In algebra, a conjugate is a binomial formed by taking the opposite of the second term of a binomial. The conjugate of <math> x+y \,</math> is <math> x-y \,</math>, where x and y are real numbers. If y is imaginary, the process is termed complex conjugation. The purpose of a conjugate is to create a perfect square, often to rationalize square roots in a denominator.
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Perfect squares
A perfect square of the form
- <math> a^2-b^2 \,</math>
can be factored to give
- <math> (a+b)(a-b) \,</math>
This can be useful when trying to rationalize a denominator by making it rational if a or b is the irrational square root of a rational number. Multiplying a binomial by -1 can be simplified to the difference between the squares of a and b.
- <math> (a+b)(a-b)=a^2-b^2 \,</math>
Rationalizing radicals in denominator
As mentioned above, 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.
- <math>\left ( \frac{1}{a+\sqrt b} \right ) \left ( \frac{a-\sqrt b}{a-\sqrt b} \right ) =\frac{a-\sqrt b}{a^2-b} \,</math>
- <math> \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} \,</math>
See also
- Conjugate (disambiguation)
- Complex conjugate
External links
- Rationalizing the Denominator from Mathwords.com
- Math glossary from Bethany Lutheran College
