anonymous
  • anonymous
What is the relationship between a complex number and its conjugate? and what is the usefulness of the conjugate and its effect on other complex numbers?
Mathematics
jamiebookeater
  • jamiebookeater
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mathmate
  • mathmate
The conjugate of a complex number is that the sign of the complex component is inverted. For example, the conjugate of a+bi is a-bi. One significance is that the product of a complex number with its conjugate is always real.
anonymous
  • anonymous
so what exactly is the relationship? a real number?
mathmate
  • mathmate
\( (a+bi)(a-bi)=a^2+b^2 \) The result is a real number for all values of a and b.

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anonymous
  • anonymous
how can i put that into words?
mathmate
  • mathmate
You can expand the last sentence of my first response. Post your version if you're not sure.
anonymous
  • anonymous
"the complex number and its conjugate have a set of real numbers in common"
anonymous
  • anonymous
this is my version
mathmate
  • mathmate
I would put it as the product of a complex and its conjugate is real. This is useful in "rationalizing" complex numbers where the denominator is a complex number. \[\frac{2+3i}{3+4i} = \frac{(2+3i)(3-4i)}{(3+4i)(3-4i)}=\frac{18+i}{25}\] This facilitates manipulation (such as addition) of complex numbers.

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