Conjugate (Square Roots)

In mathematics, the conjugate of an expression of the form [math]\displaystyle{ a + b \sqrt d }[/math] is [math]\displaystyle{ a - b \sqrt d, }[/math] provided that [math]\displaystyle{ \sqrt d }[/math] does not appear in a and b. One says also that the two expressions are conjugate.

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

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

Properties

As [math]\displaystyle{ (a + b \sqrt d)(a - b \sqrt d) = a^2 - b^2 d }[/math] and [math]\displaystyle{ (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]\displaystyle{ \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]\displaystyle{ \frac{1}{a + b \sqrt d} = \frac{a - b \sqrt d}{a^2 - db^2}. }[/math]

A corollary property is that the subtraction:

[math]\displaystyle{ (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

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