Conjugate (square roots)

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 conjugate of a root of a quadratic polynomial is the other root, obtained by changing the sign of the square root appearing in the quadratic formula.

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

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

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