Question

Find the polar form of the multiplicative inverse of the complex number: z = a + 2ib.​

Answer 1

Answer:

Use the polar form of complex numbers to show that every complex number z≠0 has multiplicative inverse z−1.

If z=a+bi, then the polar form is z=r(cos(α))+i(sin(α)).

I can do it, not using polar coordinates: Let z=a+ib.

Since 1+0i is the multiplicative identity, if x+iy is the multiplicative inverse of z=a+ib, then (a+ib)(x+iy)=1+i0 ⇒ (ax−by)+i(ay+bx)=1+i0 ⇒ ax−by=1bx+ay=0 ⇒ x=aa2+b2, y=−ba2+b2, if a2+b2≠0.

The multiplicative inverse of a+ib is therefore aa2+b2+i−ba2+b2.

Now, should I use this same process and just replace z=a+ib with z=r(cos(α))+i(sin(α))?? Or is there another way to go about this process?