A report on Algebra, Geometry, Mathematics and Calculus

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

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.

- Calculus

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

Even though some methods, which had been developed much earlier, may be considered nowadays as algebra, the emergence of algebra and, soon thereafter, of infinitesimal calculus as subfields of mathematics only dates from the 16th or 17th century.

- Algebra

This was a necessary precursor to the development of calculus and a precise quantitative science of physics.

- Geometry

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.

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