Conjugate (square roots)

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 ${\displaystyle a+b{\sqrt {d}}}$ is ${\displaystyle a-b{\sqrt {d}},}$ provided that ${\displaystyle {\sqrt {d}}}$ 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 ${\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}}$.

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

Properties

As

${\displaystyle (a+b{\sqrt {d}})(a-b{\sqrt {d}})=a^{2}-db^{2}}$ 

and

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

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). Typically, one has

${\displaystyle {\frac {a_{1}+b_{1}{\sqrt {d}}}{a_{2}+b_{2}{\sqrt {d}}}}={\frac {(a_{1}+b_{1}{\sqrt {d}})(a_{2}-b_{2}{\sqrt {d}})}{(a_{2}+b_{2}{\sqrt {d}})(a_{2}-b_{2}{\sqrt {d}})}}={\frac {a_{1}a_{2}-db_{1}b_{2}+(a_{2}b_{1}-a_{1}b_{2}){\sqrt {d}}}{a_{2}^{2}-db_{2}^{2}}}.}$ 

In particular

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

A corollary property is that the subtraction:

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

leaves only a term containing the root.

See also

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