Multiplying Complex Numbers
Multiplying complex numbers is a fundamental operation on complex numbers where two or more complex numbers are multiplied. It is a complex operation as compared to the addition and subtraction of complex numbers. A complex number is of the form a + ib, where i is an imaginary number and a, b are real numbers. The working mechanism of the multiplication of complex numbers is similar to the multiplication of binomials using the distributive property.
Let us understand the concept of multiplying complex numbers using the distributive property, its formula, multiplication of a real number, and purely imaginary number with complex numbers. We will also explore squaring complex numbers along with some solved examples for a better understanding.
What is Multiplication Of Complex Numbers?
Multiplication of complex numbers is a process of the multiplication of two or more complex numbers using the distributive property. Mathematically, if we have two complex numbers z = a + ib and w = c + id, then multiplication of complex numbers z and w is written as zw = (a + ib) (c + id). We use the distributive property of multiplication to find the product of complex numbers.
Multiplication Of Complex Numbers Formula
Multiplying complex numbers is similar to multiplying polynomials. We use the following polynomial identity to solve the multiplication of complex numbers: (a+b)(c+d) = ac + ad + bc + bd. The formula for multiplying complex numbers is given as:
(a + ib) (c + id) = ac + iad + ibc + i^{2}bd
⇒ (a + ib) (c + id) = (ac  bd) + i(ad + bc) [Because i^{2 }= 1]
Multiplication Of Complex Numbers in Polar Form
A complex number in polar form is written as z = r (cos θ + i sin θ), where r is the modulus of the complex number and θ is its argument. Now, the formula for multiplying complex numbers z_{1} = r_{1} (cos θ_{1} + i sin θ_{1}) and z_{2} = r_{2} (cos θ_{2} + i sin θ_{2}) in polar form is given as:
z_{1}z_{2} = [r_{1} (cos θ_{1} + i sin θ_{1})] [r_{2} (cos θ_{2} + i sin θ_{2})]
= r_{1 }r_{2} (cos θ_{1 }cos θ_{2} + i cos θ_{1 }sin θ_{2} + i sin θ_{1 }cos θ_{2} + i^{2} sin θ_{1 }sin θ_{2})
= r_{1 }r_{2} (cos θ_{1 }cos θ_{2} + i cos θ_{1 }sin θ_{2} + i sin θ_{1 }cos θ_{2}  sin θ_{1 }sin θ_{2}) {Because i^{2} = 1}
= r_{1 }r_{2} [cos θ_{1 }cos θ_{2}  sin θ_{1 }sin θ_{2 }+ i (cos θ_{1 }sin θ_{2} + sin θ_{1 }cos θ_{2} )]
= r_{1 }r_{2} [cos (θ_{1} + θ_{2}) + i sin (θ_{1} + θ_{2})] {Because cos a cos b  sin a sin b = cos (a + b) and sin a cos b + sin b cos a = sin (a + b)}
Hence the formula for multiplying complex numbers in polar form is [r_{1} (cos θ_{1} + i sin θ_{1})] [r_{2} (cos θ_{2} + i sin θ_{2})] = r_{1 }r_{2} [cos (θ_{1} + θ_{2}) + i sin (θ_{1} + θ_{2})]
Multiplication Of Complex Numbers with Purely Real and Imaginary Numbers
We know that the formula for multiplying complex numbers is (a + ib) (c + id) = (ac  bd) + i(ad + bc). If we have b = 0, then the two complex numbers are 'a' and 'c + id'. The formula to multiply a complex number with a real number becomes a (c + id) = ac + iad. For example, we multiply 2 with 1 + 3i as:
2 × (1 + 3i) = 2 + 6i
Now, if we multiply a purely imaginary number of the form bi with a complex number, then the formula becomes (bi) (c + id) = ibc  bd. For example, if we multiply a complex number 2 + 3i with 5i, we have:
(5i) (2 + 3i) = 10i 15i^{2}
= 10i + 15
Squaring Complex Numbers
As we know that the formula for the multiplication of complex numbers is (a + ib) (c + id) = (ac  bd) + i(ad + bc). If we have a + ib = c + id, then we have a = c and b = d, i.e., we multiply the same complex number with itself. So, the formula for multiplying a complex number with itself becomes:
(a + ib) (a + ib) = (a.a  b.b) + i(ab + ba)
= (a^{2}  b^{2}) + i 2ab
For example, square the complex number 3  7i. We have (3  7i)^{2} = (3^{2}  (7)^{2}) + i 2 × 3 × (7) = (9  49)  42i = 40  42i
Multiplicative Inverse Of Complex Numbers
The multiplicative inverse of a complex number on multiplying with the given complex number results in the multiplicative identity of 1. The multiplicative inverse of the complex number z = a + ib is z^{1} = \(\dfrac{\bar z}{z^2}\). Here \(\bar z \) = a  ib , and z = \(\sqrt {a^2 + b^2}\).
z.z^{1} = 1
z^{1} = z = \(\dfrac{\bar z}{z^2}\)
For finding the multiplicative inverse of a complex number we need the conjugate of the complex number and the modulus of the complex number. The conjugate of the complex number is \(\bar z\) = a  ib, and the modulus of the complex number is z = \(\sqrt {a^2 + b^2}\).
Let us take a simple example of finding the multiplicative inverse of a complex number z = 3 + 4i. For this complex number the conjugate complex number is \(\bar z \) = 3  4i, and the modulus of the complex number is z = \(\sqrt {3^2 + 4^2}\) = 5. And the multiplicative inverse is z^{1} = \(\dfrac{3  4i}{5}\) = 3/5  4i/5.
Important Notes on Multiplying Complex Numbers
 Multiplication of complex numbers in cartesian form: (a + ib) (c + id) = (ac  bd) + i(ad + bc)
 Multiplication of complex numbers in polar form: [r_{1} (cos θ_{1} + i sin θ_{1})] [r_{2} (cos θ_{2} + i sin θ_{2})] = r_{1 }r_{2} [cos (θ_{1} + θ_{2}) + i sin (θ_{1} + θ_{2})]
 Squaring Complex Number: (a + ib)^{2} = (a^{2}  b^{2}) + i 2ab
Topics Related to Multiplication of Complex Numbers
Multiplication Of Complex Numbers Examples

Example 1: Multiply complex numbers z = 3 − 2i and w = − 4 + 3i.
Solution: For multiplying complex numbers z and w, we will use the formula (a + ib) (c + id) = (ac  bd) + i(ad + bc). Here a = 3, b = 2, c = 4, d = 3
(3 − 2i) (− 4 + 3i) = [3 × (4)  (2) × 3) + i(3 × 3 + (2) × (4))
= (12 + 6) + i (9 + 8)
= 6 + 17i
Answer: (3 − 2i) (− 4 + 3i) = 6 + 17i

Example 2: Find the square of the complex number (4 + 6i).
Solution: To find the square of a complex number, we will use the formula (a + ib)^{2} = (a^{2}  b^{2}) + i 2ab. Here, a = 4 and b = 6
(4 + 6i)^{2} = ((4)^{2}  6^{2}) + i 2 × (4) × 6
= (16  36)  48i
= 20  48i
Answer: (4 + 6i)^{2} = 20  48i

Example 3: Multiply complex numbers z = 2 (cos 15° + i sin 15°) and w = 5 (cos 15° + i sin 15°).
Solution: For multiplying complex numbers in polar form, we will use the formula [r_{1} (cos θ_{1} + i sin θ_{1})] [r_{2} (cos θ_{2} + i sin θ_{2})] = r_{1 }r_{2} [cos (θ_{1} + θ_{2}) + i sin (θ_{1} + θ_{2})].
Here, r_{1} = 2 and r_{2} = 5, θ_{1} = 15°, θ_{2} = 15°
zw = [2 (cos 15° + i sin 15°)] × [5 (cos 15° + i sin 15°)]
= 10 [cos (15° + 15°) + i sin (15° + 15°)]
= 10 (cos 30° + i sin 30°)
= 10 (√3/2 + i (1/2))
= (10/2) (√3 + i)
= 5√3 + 5i
Answer: [2 (cos 15° + i sin 15°)] × [5 (cos 15° + i sin 15°)] = 5√3 + 5i
FAQs on Multiplying Complex Numbers
What is Multiplication of Complex Numbers in Algebra?
Multiplication of complex numbers is a process of the multiplication of two or more complex numbers using the distributive property. Mathematically, if we have two complex numbers z = a + ib and w = c + id, then multiplication of complex numbers z and w is written as zw = (a + ib) (c + id).
What is Multiplication Of Complex Numbers Formula?
The formula for multiplying complex numbers is: (a + ib) (c + id) = (ac  bd) + i(ad + bc).
What are the Steps for Multiplication Of Complex Numbers?
The steps for multiplying complex numbers are:
 Step 1: Apply the distributive property and multiply each term of the first complex number with each term of the second complex number.
 Step 2: Simplify i^{2} = 1
 Step 3: Combine real parts and imaginary parts and simplify them to get the product.
How Do You Multiplication Of Complex Numbers in Polar Form?
The formula for multiplying complex numbers in polar form is z_{1} = r_{1} (cos θ_{1} + i sin θ_{1}) and z_{2} = r_{2} (cos θ_{2} + i sin θ_{2}) in polar form is given as [r_{1} (cos θ_{1} + i sin θ_{1})] [r_{2} (cos θ_{2} + i sin θ_{2})] = r_{1 }r_{2} [cos (θ_{1} + θ_{2}) + i sin (θ_{1} + θ_{2})].
What Happens When You Multiply Two Imaginary Numbers?
When two purely imaginary numbers are multiplied, then the product is a real number. If we have two purely imaginary complex numbers ai and bi, then their product is (ai) (bi) = i^{2} ab = ab.
What Is The Multiplicative Inverse Of Complex Number?
The multiplicative inverse of a complex number z = a + ib is z^{1} = \(\dfrac{\bar z}{z^2}\). Here the complex conjugate \(\bar z \) = a  ib, and is the conjugate of the given complex number , and z = \(\sqrt {a^2 + b^2}\) is the conjugate of the complex number.
How Do You Do Multiplication Of Complex Numbers?
Two complex numbers are multiplied in following manner:
(a + ib) (c + id) = ac + iad + ibc + i^{2}bd
⇒ (a + ib) (c + id) = (ac  bd) + i(ad + bc) [Because i^{2 }= 1]
Now, we just substitute the values of a, b, c, d in the above formula.
How Is Multiplication Of Complex Numbers Related to Multiplying Two Binomials?
The working mechanism of the multiplication of complex numbers is similar to the multiplication of binomialshe distributive property. Just like we multiply two binomials (a + bx) (c + dx) = ac + (ad + bc) x + bd x^{2}. In case of complex numbers, x takes the value of i.
What is the Multiplication of Complex Numbers with a Real Number?
Multiplying complex numbers with real numbers is evaluated as a(c + id) = ac + i ad which is a complex number.
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