Conjugate (square roots)

{{About|conjugation by changing the sign of a square root|other uses|Conjugate (disambiguation)}}

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

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

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

Properties

$(a + b \sqrt d)(a - b \sqrt d) = a^2 - b^2 d$

and

$(a + b \sqrt d) + (a - b \sqrt d) = 2a,$

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:

$\frac{a + b \sqrt d}{x + y\sqrt d} = \frac{(a + b \sqrt d)(x - y \sqrt d)}{(x + y \sqrt d)(x - y \sqrt d)}
= \frac{ax - dby + (xb - ay) \sqrt d}{x^2 - y^2 d}.$

Hence:

$\frac{1}{a + b \sqrt d} = \frac{a - b \sqrt d}{a^2 - db^2}.$

A corollary property is that the subtraction:

: $(a+b\sqrt d) - (a-b\sqrt d)= 2b\sqrt d,$

leaves only a term containing the root.

Conjugate (square roots)

Properties

See also