## anonymous 4 years ago 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?

1. 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.

2. anonymous

so what exactly is the relationship? a real number?

3. mathmate

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

4. anonymous

how can i put that into words?

5. mathmate

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

6. anonymous

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

7. anonymous

this is my version

8. 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.