Multiplication of Two Complex Numbers

Multiplying complex numbers is a basic operation on complex numbers that involves multiplying two or more complex integers. When compared to the addition and subtraction of complex numbers, this is a more difficult operation. The formula for a complex number is a + ib, where i is an imaginary number and a and b are real values. The multiplication of complex numbers works in a similar way as multiplication of binomials utilising the distributive property.

Let’s look at how to multiply complex numbers using the distributive property, its formula, and how to multiply a real and purely imaginary number with complex numbers. For a better understanding, we’ll look at squaring complex numbers and look at some solved examples.

What is Complex Number Multiplication?

Complex number multiplication is the process of multiplying 2 or more complex numbers using distributive property. Multiplication of complex numbers z and w is expressed as zw = (a + ib) (c + id) in mathematics if we have two complex numbers z = a + ib and w = c + id. To find the product of complex numbers, we apply the distributive property of multiplication.

Formula For the Multiplication of Complex Numbers

Multiplying polynomials is analogous to multiplying complex numbers. To solve the multiplication of complex numbers, we apply the polynomial identity: ac + ad + bc + bd = (a+b) (c+d). (a + ib) (c + id) = ac + iad + ibc + i²bd is the formula for multiplying complex numbers. (ac – bd) + i(ad + bc) = (a + ib) (c + id) [Due to i² = -1]

Complex Numbers in Polar Form Multiplication

In polar form, a complex number is expressed as z = r (cosθ + i sinθ), where r is the complex number’s modulus and is its argument. The formula for the multiplying complex numbers in polar form z₁ = r₁

(cos ϴ₁ + i sinϴ₁) and z₂ = r₂ (cos ϴ₂ + i sin ϴ₂) is now:

z₁z₂= r₁ r₂ (cosϴ₁ cosϴ₂ + i cosϴ₁ sinϴ₂ + i sinϴ₁ cosϴ₂ + i² sinϴ₁ sinϴ₂) 

= r₁ r₂ (cosϴ₁ cosϴ₂ + i cosϴ₁ sinϴ₂ + i sinϴ₁ cosϴ₂ – sinϴ₁ sinϴ₂) ( because i² = -1)

= [cosϴ₁ cosϴ₂ – sinϴ₁ sinϴ₂ + i (cosϴ₁ sinϴ₂ + sinϴ₁ cosϴ₂)] 

= r₁ r₂ [cosϴ₁ cosϴ₂ – sinϴ₁ sinϴ₂ + i (cosϴ₁ sinϴ₂ + sinϴ₁ cosϴ₂)]

[cos (ϴ₁ + ϴ₂) + i sin (ϴ₁ + ϴ₂)] = r₁ r₂         

 Because cos a cos b – sin a sin b = cos (a + b) and sin a cos b + sin b cos a = sin (a + b),

As a result, [r₁  (cos ϴ₁  + i sin ϴ₁ )] is the formula for multiplying complex numbers in polar form.

 [r₂ isin ϴ₂ + cos ϴ₂)] [cos (ϴ₁ + ϴ₂) + isin (ϴ1 + ϴ₂)] = r₁ r₂

Complex Numbers Multiplication with Purely Real & Imaginary Numbers

We know that (a + ib) (c + id) = (ac – bd) + i(ad + bc) is the formula for multiplying complex numbers. If b = 0 is true, the two complex numbers are ‘a’ and ‘c + id’. To multiplicate a complex number by a real number, use the formula a (c + id) = ac + iad. 2 (1 + 3i) = 2 + 6i is an example of a multiplication with 1 + 3i.

When a purely imaginary number of the type bi is multiplied by a complex number, the result is (bi) (c + id) = ibc – bd. For example, multiplying a complex integer 2 + 3i by -5i yields:

(-5i) (2 + 3i) = -10i -15i2 = -10i + 15

Complex Numbers Squaring

The formula for multiplying complex numbers is (a + ib) (c + id) = (ac – bd) + i(ad + bc), as we know. If a + ib = c + id, then a = c and b = d, implying that the very same complex number is multiplied with itself. So, multiplying a complex number by itself is (a + ib) (a + ib) = (a.a – b.b) + i(ab + ba) = (a2 – b2) + i 2ab

Square the complex number 3 – 7i, for example. (3² – (-7)²) + i  2 ✕ 3 (-7)  = -40 – 42i

Conclusion

Complex number multiplication in cartesian form: (a + ib) (ac – bd) + i (ad + bc) = (c + id)

Complex number multiplication in polar form: [r₁ isin ϴ₁ + cos ϴ₁)] [r₂ isin ϴ₂ + cos ϴ₂)] [cos (ϴ₁ + ϴ₂) + isin (ϴ₁ + ϴ2)] = r₁ r₂

Complex Number Squaring: (a² – b²) + i2ab = (a + ib)².