Question

We have seen the following:

p.p¯=62+52
z.z¯=a2+b2
Now, observe that p.p¯=62+52 and z.z¯=a2+b2 are equal to the sum of the squares of the real and imaginary parts of p=6+5i and z=a+bi respectively.

So, we can represent p.p¯ and z.z¯ in terms of |p| and |z| respectively as follows:

Answer 1

Yes, that is correct. You can represent the complex conjugate of a complex number p=a+bi as p.p¯=a-bi, where the dot (.) represents the complex conjugate.

The absolute value or magnitude of a complex number p=a+bi is given by |p|=sqrt(a^2+b^2). Therefore, you can represent p.p¯ and z.z¯ in terms of |p| and |z| as follows:

p.p¯ = |p|^2

z.z¯ = |z|^2

This is because the absolute value or magnitude of a complex number is equal to the square root of the sum of the squares of its real and imaginary parts. So, |p|^2 = a^2 + b^2 and |z|^2 = a^2 + b^2.

Explanation: