Multiplication of Complex Numbers

Before getting into the multiplication of complex numbers, let’s recall what is a complex number and how to represent it. In mathematics, we can write complex numbers in the form z = x + iy, where x and y are real numbers, and i is a symbol called the imaginary unit, and satisfying the equation i² = −1. No “real” number satisfies this equation, and i was called an imaginary number by René Descartes. As we know, x is the real part and y is called the imaginary part for the complex number x + iy. Either of the symbols

Unlike numbers, it is also possible to perform algebraic operations on complex numbers. In this article, you will learn how to perform multiplication on complex numbers along with geometrical representations.

Multiplication of Complex Numbers Formula

Suppose z₁ = a + ib and z₂ = c + id are two complex numbers such that a, b, c, and d are real, then the formula for the product of two complex numbers z₁ and z₂ is derived as given below:

Go through the steps given below to perform the multiplication of two complex numbers.

Step 1: Write the given complex numbers to be multiplied.

z₁z₂ = (a + ib)(c + id)

Step 2: Distribute the terms using the FOIL technique to remove the parentheses.

z₁z₂ = ac + i(ad) + i(bc) + i²(bd)

Step 2: Simplify the powers of i and apply the formula i² = –1.

z₁z₂ = ac + i(ad) + i(bc) + (-1)(bd)

Step 3: Combine like terms that mean combine real numbers with real numbers and imaginary numbers with imaginary numbers.

z₁z₂ = (ac – bd) + i(ad + bc)

Get more information about complex numbers here.

Multiplication of Complex Numbers by FOIL Method

Multiplication of Two Complex Numbers

Algebraically, the multiplication of two complex numbers can be found using the formula derived above. Let’s understand this with the help of an example given below:

Example:

Find the product of complex numbers (6 – 5i) and(3 + 7i).

Solution:

(6 – 5i)(3 + 7i) = (6) × (3) + (6) × (7i) – (5i) × (3) – (5i) × (7i)

= 18 + 42i – 15i – 35i²

= 18 + 27i – 35(-1)

= 53 + 27i

Geometrical Representation of Multiplication of Complex Numbers

We can represent the multiplication or product of two complex numbers geometrically in the argand plane. For the above example, the geometrical representation is given in the below figure:

Multiplication of Complex Numbers in Polar Form

Suppose z₁ = r₁(cos θ₁ + i sin θ₁) and z₂ = r₂(cos θ₂ + i sin θ₂) are two complex numbers in polar form, then the product, i.e. multiplication of these two complex numbers can be found using the formula given below:

z₁z₂ = r₁ r₂ [cos (θ₁ + θ₂) + i sin (θ₁ + θ₂)]

Geometrically we can show the multiplication of polar form of complex numbers as:

Properties of Multiplication of Complex Numbers

If z₁, z₂, and z₃ are any three complex numbers, then the following properties can be defined.

• The closure law The product of two complex numbers is a complex number; the product z1z2 is a complex number for all complex numbers z1 and z2.
• Commutative law on multiplication of two complex numbers: z₁z₂ = z₂z₁
• Associative law of multiplication of complex numbers: (z₁z₂)z₃ = z₁(z₂z₃)
• The existence of multiplicative identity: There exists the complex number 1 + i 0 (denoted as 1), called the multiplicative identity such that z.1 = z, for every complex number z.
• The existence of multiplicative inverse: For every non-zero complex number z = x + iy, we have the complex number [x/(x² + y²)] – i[y/(x² + y²)] (denoted by z⁻¹ or 1/z) such that

z.(1/z) = (1/z).z = 1

Here, 1/z is called the multiplicative inverse of z.

• Multiplication of complex numbers is distributive over the addition of complex numbers.
• The distributive law: For any three complex numbers z₁, z₂ and z₃

z₁(z₂ + z₃) = z₁z₂ + z₁z₃

(z₁ + z₂)z₃ = z₁z₃ + z₂z₃

Multiplication of Complex Numbers Examples

Go through the solved examples given below for better understanding of the concept.

Example 1: Compute (7 + 2i)(4 – 6i)

Solution:

(7 + 2i)(4 – 6i)

Using the formula (a + ib)(c + id) = (ac – bd) + i(ad + bc),

a = 7, b = 2, c = 4, d = -6

(7 + 2i)(4 – 6i) = (7)(4) – (2)(-6) + i[(7)(-6) + (2)(4)]

= 28 + 12 + i(-42 + 8)

= 40 – 34i

Example 2: Find the product of three complex numbers (3 + 4i), (2 – 3i) and (5 + 4i).

Solution:

(3 + 4i)(2 – 3i)(5 + 4i)

= [(3)(2) – 3(3i) + (4i)(2) – (4i)(3i)] (5 + 4i)

= [6 – 9i + 8i – 12i²] (5 + 4i)

= [6 – i – 12(-1)](5 + 4i)

= (18 – i)(5 + 4i)

= (18)(5) + 18(4i) – 5i – (i)(4i)

= 90 + 72i – 5i – 4i²

= 90 + 67i – 4(-1)

= 90 + 67i + 4

= 94 + 67i

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