Question

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?

Answer 1

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.

Answer 2

so what exactly is the relationship? a real number?

Answer 3

\( (a+bi)(a-bi)=a^2+b^2 \)
The result is a real number for all values of a and b.

Answer 4

how can i put that into words?

Answer 5

You can expand the last sentence of my first response. Post your version if you're not sure.

Answer 6

"the complex number and its conjugate have a set of real numbers in common"

Answer 7

this is my version

Answer 8

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.