Conjugate (algebra)

This article is about binomial conjugates in algebra. For other uses, see Conjugate (disambiguation).

In algebra, a conjugate is a binomial formed by taking the opposite of the second term of a binomial. The conjugate of $x+y \,$ is $x-y \,$, where x and y are real numbers. If y is imaginary, the process is termed complex conjugation. The complex conjugate of a+bi is a-bi.

Differences of squares

Main article: Difference of two squares

An expression of the form

$a^2-b^2 \,$

can be factored to give

$(a+b)(a-b) \,$

where one factor is the conjugate of the other. This can be useful when trying to rationalize a denominator containing radicals.

Rationalizing radicals in denominator

Main article: Rationalisation (mathematics)

An irrational binomial can sometimes be made rational by multiplying by its conjugate. When rationalizing a denominator, the numerator may remain irrational, though. In order to keep the value of the fraction the same, it is multiplied by the conjugate divided by itself, as shown in the examples below.

$\left ( \frac{1}{a+\sqrt b} \right ) \left ( \frac{a-\sqrt b}{a-\sqrt b} \right ) =\frac{a-\sqrt b}{a^2-b} \,$

$\frac{1}{2+2\sqrt 3} \frac{2-2\sqrt 3}{2-2\sqrt 3} =\frac{2-2\sqrt 3}{2^2-2^2 3}=\frac{2\sqrt 3-2}{8}=\frac{\sqrt 3-1}{4} \,$

See also

• Difference of two squares
• Conjugate element (field theory)

External links

• Rationalizing the Denominator from Mathwords.com
• Math glossary from Bethany Lutheran College