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mckayla_04
Describe the usefulness of the conjugate and its effects on other complex numbers?
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The conjugate it's used to solve operations with complex numbers.Conjugation of a complex number describes an axial symmetry of the complex plane. To conjugate a complex number, reflect its position through. Conjugation of a complex number describes an axial symmetry of the complex plane. To conjugate a complex number, reflect its position through the real axis. This geometric significance is used a lot
. In mathematics, complex conjugates are a pair of complex numbers
Any complex number may be written as \[a+ib\] where \[a,b \in \mathbb{R}\] The complex conjugate of \[a+ib\] is \[a-ib\] The effect of the conjugate can be seen under the various operations: \[a+ib - (a-ib) = 2ib \in \mathbb{C}\] \[a+ib + (a-ib) = 2a \in \mathbb{R}\] \[(a+ib)(a-ib) = a^2 +b^2 \in \mathbb{R}\] \[\frac{a+ib}{a-ib} = \frac{(a+ib)^2}{(a-ib)(a+ib)} = \frac{a^2+2abi-b^2}{a^2+b^2} = \frac{1}{a^2+b^2}([a^2-b^2] + i[2ab])\]
Notice that \[\sqrt{(a+ib)(a-ib)} = \sqrt{a^2+b^2} = |a+ib|\] (the modulus)
A complex multiplied by its conjugate is reduced to a real no. So, it helps while rationalizing, especially the denominators.