Algebraic equation in the context of "Equation"

Diophantus of Alexandria (Ancient Greek: Διόφαντος, romanized: Diophantos) (/daɪoʊˈfæntəs/; fl. 250 CE) was a Greek mathematician who was the author of the Arithmetica in thirteen books, ten of which are still extant, made up of arithmetical problems that are solved through algebraic equations.

Although Joseph-Louis Lagrange called Diophantus "the inventor of algebra" he did not invent it; however, his exposition became the standard within the Neoplatonic schools of Late antiquity, and its translation into Arabic in the 9th century AD and had influence in the development of later algebra: Diophantus' method of solution matches medieval Arabic algebra in its concepts and overall procedure. The 1621 edition of Arithmetica by Bachet gained fame after Pierre de Fermat wrote his famous "Last Theorem" in the margins of his copy.

The number π (/paɪ/ ; spelled out as pi) is a mathematical constant, approximately equal to 3.14159, that is the ratio of a circle's circumference to its diameter. It appears in many formulae across mathematics and physics, and some of these formulae are commonly used for defining π, to avoid relying on the definition of the length of a curve.

The number π is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as ${\displaystyle {\tfrac {22}{7}}}$ are commonly used to approximate it. Consequently, its decimal representation never ends, nor enters a permanently repeating pattern. It is a transcendental number, meaning that it cannot be a solution of an algebraic equation involving only finite sums, products, powers, and integers. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge. The decimal digits of π appear to be randomly distributed, but no proof of this conjecture has been found.

In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.

The function is often thought of as an "unknown" that solves the equation, similar to how x is thought of as an unknown number solving, e.g., an algebraic equation like x − 3x + 2 = 0. However, it is usually impossible to write down explicit formulae for solutions of partial differential equations. There is correspondingly a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity and stability. Among the many open questions are the existence and smoothness of solutions to the Navier–Stokes equations, named as one of the Millennium Prize Problems in 2000.

In geometry, an intersection between geometric objects (seen as sets of points) is a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case in Euclidean geometry is the line–line intersection between two distinct lines, which either is one point (sometimes called a vertex) or does not exist (if the lines are parallel). Other types of geometric intersection include:

• Line–plane intersection
• Line–sphere intersection
• Intersection of a polyhedron with a line
• Line segment intersection
• Intersection curve

Determination of the intersection of flats – linear geometric objects embedded in a higher-dimensional space – is a simple task of linear algebra, namely the solution of a system of linear equations. In general the determination of an intersection leads to non-linear equations, which can be solved numerically, for example using Newton iteration. Intersection problems between a line and a conic section (circle, ellipse, parabola, etc.) or a quadric (sphere, cylinder, hyperboloid, etc.) lead to quadratic equations that can be easily solved. Intersections between quadrics lead to quartic equations that can be solved algebraically.

In mathematics, the Laplace transform, named after Pierre-Simon Laplace (/ləˈplɑːs/), is an integral transform that converts a function of a real variable (usually ${\displaystyle t}$, in the time domain) to a function of a complex variable ${\displaystyle s}$ (in the complex-valued frequency domain, also known as s-domain or s-plane). The functions are often denoted by ${\displaystyle x(t)}$ for the time-domain representation and ${\displaystyle X(s)}$ for the frequency-domain.

The transform is useful for converting differentiation and integration in the time domain into much easier multiplication and division in the Laplace domain (analogous to how logarithms are useful for simplifying multiplication and division into addition and subtraction). This gives the transform many applications in science and engineering, mostly as a tool for solving linear differential equations and dynamical systems by simplifying ordinary differential equations and integral equations into algebraic polynomial equations, and by simplifying convolution into multiplication.

In mathematics, the Tschirnhausen cubic is a cubic plane curve defined by the polar equation$${\displaystyle r=a\sec ^{3}\left({\frac {\theta }{3}}\right)}$$or the equivalent algebraic equation$${\displaystyle 27ay^{2}=(a-x)(8a+x)^{2}.}$$

It is a nodal cubic, meaning that it crosses itself at one point, its node. The angle at this crossing point, inside the loop formed by the crossing, is 60°. Because the Tschirnhausen cubic has this singularity, it can be given a parametric equation, and any arc of it can be drawn as a cubic Bézier curve. It is a special case of a sinusoidal spiral, of a pursuit curve, and of a Pythagorean hodograph curve.

In geometry, an intersection between geometric objects (seen as sets of points) is a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case in Euclidean geometry is the line–line intersection between two distinct lines, which either is one point (sometimes called a vertex) or empty (if the lines are parallel). Other types of geometric intersection include:

• Line–plane intersection
• Line–sphere intersection
• Intersection of a polyhedron with a line
• Line segment intersection
• Intersection curve

Determination of the intersection of flats – linear geometric objects embedded in a higher-dimensional space – is a simple task of linear algebra, namely the solution of a system of linear equations. In general the determination of an intersection leads to non-linear equations, which can be solved numerically, for example using Newton iteration. Intersection problems between a line and a conic section (circle, ellipse, parabola, etc.) or a quadric (sphere, cylinder, hyperboloid, etc.) lead to quadratic equations that can be easily solved. Intersections between quadrics lead to quartic equations that can be solved algebraically.