# Problems on Complex Numbers

We will learn step-by-step how to solve different types of problems on complex numbers using the formulas.

1. Express $$(\frac{1 + i}{1 - i})^{3}$$ in the form A + iB where A and B are real numbers.

Solution:

Given $$(\frac{1 + i}{1 - i})^{3}$$

Now $$\frac{1 + i}{1 - i}$$

= $$\frac{(1 + i)(1 + i)}{(1 - i)(1 + i)}$$

= $$\frac{(1 + i)^{2}}{(1^{2} - i^{2}}$$

= $$\frac{1 + 2i + iˆ{2}}{1 - (-1)}$$

= $$\frac{1 + 2i - 1}{2}$$

= $$\frac{2i}{2}$$

= i

Therefore, $$(\frac{1 + i}{1 - i})^{3}$$ = i$$^{3}$$= i$$^{2}$$ ∙  i = - i = 0 + i (-1), which is the required form A + iB where A = 0 and B = -1.

2. Find the modulus of the complex quantity (2 - 3i)(-1 + 7i).

Solution:

The given complex quantity is (2 - 3i)(-1 + 7i)

Let z$$_{1}$$ = 2 - 3i and z$$_{2}$$ = -1 + 7i

Therefore, |z$$_{1}$$| = $$\sqrt{2^{2} + (-3)^{2}}$$ = $$\sqrt{4 + 9}$$ = $$\sqrt{13}$$

And |z$$_{2}$$| = $$\sqrt{(-1)^{2} + 7^{2}}$$ = $$\sqrt{1 + 49}$$ = $$\sqrt{50}$$ = 5$$\sqrt{2}$$

Therefore, the required modulus of the given complex quantity = |z$$_{1}$$z$$_{1}$$| = |z$$_{1}$$||z$$_{1}$$| = $$\sqrt{13}$$  ∙ 5$$\sqrt{2}$$ = 5$$\sqrt{26}$$

3. Find the modulus and principal amplitude of -4.

Solution:

Let z = -4 + 0i.

Then, modulus of z = |z| = $$\sqrt{(-4)^{2} + 0^{2}}$$ = $$\sqrt{16}$$ = 4.

Clearly, the point in the z-plane the point z = - 4 + 0i = (-4, 0) lies on the negative side of real axis.

Therefore, the principle amplitude of z is π.

4. Find the amplitude and modulus of the complex number -2 + 2√3i.

Solution:

The given complex number is -2 + 2√3i.

The modulus of -2 + 2√3i = $$\sqrt{(-2)^{2} + (2√3)^{2}}$$ = $$\sqrt{4 + 12}$$ = $$\sqrt{16}$$ = 4.

Therefore, the modulus of -2 + 2√3i = 4

Clearly, in the z-plane the point z = -2 + 2√3i = (-2, 2√3) lies in the second quadrant. Hence, if amp z = θ then,

tan θ = $$\frac{2√3}{-2}$$ = - √3 where, $$\frac{π}{2}$$ < θ ≤ π.

Therefore, tan θ = - √3 = tan (π - $$\frac{π}{3}$$) = tan $$\frac{2π}{3}$$

Therefore, θ = $$\frac{2π}{3}$$

Therefore, the required amplitude of -2 + 2√3i is $$\frac{2π}{3}$$.

5. Find the multiplicative inverse of the complex number z = 4 - 5i.

Solution:

The given complex number is z = 4 - 5i.

We know that every non-zero complex number z = x +iy possesses multiplicative inverse given by

$$(\frac{x}{x^{2} + y^{2}}) + i (\frac{-y}{x^{2} + y^{2}})$$

Therefore, using the above formula, we get

z$$^{-1}$$ = $$(\frac{4}{4^{2} + (-5)^{2}}) + i (\frac{-(-5)}{4^{2} + (-5)^{2}})$$

= $$(\frac{4}{16 + 25}) + i (\frac{5)}{16 + 25})$$

= $$(\frac{4}{41}) + (\frac{5}{41})$$i

Therefore, the multiplicative inverse of the complex number z = 4 - 5i is $$(\frac{4}{41}) + (\frac{5}{41})$$i

6. Factorize: x$$^{2}$$ + y$$^{2}$$

Solution:

x$$^{2}$$ - (-1) y$$^{2}$$ = x$$^{2}$$ - i$$^{2}$$y$$^{2}$$ = (x + iy)(x - iy)