Multiplication of Complex Numbers

Multiplication of Complex Numbers

A complex number is the sum of an actual number and an imaginary number. This combination is in the form of a + bi. A complex number is represented by ‘z,’ where ‘a’ and ‘b’ are real numbers and ‘i’ is the imaginary number, known as ‘iota.’ The value of i = (√-1), the square-root of -1.

FOIL (Firsts, Outers, Inners, Lasts) is used for the multiplication of complex numbers.

Consider (a + bi) and (c + di) are complex numbers. Firsts mean a * c, outers mean a * di, inners mean bi * c, and lasts means bi * di.  

(a + bi) * (c + di) = ac + adi + bci + bdi2

Understanding multiplication of complex numbers

Let’s start from (ac + adi + bci + bdi2) which can be simplified to ac + (ad + bc)i + bdi2..

Take an example: (5 + 3i)(1 + 7i) = 5×1 + 5×7i + 3i×1+ 3i×7i

  = 5 + 35i + 3i + 21i2

  = 3 + 35i + 3i – 21 (because i2 = -1)

  = -18 + 38i

Another example: (1 + i) 2

(1 + i)(1 + i) = 1×1 + 1×i + 1×i + i2

  = 1 + 2i – 1 (because i2 = -1)

  = 0 + 2i

There can be three possible cases while multiplying complex numbers.  

• Multiplying a complex number by an actual number 

Say (b + ci) is multiplied by y:

 (b + ci)y = by + cyi

Let us try to see an example. If ‘y’ is 2, b is 3 and c is 4, then:  (3 + 4i)2 = 6 + 8i 

Example: 

Multiply -4 (13+5i). Write the resulting number in the form of a+bi.

If -4 needs to be distributed, let’s do it this way. -4(13 + 5i) = -4(13) + -4(5i)= -52 – 20i

So, here, we have used the distributive property to multiply a real number by a complex number. After all, this is not a complex multiplication if we follow the rules.

• Multiplying a complex number by an imaginary number

Say (b + c i) is a complex number, if we multiply (b + ci) by i: (b + c i)i = bi – c (Because i = √-1 and i2 = -1)    

Let’s try to multiply 5i(3-4i) and find the solution in the form of a+bi. 5i(3-4i) = 5i(3) – 5i(4i) = 15i – 20i2

But, the answer is not of the form a+bi since it contains i2.

We already know that i2 =-1. So let’s use this knowledge and see where it leads us.

 = 15i – 20i2

= 15i – 20(-1)

= 15i + 20

Here, we use the commutative property while solving the equation. And we also have the form of a+bi as 20 + 15i.

• It is multiplying a complex number by another complex number.

If we take a complex number (b + ci) represented by z and multiply it by i, we get:

zi = (b + ci)i = bi – y 

Take another example, when we take two complex numbers, z1 = a + ib and z2 = c + id, the product is: 

z1. z2 = (ca – bd) + i (ad + bc). 

Let’s understand this with the help of an example.

Say z1 = 4+5i

       z2= 7+2i

So, z1.z2 = (4+5i).(7+2i)

Here a=4, b=5, c=7, d=2

                                 Now as per above formula, Z1.Z2 = (ca – bd) +i (ad+bc)= (7×4 – 5×2) + i(4×2+5×7)

= 18+43i

The complex conjugate

The complex conjugate of a complex number is found by changing the sign of the imaginary part of the complex number. The real part of the number is left unchanged.

For example, if a+ib is a complex number, its complex conjugate will be a – ib.  

When a complex number is multiplied by its complex conjugate, it is a real number.

Let’s understand this by an example.

If z= 6+8i, then conjugate of z’ would be 6-8i

So, z.z’ = (6+8i).(6-8i)

Following the formula, we read above 

z.z’ =(ca-bd)+i(ad+bc)

=(6×6 -8×(-8))+i(6×(-8)+8×6) =36+64=100

As discussed above, multiplying complex numbers/multiplying a complex number by its conjugate will give a real number, as we saw in the above example. 100 is a real number.

Properties of multiplication of complex numbers

If z1, z2, and z3 are any three complex numbers, multiplying these complex numbers should have these rules: 

• The complex numbers adhere to the commutative law of multiplication.

   z1z2 = z2z1

• The complex numbers adhere to the associative law of multiplication.

  (z1z2) z3 = z1(z2z3) 

• z · 1 = z = 1· z, so 1 acts as the multiplicative identity for the set of complex numbers.
• Multiplication of complex numbers is distributive over the addition of complex numbers.

   z1(z2 + z3) = z1z2 + z1z3 

 and (z1 + z2)z3 = z1z3 + z2z3

CONCLUSION

If you are looking at complex numbers and wondering how they are used in our lives, they are used in spectrum analyzers, electricity, Mandelbrot Set, and quadratic equations. In addition, complex numbers are in scientific research, signal processing, electromagnetism, fluid dynamics, quantum mechanics, and vibration analysis. Multiplying complex numbers requires that these complex numbers follow the commutative law of multiplication ,the associative law of multiplication and distributive law. Multiplication of complex numbers is not a complex multiplication. We need to solve it with logical thinking and within the written rules of mathematics.