In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is
where a ≠ 0.
The quadratic equation on a number can be solved using the well-known quadratic formula, which can be derived by completing the square. That formula always gives the roots of the quadratic equation, but the solutions are expressed in a form that often involves a quadratic irrational number, which is an algebraic fraction that can be evaluated as a decimal fraction only by applying an additional root extraction algorithm.
If the roots are real, there is an alternative technique that obtains a rational approximation to one of the roots by manipulating the equation directly. The method works in many cases, and long ago it stimulated further development of the analytical theory of continued fractions.
Here is a simple example to illustrate the solution of a quadratic equation using continued fractions. We begin with the equation
and manipulate it directly. Subtracting one from both sides we obtain
This is easily factored into
from which we obtain
and finally
Now comes the crucial step. We substitute this expression for x back into itself, recursively, to obtain
But now we can make the same recursive substitution again, and again, and again, pushing the unknown quantity x as far down and to the right as we please, and obtaining in the limit the infinite simple continued fraction
By applying the fundamental recurrence formulas we may easily compute the successive convergents of this continued fraction to be 1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, ..., where each successive convergent is formed by taking the numerator plus the denominator of the preceding term as the denominator in the next term, then adding in the preceding denominator to form the new numerator. This sequence of denominators is a particular Lucas sequence known as the Pell numbers.
We can gain further insight into this simple example by considering the successive powers of
That sequence of successive powers is given by
and so forth. Notice how the fractions derived as successive approximants to √2 appear in this geometric progression.
Since 0 <ω< 1, the sequence {ωn} clearly tends toward zero, by well-known properties of the positive real numbers. This fact can be used to prove, rigorously, that the convergents discussed in the simple example above do in fact converge to √2, in the limit.
We can also find these numerators and denominators appearing in the successive powers of
The sequence of successive powers {ω−n} does not approach zero; it grows without limit instead. But it can still be used to obtain the convergents in our simple example.
Notice also that the set obtained by forming all the combinations a + b√2, where a and b are integers, is an example of an object known in abstract algebra as a ring, and more specifically as an integral domain. The number ω is a unit in that integral domain. See also algebraic number field.
Continued fractions are most conveniently applied to solve the general quadratic equation expressed in the form of a monic polynomial
which can always be obtained by dividing the original equation by its leading coefficient. Starting from this monic equation we see that
But now we can apply the last equation to itself recursively to obtain
If this infinite continued fraction converges at all, it must converge to one of the roots of the monic polynomial x2 + bx + c = 0. Unfortunately, this particular continued fraction does not converge to a finite number in every case. We can easily see that this is so by considering the quadratic formula and a monic polynomial with real coefficients. If the discriminant of such a polynomial is negative, then both roots of the quadratic equation have imaginary parts. In particular, if b and c are real numbers and b2 − 4c< 0, all the convergents of this continued fraction "solution" will be real numbers, and they cannot possibly converge to a root of the form u + iv (where v ≠ 0), which does not lie on the real number line.
By applying a result obtained by Euler in 1748 it can be shown that the continued fraction solution to the general monic quadratic equation with real coefficients
given by
either converges or diverges depending on both the coefficient b and the value of the discriminant, b2− 4c.
If b = 0 the general continued fraction solution is totally divergent; the convergents alternate between 0 and . If b ≠ 0 we distinguish three cases.
When the monic quadratic equation with real coefficients is of the form x2 = c, the general solution described above is useless because division by zero is not well defined. As long as c is positive, though, it is always possible to transform the equation by subtracting a perfect square from both sides and proceeding along the lines illustrated with √2 above. In symbols, if
just choose some positive real number p such that
Then by direct manipulation we obtain
and this transformed continued fraction must converge because all the partial numerators and partial denominators are positive real numbers.
By the fundamental theorem of algebra, if the monic polynomial equation x2 + bx + c = 0 has complex coefficients, it must have two (not necessarily distinct) complex roots. Unfortunately, the discriminant b2 − 4c is not as useful in this situation, because it may be a complex number. Still, a modified version of the general theorem can be proved.
The continued fraction solution to the general monic quadratic equation with complex coefficients
given by
converges or not depending on the value of the discriminant, b2− 4c, and on the relative magnitude of its two roots.
Denoting the two roots by r1 and r2 we distinguish three cases.
In case 2, the rate of convergence depends on the absolute value of the ratio between the two roots: the farther that ratio is from unity, the more quickly the continued fraction converges.
This general solution of monic quadratic equations with complex coefficients is usually not very useful for obtaining rational approximations to the roots, because the criteria are circular (that is, the relative magnitudes of the two roots must be known before we can conclude that the fraction converges, in most cases). But this solution does find useful applications in the further analysis of the convergence problem for continued fractions with complex elements.
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer coefficients. For example, the golden ratio, , is an algebraic number, because it is a root of the polynomial x2 − x − 1. That is, it is a value for x for which the polynomial evaluates to zero. As another example, the complex number is algebraic because it is a root of x4 + 4.
In mathematics, a quadratic equation is an equation that can be rearranged in standard form as where the variable x represents an unknown number, and a, b, and c represent known numbers, where a ≠ 0. The numbers a, b, and c are the coefficients of the equation and may be distinguished by respectively calling them, the quadratic coefficient, the linear coefficient and the constant coefficient or free term.
In mathematics, a square root of a number x is a number y such that ; in other words, a number y whose square is x. For example, 4 and −4 are square roots of 16 because .
A simple or regular continued fraction is a continued fraction with numerators all equal one, and denominators built from a sequence of integer numbers. The sequence can be finite or infinite, resulting in a finite continued fraction like
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factoring, number theory, and algebraic geometry.
In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation. Other ways of solving quadratic equations, such as completing the square, yield the same solutions.
In algebra, a cubic equation in one variable is an equation of the form in which a is not zero.
In mathematics, an nth root of a number x is a number r which, when raised to the power of the positive integer n, yields x:
In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form to the form for some values of and . In terms of a new quantity , this expression is a quadratic polynomial with no linear term. By subsequently isolating and taking the square root, a quadratic problem can be reduced to a linear problem.
In mathematics, a quadratic irrational number 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
In algebra, a quartic function is a function of the form
A continued fraction is a mathematical expression that can be writen as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, the continued fraction is finite or infinite.
Methods of computing square roots are algorithms for approximating the non-negative square root of a positive real number . Since all square roots of natural numbers, other than of perfect squares, are irrational, square roots can usually only be computed to some finite precision: these methods typically construct a series of increasingly accurate approximations.
In the analytic theory of continued fractions, the convergence problem is the determination of conditions on the partial numeratorsai and partial denominatorsbi that are sufficient to guarantee the convergence of the infinite continued fraction
In mathematics, an infinite periodic continued fraction is a simple continued fraction that can be placed in the form
In algebra, casus irreducibilis is one of the cases that may arise in solving polynomials of degree 3 or higher with integer coefficients algebraically, i.e., by obtaining roots that are expressed with radicals. It shows that many algebraic numbers are real-valued but cannot be expressed in radicals without introducing complex numbers. The most notable occurrence of casus irreducibilis is in the case of cubic polynomials that have three real roots, which was proven by Pierre Wantzel in 1843. One can see whether a given cubic polynomial is in the so-called casus irreducibilis by looking at the discriminant, via Cardano's formula.
The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. It is more precisely called the principal square root of 5, to distinguish it from the negative number with the same property. This number appears in the fractional expression for the golden ratio. It can be denoted in surd form as:
In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form
In mathematics, a Brjuno number is a special type of irrational number named for Russian mathematician Alexander Bruno, who introduced them in Brjuno (1971).
The square root of 6 is the positive real number that, when multiplied by itself, gives the natural number 6. It is more precisely called the principal square root of 6, to distinguish it from the negative number with the same property. This number appears in numerous geometric and number-theoretic contexts. It can be denoted in surd form as: