15 = 71 in modern notation. From The Whetstone of Witte by Robert Recorde of Wales (1557).

Equation

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The first use of an equals sign, equivalent to 14x + 15 = 71 in modern notation. From The Whetstone of Witte by Robert Recorde of Wales (1557).
Illustration of a simple equation; x, y, z are real numbers, analogous to weights.
The solutions –1 and 2 of the polynomial equation x2 – x + 2 = 0 are the points where the graph of the quadratic function y = x2 – x + 2 cuts the x-axis.
The Nine Chapters on the Mathematical Art is an anonymous 2nd-century Chinese book proposing a method of resolution for linear equations.
The blue and red line is the set of all points (x,y) such that x+y=5 and -x+2y=4, respectively. Their intersection point, (2,3), satisfies both equations.
A conic section is the intersection of a plane and a cone of revolution.
Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red. The equation of a circle is (x − a)2 + (y − b)2 = r2 where a and b are the coordinates of the center (a, b) and r is the radius.
A strange attractor, which arises when solving a certain differential equation

In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.

Algebra studies two main families of equations: polynomial equations and, among them, the special case of linear equations.

Modern algebraic geometry is based on more abstract techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry.

Addition of functions: The sum of the sine and the exponential function is

Vector space

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[[File: Vector add scale.svg|200px|thumb|right|Vector addition and scalar multiplication: a vector

[[File: Vector add scale.svg|200px|thumb|right|Vector addition and scalar multiplication: a vector

Addition of functions: The sum of the sine and the exponential function is
A typical matrix
Commutative diagram depicting the universal property of the tensor product.
The succeeding snapshots show summation of 1 to 5 terms in approximating a periodic function (blue) by finite sum of sine functions (red).
An affine plane (light blue) in R3. It is a two-dimensional subspace shifted by a vector x (red).

In mathematics, physics, and engineering, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars.

Finite-dimensional vector spaces occur naturally in geometry and related areas.

Actually Grassmann's 1844 work exceeds the framework of vector spaces, since his considering multiplication, too, led him to what are today called algebras.

An illustration of Euclid's proof of the Pythagorean theorem.

Greek mathematics

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An illustration of Euclid's proof of the Pythagorean theorem.
Detail of Pythagoras with a tablet of ratios, from The School of Athens by Raphael. Vatican Palace, Rome, 1509.
A fragment from Euclid's Elements (c. 300 BC), widely considered the most influential mathematics textbook of all time.
Cover of Arithmetica written by Greek Mathematician Diophantus

Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean.

Greek mathematics constitutes an important period in the history of mathematics: fundamental in respect of geometry and for the idea of formal proof.

The methods employed made no explicit use of algebra, nor trigonometry, the latter appearing around the time of Hipparchus.