Английская Википедия:Conjugate (square roots)

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Шаблон:Short description Шаблон:About In mathematics, the conjugate of an expression of the form <math>a + b \sqrt d</math> is <math>a - b \sqrt d,</math> provided that <math>\sqrt d</math> does not appear in Шаблон:Mvar and Шаблон:Mvar. One says also that the two expressions are conjugate.

In particular, the two solutions of a quadratic equation are conjugate, as per the <math>\pm</math> in the quadratic formula <math>x = \frac{-b \pm \sqrt{b^2 - 4ac} }{2a}</math>.

Complex conjugation is the special case where the square root is <math>i = \sqrt{-1},</math> the imaginary unit.

Properties

As <math display="block">(a + b \sqrt d)(a - b \sqrt d) = a^2 - b^2 d</math> and <math display="block">(a + b \sqrt d) + (a - b \sqrt d) = 2a,</math> the sum and the product of conjugate expressions do not involve the square root anymore.

This property is used for removing a square root from a denominator, by multiplying the numerator and the denominator of a fraction by the conjugate of the denominator (see Rationalisation). An example of this usage is: <math display="block">\frac{a + b \sqrt d}{x + y\sqrt d} = \frac{(a + b \sqrt d)(x - y \sqrt d)}{(x + y \sqrt d)(x - y \sqrt d)} = \frac{ax - dby + (xb - ay) \sqrt d}{x^2 - y^2 d}.</math> Hence: <math display="block">\frac{1}{a + b \sqrt d} = \frac{a - b \sqrt d}{a^2 - db^2}.</math>

A corollary property is that the subtraction:

<math display="block">(a+b\sqrt d) - (a-b\sqrt d)= 2b\sqrt d,</math>

leaves only a term containing the root.

See also

• Conjugate element (field theory), the generalization to the roots of a polynomial of any degree