1.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times

Mathematics
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Euclid (holding calipers), Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens.

Mathematics
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Greek mathematician Pythagoras (c. 570 – c. 495 BC), commonly credited with discovering the Pythagorean theorem

Mathematics
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Leonardo Fibonacci, the Italian mathematician who established the Hindu–Arabic numeral system to the Western World

Mathematics
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Carl Friedrich Gauss, known as the prince of mathematicians

2.
Equation
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In mathematics, an equation is a statement of an equality containing one or more variables. Solving the equation consists of determining which values of the make the equality true. Variables are also called unknowns and the values of the unknowns which satisfy the equality are called solutions of the equation, there are two kinds of equations, identity equations and conditional equations. An identity equation is true for all values of the variable, a conditional equation is true for only particular values of the variables. Each side of an equation is called a member of the equation, each member will contain one or more terms. The equation, A x 2 + B x + C = y has two members, A x 2 + B x + C and y, the left member has three terms and the right member one term. The variables are x and y and the parameters are A, B, an equation is analogous to a scale into which weights are placed. When equal weights of something are place into the two pans, the two weights cause the scale to be in balance and are said to be equal. If a quantity of grain is removed from one pan of the balance, likewise, to keep an equation in balance, the same operations of addition, subtraction, multiplication and division must be performed on both sides of an equation for it to remain an equality. In geometry, equations are used to describe geometric figures and this is the starting idea of algebraic geometry, an important area of mathematics. Algebra studies two main families of equations, polynomial equations and, among them the case of linear equations. Polynomial equations have the form P =0, where P is a polynomial, linear equations have the form ax + b =0, where a and b are parameters. To solve equations from either family, one uses algorithmic or geometric techniques, algebra also studies Diophantine equations where the coefficients and solutions are integers. The techniques used are different and come from number theory and these equations are difficult in general, one often searches just to find the existence or absence of a solution, and, if they exist, to count the number of solutions. Differential equations are equations that involve one or more functions and their derivatives and they are solved by finding an expression for the function that does not involve derivatives. Differential equations are used to model processes that involve the rates of change of the variable, and are used in such as physics, chemistry, biology. The = symbol, which appears in equation, was invented in 1557 by Robert Recorde. An equation is analogous to a scale, balance, or seesaw

Equation
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A strange attractor which arises when solving a certain differential equation.

3.
System of linear equations
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In mathematics, a system of linear equations is a collection of two or more linear equations involving the same set of variables. A solution to a system is an assignment of values to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by x =1 y = −2 z = −2 since it all three equations valid. The word system indicates that the equations are to be considered collectively, in mathematics, the theory of linear systems is the basis and a fundamental part of linear algebra, a subject which is used in most parts of modern mathematics. Computational algorithms for finding the solutions are an important part of linear algebra, and play a prominent role in engineering, physics, chemistry, computer science. A system of equations can often be approximated by a linear system. For solutions in an integral domain like the ring of the integers, or in other structures, other theories have been developed. Integer linear programming is a collection of methods for finding the best integer solution, gröbner basis theory provides algorithms when coefficients and unknowns are polynomials. Also tropical geometry is an example of linear algebra in a more exotic structure, the simplest kind of linear system involves two equations and two variables,2 x +3 y =64 x +9 y =15. One method for solving such a system is as follows, first, solve the top equation for x in terms of y, x =3 −32 y. Now substitute this expression for x into the equation,4 +9 y =15. This results in an equation involving only the variable y. Solving gives y =1, and substituting this back into the equation for x yields x =3 /2. Here x 1, x 2, …, x n are the unknowns, a 11, a 12, …, a m n are the coefficients of the system, and b 1, b 2, …, b m are the constant terms. Often the coefficients and unknowns are real or complex numbers, but integers and rational numbers are seen, as are polynomials. One extremely helpful view is that each unknown is a weight for a vector in a linear combination. X1 + x 2 + ⋯ + x n = This allows all the language, If every vector within that span has exactly one expression as a linear combination of the given left-hand vectors, then any solution is unique. This is important because if we have m independent vectors a solution is guaranteed regardless of the right-hand side, and otherwise not guaranteed

System of linear equations
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A linear system in three variables determines a collection of planes. The intersection point is the solution.

4.
System of ordinary differential equations
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In mathematics, an ordinary differential equation is a differential equation containing one or more functions of one independent variable and its derivatives. The term ordinary is used in contrast with the partial differential equation which may be with respect to more than one independent variable. ODEs that are linear equations have exact closed-form solutions that can be added and multiplied by coefficients. Graphical and numerical methods, applied by hand or by computer, may approximate solutions of ODEs and perhaps yield useful information, often sufficing in the absence of exact, Ordinary differential equations arise in many contexts of mathematics and science. Mathematical descriptions of change use differentials and derivatives, often, quantities are defined as the rate of change of other quantities, or gradients of quantities, which is how they enter differential equations. Specific mathematical fields include geometry and analytical mechanics, scientific fields include much of physics and astronomy, meteorology, chemistry, biology, ecology and population modelling, economics. Many mathematicians have studied differential equations and contributed to the field, including Newton, Leibniz, the Bernoulli family, Riccati, Clairaut, dAlembert, in general, F is a function of the position x of the particle at time t. The unknown function x appears on both sides of the equation, and is indicated in the notation F. In what follows, let y be a dependent variable and x an independent variable, the notation for differentiation varies depending upon the author and upon which notation is most useful for the task at hand. Given F, a function of x, y, and derivatives of y, then an equation of the form F = y is called an explicit ordinary differential equation of order n. The function r is called the term, leading to two further important classifications, Homogeneous If r =0, and consequently one automatic solution is the trivial solution. The solution of a homogeneous equation is a complementary function. The additional solution to the function is the particular integral. The general solution to an equation can be written as y = yc + yp. Non-linear A differential equation that cannot be written in the form of a linear combination, a number of coupled differential equations form a system of equations. In column vector form, = These are not necessarily linear, the implicit analogue is, F =0 where 0 = is the zero vector. In the same sources, implicit ODE systems with a singular Jacobian are termed differential algebraic equations and this distinction is not merely one of terminology, DAEs have fundamentally different characteristics and are generally more involved to solve than ODE systems. Given a differential equation F =0 a function u, I ⊂ R → R is called the solution or integral curve for F, if u is n-times differentiable on I, and F =0 x ∈ I

System of ordinary differential equations
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Navier–Stokes differential equations used to simulate airflow around an obstruction.

5.
Partial differential equation
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In mathematics, a partial differential equation is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, PDEs can be used to describe a wide variety of phenomena such as sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, or quantum mechanics. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs, just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. PDEs find their generalisation in stochastic partial differential equations, Partial differential equations are equations that involve rates of change with respect to continuous variables. The dynamics for the body take place in a finite-dimensional configuration space. This distinction usually makes PDEs much harder to solve ordinary differential equations. Classic domains where PDEs are used include acoustics, fluid dynamics, electrodynamics, a partial differential equation for the function u is an equation of the form f =0. If f is a function of u and its derivatives. Common examples of linear PDEs include the equation, the wave equation, Laplaces equation, Helmholtz equation, Klein–Gordon equation. A relatively simple PDE is ∂ u ∂ x =0 and this relation implies that the function u is independent of x. However, the equation gives no information on the dependence on the variable y. Hence the general solution of equation is u = f. The analogous ordinary differential equation is d u d x =0, which has the solution u = c and these two examples illustrate that general solutions of ordinary differential equations involve arbitrary constants, but solutions of PDEs involve arbitrary functions. A solution of a PDE is generally not unique, additional conditions must generally be specified on the boundary of the region where the solution is defined. For instance, in the example above, the function f can be determined if u is specified on the line x =0. Even if the solution of a differential equation exists and is unique. The mathematical study of questions is usually in the more powerful context of weak solutions. The derivative of u with respect to y approaches 0 uniformly in x as n increases and this solution approaches infinity if nx is not an integer multiple of π for any non-zero value of y

Partial differential equation
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Navier–Stokes differential equations used to simulate airflow around an obstruction.

6.
Elementary algebra
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Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to school students and builds on their understanding of arithmetic. Whereas arithmetic deals with specified numbers, algebra introduces quantities without fixed values and this use of variables entails a use of algebraic notation and an understanding of the general rules of the operators introduced in arithmetic. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real, the use of variables to denote quantities allows general relationships between quantities to be formally and concisely expressed, and thus enables solving a broader scope of problems. Many quantitative relationships in science and mathematics are expressed as algebraic equations, algebraic notation describes how algebra is written. It follows certain rules and conventions, and has its own terminology, a term is an addend or a summand, a group of coefficients, variables, constants and exponents that may be separated from the other terms by the plus and minus operators. By convention, letters at the beginning of the alphabet are used to represent constants. They are usually written in italics, algebraic operations work in the same way as arithmetic operations, such as addition, subtraction, multiplication, division and exponentiation. and are applied to algebraic variables and terms. Multiplication symbols are usually omitted, and implied when there is no space between two variables or terms, or when a coefficient is used. For example,3 × x 2 is written as 3 x 2, usually terms with the highest power, are written on the left, for example, x 2 is written to the left of x. When a coefficient is one, it is usually omitted, likewise when the exponent is one. When the exponent is zero, the result is always 1, however 00, being undefined, should not appear in an expression, and care should be taken in simplifying expressions in which variables may appear in exponents. Other types of notation are used in algebraic expressions when the required formatting is not available, or can not be implied, such as where only letters, for example, exponents are usually formatted using superscripts, e. g. x 2. In plain text, and in the TeX mark-up language, the symbol ^ represents exponents. In programming languages such as Ada, Fortran, Perl, Python and Ruby, many programming languages and calculators use a single asterisk to represent the multiplication symbol, and it must be explicitly used, for example,3 x is written 3*x. Elementary algebra builds on and extends arithmetic by introducing letters called variables to represent general numbers and this is useful for several reasons. Variables may represent numbers whose values are not yet known, for example, if the temperature of the current day, C, is 20 degrees higher than the temperature of the previous day, P, then the problem can be described algebraically as C = P +20. Variables allow one to describe general problems, without specifying the values of the quantities that are involved, for example, it can be stated specifically that 5 minutes is equivalent to 60 ×5 =300 seconds

Elementary algebra
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A typical algebra problem.

Elementary algebra
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Two-dimensional plot (magenta curve) of the algebraic equation

7.
Non-linear system
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In mathematics and physical sciences, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, physicists and mathematicians, nonlinear systems may appear chaotic, unpredictable or counterintuitive, contrasting with the much simpler linear systems. In other words, in a system of equations, the equation to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. Systems can be defined as non-linear, regardless of whether or not known linear functions appear in the equations. In particular, an equation is linear if it is linear in terms of the unknown function and its derivatives. As nonlinear equations are difficult to solve, nonlinear systems are approximated by linear equations. This works well up to some accuracy and some range for the input values and it follows that some aspects of the behavior of a nonlinear system appear commonly to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is not random. For example, some aspects of the weather are seen to be chaotic and this nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology. Some authors use the term nonlinear science for the study of nonlinear systems and this is disputed by others, Using a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals. In mathematics, a function f is one which satisfies both of the following properties, Additivity or superposition, f = f + f, Homogeneity. Additivity implies homogeneity for any rational α, and, for continuous functions, for a complex α, homogeneity does not follow from additivity. For example, a map is additive but not homogeneous. The equation is called homogeneous if C =0, if f contains differentiation with respect to x, the result will be a differential equation. Nonlinear algebraic equations, which are also called polynomial equations, are defined by equating polynomials to zero, for example, x 2 + x −1 =0. For a single equation, root-finding algorithms can be used to find solutions to the equation. However, systems of equations are more complicated, their study is one motivation for the field of algebraic geometry. It is even difficult to decide whether a given system has complex solutions

Non-linear system
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Illustration of a pendulum