Multiplying Complex Numbers

The multiplication of complex numbers involves finding the product of two or more complex numbers using the distributive property.

That is, for two complex numbers: z₁ = a + ib and z₂ = c + id, the product of z₁ and z₂ is z₁z₂ = (a + ib) (c + id).

The multiplication of complex numbers follows similar properties to the real numbers. The properties are 

• Commutative property
   
• Associative property
   
• Distributive property

Commutative property: It can be stated as the order of multiplication does not affect the result. For example, z₁ × z₂ = z₂ × z₁. 

Associative property: The associative property of complex numbers states that the order of grouping the complex numbers doesn't change the result. That is, z₁ × (z₂ × z₃) = (z₁ × z₂) × z₃. 

Distributive property: The distributive property states that z₁ (z₂ + z₃) = z1z₂ + z₁z₃

Now, let’s learn how to multiply complex numbers in Cartesian form. In this form, we multiply the complex number term by term. 
 

That is, if z₁ = a + ib and z₂ = c + id. 

Then z₁ × z₂ = (a + ib) (c + id) 

= ac + a(id) + ib(c) + i²bd

= (ac - bd) +  i(ad + bc)

So, (a + ib) (c + id) = (ac - bd) +  i(ad + bc)

For example, find the product of Z₁ = 3 + 2i and z₂ = 1 + 4i
Z₁z₂ = (3 + 2i) (1 + 4i)

Using the formula for multiplying complex numbers, (a + ib) (c + id) = (ac - bd) +  i(ad + bc)

(3 + 2i) (1 + 4i) = ((3 × 1) - (2 × 4)) + i((3 × 4) + (2 × 1))

= (3 - 8) + i (12 + 2)

= -5 + 14i

When multiplying complex numbers in polar form, we first multiply the moduli and then add the arguments. That is, when multiplying z₁ = r₁e^iΘ1 and z₂ = r2e^iΘ2

Z₁z₂ = r₁e^iΘ1 × r₂e^iΘ2

= (r₁ r₂) e^{iΘ1 + iΘ2}

= r₁r₂ e^{i(Θ1 + Θ2)}
  

For example, multiplying z₁ = 2e^iπ/6 and z₂ = 3e^iπ/4

z1z₂ = r₁r₂ e^{i(Θ1 + Θ2)}

Here, r₁ = 2

r₂ = 3

Θ₁ = π/6 

Θ₂ = π/4

Multiplying moduli (r₁r₂) = 2 × 3 = 6

Adding the arguments = π/6 + π/4 

= 2π/12 + 3π/12

= 5π/12
 

So, 2e^iπ/6 × 3e^iπ/4 = 6e^i5π/12

The multiplication formula for complex numbers is  (a + ib) (c + id) = (ac - bd) +  i (ad + bc). When multiplying the complex number by a real number, let's consider b as 0. That is a (c + id) = ac + iad. 

For example: -5i (2 + 3i)

Using the formula, a (c + id) = ac + iad. 

Here, a = -5i

c = 2

d = 3i

Substituting the value, -5i (2 + 3i) = (-5i × 2) + (-5i × 3i)
= -10i + (-15i²)

Substituting i² = -1

= -10i + 15 = 15 - 10i. 

The formula for multiplying complex numbers is (a + ib) (c + id) = (ac - bd) +  i(ad + bc). When squaring a complex number, consider a = c and b = d, and then the formula becomes;

(a + ib)² = (a×a - b × b) + i(ab + ba)

= (a² - b²) + i (2ab)

For example, squaring 4 + 5i

(a + ib)² = (a² - b²) + i(2ab).

So, (4 + 5i) = (4² - (5i)²) + i(2 × 4 × 5)

= (16 - 25 × i²) + 40i

= 16 - 25 + 40i

= -9 + 40i

The multiplicative inverse of a complex number z = a + ib is another complex number z^-1, that is, z × z^-1 = 1. For a complex number z = a + ib, the multiple inverse z^-1 = z/|z|²

Here, z = a - ib, so, |z| = √a² + b2

The formula for the multiplicative inverse: z^-1 = z/|z|²

Where, conjugate: z = a - ib, and the modulus of the complex number is |z| = √a² + b²

Complex number multiplication is used in various fields, like engineering, quantum mechanics, computer graphics, etc. In this section, we will discuss them in detail. 

• In electrical engineering, complex number multiplication is used to calculate power, impedance, and phase shifts.

• In quantum mechanics, it is used to describe and calculate quantum states. 

• In 2D graphics, complex numbers are used in transformations such as rotation, scaling, and reflections. 

• Cryptography involves complex numbers, especially in elliptic curve cryptography and quantum cryptography.