Question

Describe the usefulness of the conjugate and its effects on other complex numbers?

Answer 1

i got it hold on

Answer 2

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

Answer 3

. In mathematics, complex conjugates are a pair of complex numbers

Answer 4

done

Answer 5

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:
\[a+ib - (a-ib) = 2ib \in \mathbb{C}\]
\[a+ib + (a-ib) = 2a \in \mathbb{R}\]
\[(a+ib)(a-ib) = a^2 +b^2 \in \mathbb{R}\]
\[\frac{a+ib}{a-ib} = \frac{(a+ib)^2}{(a-ib)(a+ib)} = \frac{a^2+2abi-b^2}{a^2+b^2} = \frac{1}{a^2+b^2}([a^2-b^2] + i[2ab])\]

Answer 6

Notice that \[\sqrt{(a+ib)(a-ib)} = \sqrt{a^2+b^2} = |a+ib|\]
(the modulus)

Answer 7

A complex multiplied by its conjugate is reduced to a real no. So, it helps while rationalizing, especially the denominators.