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+b\sqrt{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}=1.732\ldots=[1;1,2,1,2,1,2,\ldots]

Square 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. 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 if an integer is not an exact square of another integer its square root must be 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.^[1]

The following proof by Richard Dedekind assumes nothing more than the ordering of the natural numbers and makes no use of prime numbers.^[2] 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

< (n+1)q

Let s = p-nq then 0 < 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

² - 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.

See also

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

External links

• Continued fraction calculator for quadratic irrationals
• Proof that e is not a quadratic irrational

References

de:Quadratisch irrationale Zahl fr:Irrationnel quadratique sl:kvadratno iracionalno ?tevilo zh:?????

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