for integers a, b, c, d; with b, c and d non-zero, and with csquare-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 fieldQ(√c). For example, the inverses of elements of Q(√c) are of the same form as the above algebraic numbers:
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
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 for continued fractions.
Real quadratic irrational numbers and indefinite binary quadratic forms
We may rewrite a quadratic irrationality as follows:
It follows that every quadratic irrational number can be written in the form
This expression is not unique.
Fix a non-square, positive integer congruent to or modulo , and define a set as
Every quadratic irrationality is in some set , since the congruence conditions can be met by scaling the numerator and denominator by an appropriate factor.
with integer entries and can be used to transform a number in . The transformed number is
If is in , then is too.
The relation between and 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 .) Thus, 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 . The standard proof of this involves considering the map from binary quadratic forms of discriminant to given by
A computation shows that 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 , expanding a number in
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
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.
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.
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.
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.
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.
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 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:
Homogeneous second-order linear ordinary differential equation:
Homogeneous second-order linear constant coefficient ordinary differential equation describing the harmonic oscillator:
Second-order nonlinear (due to sine function) ordinary differential equation describing the motion of a pendulum of length L:
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:
Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the Laplace equation:
Homogeneous third-order non-linear partial differential equation, the KdV equation:
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 in the xy-plane, define some rectangular region , such that and is in the interior of . If we are given a differential equation and the condition that when , then there is locally a solution to this problem if and are both continuous on . This solution exists on some interval with its center at . 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:
such that
For any nonzero , if and are continuous on some interval containing , 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.
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: