mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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, gr ...

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 . E ...

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