Multiplication of complex numbers

In this publication, we will consider formulas with which you can find the product of two complex numbers presented in algebraic or trigonometric form. Examples are also given for a better understanding of the theoretical material.

Multiplication in algebraic form

The product of two complex numbers x = a₁ + b₁i и y = to₂ + b₂i is also a complex number z:

z = x ⋅ y = (a₁a₂ – b₁b₂) + (a₁b₂ + b₁a₂) ⋅ i

The formula is obtained by multiplying the binomials (a₁ + b₁i)(a₂ + b₂i). At the same time, do not forget that i² =-1.

Example 1

Find the product of complex numbers: x = 3 + 7i и y = 2 – i.

Decision:

x ⋅ y = (3 + 7i)(2 – i) = 3 ⋅ 2 – 3 ⋅ i + 7i ⋅ 2 – 7i ⋅ i = 6 – 3i + 14i – 7i² = 6 + 11i – 7 ⋅ (-1) = 13 + 11i.

Product in trigonometric form

Complex numbers can be given in trigonometric form, for example x = |x| ⋅ (cos φ₁ + i ⋅ without φ₁) и y = |y| ⋅ (cos φ₂ + i ⋅ without φ₂).

In this case, the product formula looks like this:

x ⋅ y = |x| ⋅ |y| ⋅ [cos(Phi₁ + f₂) + i ⋅ without(Phi₁ + f₂)]

Example 2

Let’s do the multiplication of two complex numbers: x = 2 ⋅ (cos 15° + i ⋅ without 15 °) и y = 5 ⋅ (cos 30° + i ⋅ without 30 °).

Decision:

|x| ⋅ |y| = 2 ⋅ 5 = 10

φ₁ + f₂ = 15° + 30° = 45°

x ⋅ y = 10 ⋅ (cos 45° + i ⋅ without 45 °)