Question

if z is any non-zero complex number; prove that the multiplicative inverse of z is z bar/|z^2| .Hence express the following numbers in the form of X+iy.​

Answer 1

Answer:

written the reason in attachment please refer it

Answer 2

Answer:

The multiplicative inverse of 'z' is $\bar z$/|z|².

Step-by-step explanation:

Let the multiplicative inverse of z be p.

The multiplicative inverse is defined as when z and p are multiplied we get 1.

i.e. z.p = 1

⇒ p = 1/z

But we know that z = x + iy for a complex function

⇒ p = 1/(x + iy)

$\implies p = \frac{x-iy}{(x+ iy)(x-iy)}$ (We multiplied the numerator and denominator by x - iy)

⇒ p = (x - iy)/(x²  + y²)  (since, (x + iy)(x - iy) = (x² + y²) = |z|²)

But we know that (x - iy) = $\bar z$ i.e. $\bar z$ is the conjugate of 'z'.

So the equation becomes:

p =  $\bar z$/|z|²

Therefore, the multiplicative inverse of 'z' is $\bar z$/|z|².

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