**How to Find Inverse of the Given Complex Number :**

Here we are going to see some example problems to understand how to find the inverse of the given complex number.

**Question 1 :**

If z_{1} = 2 − i and z_{2} = −4 + 3i , find the inverse of z_{1} z_{2 }and z_{1}/z_{2}

**Solution : **

(i)

z_{1} z_{2 }= (2 - i)(-4 + 3i)

= -8 + 6i + 4i - 3i^{2}

= -8 + 10i - 3(-1)

= -8 + 10i + 3

= -5 + 10i

z_{1} z_{2} = -5(1 - 2i)

Inverse of z_{1} z_{2 }= 1/z_{1} z_{2}

= (-1/5)(1/(1 - 2i))

= (-1/5)(1/(1 - 2i))((1 + 2i)/(1 + 2i))

= (-1/5)(1 + 2i)/(1 - 4(-1))

= (-1/5)(1 + 2i)/(1 + 4)

= (-1/25)(1 + 2i)

= (-1 - 2i)/25

(ii) Inverse of z_{1}/z_{2 }= z_{2}/z_{1}

= [(-4 + 3i)/(2 - i)][(2 + i)/(2 + i)]

= (-4 + 3i)(2 + i) / (2 - i)(2 +i)

= (-8 - 4i + 6i + 3i^{2})/(4 -(-1))

= (-8 + 2i - 3)/(4 + 1)

= (-11 + 2i)/5

= (1/5) (-11 + 2i)

Let us look in to the next problem on "How to Find Inverse of the Given Complex Number".

**Question 2 :**

The complex numbers u,v , and w are related by (1/u) = (1/v) + (1/w)

if v = 3 - 4i and w = 4 + 3i, find u in rectangular form.

**Solution :**

Given that

(1/u) = (1/v) + (1/w)

1/v = [1/(3 - 4i)][(3 + 4i)/(3 + 4i)]

= (3 + 4i)/(9 - 16(-1))

= (3 + 4i)/(9 + 16)

= (3 + 4i)/25

1/w = [1/(4 + 3i)][(4 - 3i)/(4 - 3i)]

= (4 - 3i)/(16 - 9(-1))

= (4 - 3i)/(16 + 9)

= (4 - 3i)/25

1/u = [(3 + 4i)/25] + [(4 - 3i)/25]

(1/u) = (7 + i)/25

By finding the inverse of 1/u, we get u

= [25/(7 + i)][(7 - i)/(7 - i)]

= 25(7 - i)/(49 + 1)

= 25(7 - i)/50

= (7 - i)/2

Hence the value of u in rectangular form is (1/2)(7 - i).

**Question 4 :**

Prove the following properties:

(i) z is real if and only if z = z bar

**Solution :**

Let us consider the complex number 4 + 3i

Here 4 is real part and 3 is the imaginary part.

z bar = 4 - 3i

There is no changes in real parts of the above complex numbers.

Hence z is real if and only if z = z bar.

(ii) Re(z) = (z + z bar)/2

**Solution :**

z = 4 + 3i

z bar = 4 - 3i

z + z bar = (4 + 3i) + (4 - 3i)

= (4 + 4) + (3i - 3i)

= 8

(z + z bar) / 2 = 8/2 = 4 ----(1)

Re(z) = 4 ----(2)

(1) = (2)

Hence proved.

(iii) im(z) = (z - z bar) / 2

z = 4 + 3i

z bar = 4 - 3i

z - z bar = (4 + 3i) - (4 - 3i)

= (4 - 4) + (3i + 3i)

= 6i

(z + z bar) / 2 = 6i/2 = 3 ----(1)

im(z) = 3 ----(2)

(1) = (2)

After having gone through the stuff given above, we hope that the students would have understood, "How to Find Inverse of the Given Complex Number".

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