In mathematics, the conjugate of an expression of the form

a+b\sqrt d

a-b\sqrt d,

provided that

\sqrt d

does not appear in and . One says also that the two expressions are conjugate. In particular, the two solutions of a

quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown (mathematics), unknown value, and , , and represent known numbers, where . (If and then the equati ...

are conjugate, as per the

\pm

in the

quadratic formula In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, g ...

x=\frac

Complex conjugation In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a ...

is the special case where the

square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...

i=\sqrt.

Properties

As :

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

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 A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...

, by multiplying the

numerator A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...

and the denominator of a

fraction A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...

by the conjugate of the denominator (see Rationalisation). Typically, one has :

\frac = \frac 
= \frac.

In particular :

\frac = \frac.

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.

Properties

See also