Quadratic irrational

In mathematics, a quadratic irrational, also known as a quadratic irrationality or quadratic surd, is an irrational number that is the solution to some quadratic equation with rational coefficients. Since fractions can be cleared from a quadratic equation by multiplying both sides by their common denominator, this is the same as saying it is an irrational root of some quadratic equation whose coefficients are integers. They form the real number subset of the algebraic numbers of degree 2. The quadratic irrationals, therefore, are all those numbers that can be expressed in this form:

:$\{a+bsqrt\{c\}\; over\; d\}$

for integers "a", "b", "c", "d"; with "b" and "d" non-zero, and with "c">1 and square free . This implies that the quadratic irrationals have the same cardinality as ordered quadruples of integers, and are therefore countable.

The quadratic irrationals with a given "c" form a field, called a quadratic field.

Quadratic irrationals have useful properties, especially in relation to continued fractions, where we have the result that "all" quadratic irrationals, and "only" quadratic irrationals, have periodic continued fraction forms. For example:$sqrt\{3\}=1.732ldots=\; [1;1,2,1,2,1,2,ldots]$

quare root of non-square is irrational

The square root of any natural number that is not a square is irrational. The square root of 2 was the first to be proved irrational. Theodorus of Cyrene proved the irrationality of the surds of whole numbers up to 17, but stopped there, probably because the algebra he used couldn't be applied to the square root of 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 people when they try to prove the irrationality of the non-square natural numbers implicitly assume the fundamental theorem of arithmetic which was first proven by Carl Friedrich Gauss in his Disquisitiones Arithmeticae.

The following proof by Richard Dedekind assumes nothing more than the ordering of the natural numbers and makes no use of prime numbers. [ [http://www.cut-the-knot.org/proofs/sq_root.shtml Cut the knot: Square root of 2 is irrational] ] Like most proofs of this theorem it uses recursive descent.

Assume D is a non square natural number then there is a number n such that:

:n² < D < (n+1)²

Assume the square root of D is a rational number p/q, assume the q here is the smallest for which this is true, then:

:Dq² = p²

Multiplying through by q² we get

:n²q² < Dq² < (n+1)²q²

then substituting for the middle term and removing the squares:

:nq < p < (n+1)q

Let s = p-nq then 0 < s < q

Furthermore multiplying through by p² we get:

:n²p² < Dp² < (n+1)²p²

again substituting for the middle term and removing the squares we get:

:np < Dq < (n+1)p

Let r = Dq-np then 0 < r < p

Then we get to the recursive descent part of the proof

:Ds² - r² = D(p-nq)² - (Dq-np)² = Dp² - 2Dnpq + Dn²q² - D²q² + 2Dnpq - n²p²

Using Dq² = p² three times this all disappears so we get

:Ds² - r² = 0

But s is greater than zero and less than q which contradicts the assumption about q and the result follows by reductio ad absurdum.

ee also

* Algebraic number field
* Periodic continued fraction
* Restricted partial quotients

External links

*mathworld|QuadraticSurd
* [http://www.numbertheory.org/php/surd.html Continued fraction calculator for quadratic irrationals]
* [http://planetmath.org/encyclopedia/EIsIrrational.html Proof that e is not a quadratic irrational]

References