Conjugate (square roots)

Change of the sign of a square root

This article is about conjugation by changing the sign of a square root. For other uses, see Conjugate (disambiguation).

In mathematics, the conjugate of an expression of the form  a + b d {\displaystyle a+b{\sqrt {d}}} is  a − b d , {\displaystyle a-b{\sqrt {d}},} provided that  d {\displaystyle {\sqrt {d}}} 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  ± {\displaystyle \pm } in the quadratic formula x = − b ± b 2 − 4 a c 2 a {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}} .

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

## Properties

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As  ( a + b d ) ( a − b d ) = a 2 − b 2 d {\displaystyle (a+b{\sqrt {d}})(a-b{\sqrt {d}})=a^{2}-b^{2}d} and  ( a + b d ) + ( a − b d ) = 2 a , {\displaystyle (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:  a + b d x + y d = ( a + b d ) ( x − y d ) ( x + y d ) ( x − y d ) = a x − d b y + ( x b − a y ) d x 2 − y 2 d . {\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}}.} Hence:  1 a + b d = a − b d a 2 − d b 2 . {\displaystyle {\frac {1}{a+b{\sqrt {d}}}}={\frac {a-b{\sqrt {d}}}{a^{2}-db^{2}}}.}

A corollary property is that the subtraction:

( a + b d ) − ( a − b d ) = 2 b d , {\displaystyle (a+b{\sqrt {d}})-(a-b{\sqrt {d}})=2b{\sqrt {d}},}

leaves only a term containing the root.

## See also

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