Question

Give multiplicative inverse of z=a+ib

Answer 1

multiplicative inverse of z=(a-ib)/a square +b square

Answer 2

Concept:

The multiplicative inverse is a value which when multiplied by the original number results in 1.

Given:

z = a + ib

Find:

We are asked to find the multiplicative inverse.

Solution:

We have,

z = a + ib

So,

The reciprocal of z is z⁻¹,

i.e.

z⁻¹ = 1 /z

So,

z⁻¹ = 1 /z = 1 / (a + ib)

NOw,

z⁻¹ = 1 /z = [1 / (a + ib)] × [(a - ib) / (a - ib)]

i.e.

z⁻¹ =  [(a - ib) / (a² - i²b²)]

We get,

z⁻¹ =  [(a - ib) / (a² + b²)]   (Because i² = -1)

So,

Now,

z⁻¹ =  [(a - ib) / (a² + b²)]

So, this is the multiplicative inverse.

Hence, the multiplicative inverse of z = a + ib is z⁻¹ =  [(a - ib) / (a² + b²)] .

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