Quadratic irrational number

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Quadratic_irrational_number

In mathematics, a quadratic irrational number (also known as a quadratic irrational or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the rational numbers. Since fractions in the coefficients of a quadratic equation can be cleared by multiplying both sides by their least common denominator, a quadratic irrational is an irrational root of some quadratic equation with integer coefficients. The quadratic irrational numbers, a subset of the complex numbers, are algebraic numbers of degree 2, and can therefore be expressed as

${\displaystyle \frac{a+b\sqrt{c}}{d},}$

for integers a, b, c, d; with b, c and d non-zero, and with c square-free. When c is positive, we get real quadratic irrational numbers, while a negative c gives complex quadratic irrational numbers which are not real numbers. This defines an injection from the quadratic irrationals to quadruples of integers, so their cardinality is at most countable; since on the other hand every square root of a prime number is a distinct quadratic irrational, and there are countably many prime numbers, they are at least countable; hence the quadratic irrationals are a countable set.

Quadratic irrationals are used in field theory to construct field extensions of the field of rational numbers Q. Given the square-free integer c, the augmentation of Q by quadratic irrationals using √c produces a quadratic field Q(√c). For example, the inverses of elements of Q(√c) are of the same form as the above algebraic numbers:

${\displaystyle \frac{d}{a+b\sqrt{c}}=\frac{ad-bd\sqrt{c}}{a^2-b^{2}c}.}$

Quadratic irrationals have useful properties, especially in relation to continued fractions, where we have the result that all real quadratic irrationals, and only real quadratic irrationals, have periodic continued fraction forms. For example

${\displaystyle \sqrt{3}=1.732\dots =[1;1,2,1,2,1,2,\dots ]}$

The periodic continued fractions can be placed in one-to-one correspondence with the rational numbers. The correspondence is explicitly provided by Minkowski's question mark function, and an explicit construction is given in that article. It is entirely analogous to the correspondence between rational numbers and strings of binary digits that have an eventually-repeating tail, which is also provided by the question mark function. Such repeating sequences correspond to periodic orbits of the dyadic transformation (for the binary digits) and the Gauss map ${\displaystyle h(x)=1/x-\lfloor 1/x\rfloor }$ for continued fractions.

Real quadratic irrational numbers and indefinite binary quadratic forms

We may rewrite a quadratic irrationality as follows:

${\displaystyle \frac{a+b\sqrt{c}}{d}=\frac{a+\sqrt{b^{2}c}}{d}.}$

It follows that every quadratic irrational number can be written in the form

${\displaystyle \frac{a+\sqrt{c}}{d}.}$

This expression is not unique.

Fix a non-square, positive integer ${\displaystyle c}$ congruent to ${\displaystyle 0}$ or ${\displaystyle 1}$ modulo ${\displaystyle 4}$, and define a set ${\displaystyle S_c}$ as

${\displaystyle S_c=\left\{\frac{a+\sqrt{c}}{d}:a,d\text{ integers, }d\text{ even},a^2\equiv c(\mathrm{mod}2d)\right\}.}$

Every quadratic irrationality is in some set ${\displaystyle S_c}$, since the congruence conditions can be met by scaling the numerator and denominator by an appropriate factor.

A matrix

${\displaystyle \left(\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right)}$

with integer entries and ${\displaystyle \alpha \delta -\beta \gamma =1}$ can be used to transform a number ${\displaystyle y}$ in ${\displaystyle S_c}$. The transformed number is

${\displaystyle z=\frac{\alpha y+\beta }{\gamma y+\delta }}$

If ${\displaystyle y}$ is in ${\displaystyle S_c}$, then ${\displaystyle z}$ is too.

The relation between ${\displaystyle y}$ and ${\displaystyle z}$ above is an equivalence relation. (This follows, for instance, because the above transformation gives a group action of the group of integer matrices with determinant 1 on the set ${\displaystyle S_c}$.) Thus, ${\displaystyle S_c}$ partitions into equivalence classes. Each equivalence class comprises a collection of quadratic irrationalities with each pair equivalent through the action of some matrix. Serret's theorem implies that the regular continued fraction expansions of equivalent quadratic irrationalities are eventually the same, that is, their sequences of partial quotients have the same tail. Thus, all numbers in an equivalence class have continued fraction expansions that are eventually periodic with the same tail.

There are finitely many equivalence classes of quadratic irrationalities in ${\displaystyle S_c}$. The standard proof of this involves considering the map ${\displaystyle \varphi }$ from binary quadratic forms of discriminant ${\displaystyle c}$ to ${\displaystyle S_c}$ given by

${\displaystyle \varphi (t{x}^2+uxy+v{y}^2)=\frac{-u+\sqrt{c}}{2t}}$

A computation shows that ${\displaystyle \varphi }$ is a bijection that respects the matrix action on each set. The equivalence classes of quadratic irrationalities are then in bijection with the equivalence classes of binary quadratic forms, and Lagrange showed that there are finitely many equivalence classes of binary quadratic forms of given discriminant.

Through the bijection ${\displaystyle \varphi }$, expanding a number in ${\displaystyle S_c}$ in a continued fraction corresponds to reducing the quadratic form. The eventually periodic nature of the continued fraction is then reflected in the eventually periodic nature of the orbit of a quadratic form under reduction, with reduced quadratic irrationalities (those with a purely periodic continued fraction) corresponding to reduced quadratic forms.

Square root of non-square is irrational

The definition of quadratic irrationals requires them to satisfy two conditions: they must satisfy a quadratic equation and they must be irrational. The solutions to the quadratic equation ax² + bx + c = 0 are

${\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}.}$

Thus quadratic irrationals are precisely those real numbers in this form that are not rational. Since b and 2a are both integers, asking when the above quantity is irrational is the same as asking when the square root of an integer is irrational. The answer to this is that the square root of any natural number that is not a square number is irrational.

The square root of 2 was the first such number to be proved irrational. Theodorus of Cyrene proved the irrationality of the square roots of non-square natural numbers up to 17, but stopped there, probably because the algebra he used could not be applied to the square root of numbers greater than 17. Euclid's Elements Book 10 is dedicated to classification of irrational magnitudes. The original proof of the irrationality of the non-square natural numbers depends on Euclid's lemma.

Many proofs of the irrationality of the square roots of non-square natural numbers implicitly assume the fundamental theorem of arithmetic, which was first proven by Carl Friedrich Gauss in his Disquisitiones Arithmeticae. This asserts that every integer has a unique factorization into primes. For any rational non-integer in lowest terms there must be a prime in the denominator which does not divide into the numerator. When the numerator is squared that prime will still not divide into it because of the unique factorization. Therefore, the square of a rational non-integer is always a non-integer; by contrapositive, the square root of an integer is always either another integer, or irrational.

Euclid used a restricted version of the fundamental theorem and some careful argument to prove the theorem. His proof is in Euclid's Elements Book X Proposition 9.

The fundamental theorem of arithmetic is not actually required to prove the result, however. There are self-contained proofs by Richard Dedekind, among others. The following proof was adapted by Colin Richard Hughes from a proof of the irrationality of the square root of 2 found by Theodor Estermann in 1975.

If D is a non-square natural number, then there is a natural number n such that:

n² < D < (n + 1)²,

so in particular

0 < √D − n < 1.

If the square root of D is rational, then it can be written as the irreducible fraction p/q, so that q is the smallest possible denominator, and hence the smallest number for which q√D is also an integer. Then:

(√D − n)q√D = qD − nq√D

which is thus also an integer. But 0 < (√D − n) < 1 so (√D − n)q < q. Hence (√D − n)q is an integer smaller than q which multiplied by √D makes an integer. This is a contradiction, because q was defined to be the smallest such number. Therefore, √D cannot be rational.

Differential equation

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Quadratic_irrational_number

Visualization of heat transfer in a pump casing, created by solving the heat equation. Heat is being generated internally in the casing and being cooled at the boundary, providing a steady state temperature distribution.

In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

The study of differential equations consists mainly of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are soluble by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly.

Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

History

Differential equations came into existence with the invention of calculus by Isaac Newton and Gottfried Leibniz. In Chapter 2 of his 1671 work Methodus fluxionum et Serierum Infinitarum, Newton listed three kinds of differential equations:

${\displaystyle \begin{array}{rl}\frac{dy}{dx}& =f(x)\\ \frac{dy}{dx}& =f(x,y)\\ x_1\frac{\mathrm{\partial }y}{\mathrm{\partial }{x}_1}& +x_2\frac{\mathrm{\partial }y}{\mathrm{\partial }{x}_2}=y\end{array}}$

In all these cases, y is an unknown function of x (or of x₁ and x₂), and f is a given function.

He solves these examples and others using infinite series and discusses the non-uniqueness of solutions.

Jacob Bernoulli proposed the Bernoulli differential equation in 1695. This is an ordinary differential equation of the form

${\displaystyle y^{\prime }+P(x)y=Q(x)y^n}$

for which the following year Leibniz obtained solutions by simplifying it.

Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange.^[5][6][7][8] In 1746, d’Alembert discovered the one-dimensional wave equation, and within ten years Euler discovered the three-dimensional wave equation.

The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point. Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it to mechanics, which led to the formulation of Lagrangian mechanics.

In 1822, Fourier published his work on heat flow in Théorie analytique de la chaleur (The Analytic Theory of Heat), in which he based his reasoning on Newton's law of cooling, namely, that the flow of heat between two adjacent molecules is proportional to the extremely small difference of their temperatures. Contained in this book was Fourier's proposal of his heat equation for conductive diffusion of heat. This partial differential equation is now a common part of mathematical physics curriculum.

Example

In classical mechanics, the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow these variables to be expressed dynamically (given the position, velocity, acceleration and various forces acting on the body) as a differential equation for the unknown position of the body as a function of time.

In some cases, this differential equation (called an equation of motion) may be solved explicitly.

An example of modeling a real-world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the deceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the ball's velocity. This means that the ball's acceleration, which is a derivative of its velocity, depends on the velocity (and the velocity depends on time). Finding the velocity as a function of time involves solving a differential equation and verifying its validity.

Types

Differential equations can be divided into several types. Apart from describing the properties of the equation itself, these classes of differential equations can help inform the choice of approach to a solution. Commonly used distinctions include whether the equation is ordinary or partial, linear or non-linear, and homogeneous or heterogeneous. This list is far from exhaustive; there are many other properties and subclasses of differential equations which can be very useful in specific contexts.

Ordinary differential equations

Main articles: Ordinary differential equation and Linear differential equation

An ordinary differential equation (ODE) is an equation containing an unknown function of one real or complex variable x, its derivatives, and some given functions of x. The unknown function is generally represented by a variable (often denoted y), which, therefore, depends on x. Thus x is often called the independent variable of the equation. The term "ordinary" is used in contrast with the term partial differential equation, which may be with respect to more than one independent variable.

Linear differential equations are the differential equations that are linear in the unknown function and its derivatives. Their theory is well developed, and in many cases one may express their solutions in terms of integrals.

Most ODEs that are encountered in physics are linear. Therefore, most special functions may be defined as solutions of linear differential equations (see Holonomic function).

As, in general, the solutions of a differential equation cannot be expressed by a closed-form expression, numerical methods are commonly used for solving differential equations on a computer.

Partial differential equations

Main article: Partial differential equation

A partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. (This is in contrast to ordinary differential equations, which deal with functions of a single variable and their derivatives.) PDEs are used to formulate problems involving functions of several variables, and are either solved in closed form, or used to create a relevant computer model.

PDEs can be used to describe a wide variety of phenomena in nature such as sound, heat, electrostatics, electrodynamics, fluid flow, elasticity, or quantum mechanics. These seemingly distinct physical phenomena can be formalized similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. Stochastic partial differential equations generalize partial differential equations for modeling randomness.

Non-linear differential equations

Main article: Non-linear differential equations

A non-linear differential equation is a differential equation that is not a linear equation in the unknown function and its derivatives (the linearity or non-linearity in the arguments of the function are not considered here). There are very few methods of solving nonlinear differential equations exactly; those that are known typically depend on the equation having particular symmetries. Nonlinear differential equations can exhibit very complicated behaviour over extended time intervals, characteristic of chaos. Even the fundamental questions of existence, uniqueness, and extendability of solutions for nonlinear differential equations, and well-posedness of initial and boundary value problems for nonlinear PDEs are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory (cf. Navier–Stokes existence and smoothness). However, if the differential equation is a correctly formulated representation of a meaningful physical process, then one expects it to have a solution.

Linear differential equations frequently appear as approximations to nonlinear equations. These approximations are only valid under restricted conditions. For example, the harmonic oscillator equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations.

Equation order and degree

The order of the differential equation is the highest order of derivative of the unknown function that appears in the differential equation. For example, an equation containing only first-order derivatives is a first-order differential equation, an equation containing the second-order derivative is a second-order differential equation, and so on.

When it is written as a polynomial equation in the unknown function and its derivatives, its degree of the differential equation is, depending on the context, the polynomial degree in the highest derivative of the unknown function, or its total degree in the unknown function and its derivatives. In particular, a linear differential equation has degree one for both meanings, but the non-linear differential equation ${\displaystyle y^{\prime }+y^2=0}$ is of degree one for the first meaning but not for the second one.

Differential equations that describe natural phenomena almost always have only first and second order derivatives in them, but there are some exceptions, such as the thin-film equation, which is a fourth order partial differential equation.

Examples

In the first group of examples u is an unknown function of x, and c and ω are constants that are supposed to be known. Two broad classifications of both ordinary and partial differential equations consist of distinguishing between linear and nonlinear differential equations, and between homogeneous differential equations and heterogeneous ones.

• Heterogeneous first-order linear constant coefficient ordinary differential equation:

  ${\displaystyle \frac{du}{dx}=cu+x^2.}$

• Homogeneous second-order linear ordinary differential equation:

  ${\displaystyle \frac{d^{2}u}{d{x}^2}-x\frac{du}{dx}+u=0.}$

• Homogeneous second-order linear constant coefficient ordinary differential equation describing the harmonic oscillator:

  ${\displaystyle \frac{d^{2}u}{d{x}^2}+{\omega }^{2}u=0.}$

• Heterogeneous first-order nonlinear ordinary differential equation:

  ${\displaystyle \frac{du}{dx}=u^2+4.}$

• Second-order nonlinear (due to sine function) ordinary differential equation describing the motion of a pendulum of length L:

  ${\displaystyle L\frac{d^{2}u}{d{x}^2}+g\sin u=0.}$

In the next group of examples, the unknown function u depends on two variables x and t or x and y.

• Homogeneous first-order linear partial differential equation:

  ${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}+t\frac{\mathrm{\partial }u}{\mathrm{\partial }x}=0.}$

• Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the Laplace equation:

  ${\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}=0.}$

• Homogeneous third-order non-linear partial differential equation, the KdV equation:

  ${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}=6u\frac{\mathrm{\partial }u}{\mathrm{\partial }x}-\frac{{\mathrm{\partial }}^{3}u}{\mathrm{\partial }{x}^3}.}$

Existence of solutions

Solving differential equations is not like solving algebraic equations. Not only are their solutions often unclear, but whether solutions are unique or exist at all are also notable subjects of interest.

For first order initial value problems, the Peano existence theorem gives one set of circumstances in which a solution exists. Given any point ${\displaystyle (a,b)}$ in the xy-plane, define some rectangular region ${\displaystyle Z}$, such that ${\displaystyle Z=[l,m]\times [n,p]}$ and ${\displaystyle (a,b)}$ is in the interior of ${\displaystyle Z}$. If we are given a differential equation $\frac{dy}{dx}=g(x,y)$ and the condition that ${\displaystyle y=b}$ when ${\displaystyle x=a}$, then there is locally a solution to this problem if ${\displaystyle g(x,y)}$ and $\frac{\mathrm{\partial }g}{\mathrm{\partial }x}$ are both continuous on ${\displaystyle Z}$. This solution exists on some interval with its center at ${\displaystyle a}$. The solution may not be unique. (See Ordinary differential equation for other results.)

However, this only helps us with first order initial value problems. Suppose we had a linear initial value problem of the nth order:

${\displaystyle f_n(x)\frac{d^{n}y}{d{x}^n}+\cdots +f_1(x)\frac{dy}{dx}+f_0(x)y=g(x)}$

such that

${\displaystyle \begin{array}{rlrlrlr}y(x_0)& =y_0,& y^{\prime }(x_0)& =y_0^{\prime },& y^{″}(x_0)& =y_0^{″},& \dots \end{array}}$

For any nonzero ${\displaystyle f_n(x)}$, if ${\displaystyle \{f_0,f_1,\dots \}}$ and ${\displaystyle g}$ are continuous on some interval containing ${\displaystyle x_0}$, ${\displaystyle y}$ is unique and exists.

Related concepts

• A delay differential equation (DDE) is an equation for a function of a single variable, usually called time, in which the derivative of the function at a certain time is given in terms of the values of the function at earlier times.
• Integral equations may be viewed as the analog to differential equations where instead of the equation involving derivatives, the equation contains integrals.
• An integro-differential equation (IDE) is an equation that combines aspects of a differential equation and an integral equation.
• A stochastic differential equation (SDE) is an equation in which the unknown quantity is a stochastic process and the equation involves some known stochastic processes, for example, the Wiener process in the case of diffusion equations.
• A stochastic partial differential equation (SPDE) is an equation that generalizes SDEs to include space-time noise processes, with applications in quantum field theory and statistical mechanics.
• An ultrametric pseudo-differential equation is an equation which contains p-adic numbers in an ultrametric space. Mathematical models that involve ultrametric pseudo-differential equations use pseudo-differential operators instead of differential operators.
• A differential algebraic equation (DAE) is a differential equation comprising differential and algebraic terms, given in implicit form.

Connection to difference equations

See also: Time scale calculus

The theory of differential equations is closely related to the theory of difference equations, in which the coordinates assume only discrete values, and the relationship involves values of the unknown function or functions and values at nearby coordinates. Many methods to compute numerical solutions of differential equations or study the properties of differential equations involve the approximation of the solution of a differential equation by the solution of a corresponding difference equation.

Applications

The study of differential equations is a wide field in pure and applied mathematics, physics, and engineering. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions. Differential equations play an important role in modeling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not have closed form solutions. Instead, solutions can be approximated using numerical methods.

Many fundamental laws of physics and chemistry can be formulated as differential equations. In biology and economics, differential equations are used to model the behavior of complex systems. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations. Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. As an example, consider the propagation of light and sound in the atmosphere, and of waves on the surface of a pond. All of them may be described by the same second-order partial differential equation, the wave equation, which allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat, the theory of which was developed by Joseph Fourier, is governed by another second-order partial differential equation, the heat equation. It turns out that many diffusion processes, while seemingly different, are described by the same equation; the Black–Scholes equation in finance is, for instance, related to the heat equation.

The number of differential equations that have received a name, in various scientific areas is a witness of the importance of the topic. See List of named differential equations.

Software

Some CAS software can solve differential equations. These are the commands used in the leading programs:

• Maple: dsolve
• Mathematica: DSolve[]
• Maxima: ode2(equation, y, x)
• SageMath: desolve()
• SymPy: sympy.solvers.ode.dsolve(equation)
• Xcas: desolve(y'=k*y,y)