Adding and Subtracting Complex Numbers

Adding and Subtracting Complex Numbers: A complex number comprises a real number and an imaginary number. It is usually represented in the form of z = a + ib, where a is the real part and b is the imaginary part. Here, i represents an imaginary unit number, whose value is equal to √-1. Thus, i = √-1.

This article provides the steps to add or subtract complex numbers, with solved examples and properties of addition or subtraction of complex numbers.

Addition of Complex Numbers

Let us consider two complex numbers z₁ = a + ib and z₂ = c + id. For adding the complex numbers, we simply combine the real and imaginary parts of the two complex numbers and then apply the addition operation. The formula for adding the complex numbers is given by:

z₁ + z₂ = (a + ib) + (c + id) = (a + c) + i (b + d)

If z = z₁ + z₂, then  

z = (a + c) + i (b + d)

Subtraction of Complex Numbers

Two complex numbers z₁ = a + ib and z₂ = c + id can be subtracted by combining the real and imaginary parts of both the complex numbers and applying the subtraction operation separately on each of them. The formula for subtracting the complex numbers is given by:

z₁ – z₂ = (a + ib) – (c + id) = (a – c) + i (b – d)

If z = z₁ – z₂, then

z = (a – c) + i (b – d)

How to Add or Subtract Complex Numbers?

• Adding or subtracting complex numbers is simply combining the real and imaginary parts of the complex numbers and applying the operations separately on each of the combinations.
• All real numbers are complex numbers with imaginary parts equal to zero, but all complex numbers are not real numbers
• If the complex numbers are in polar form, we first convert them into cartesian form and apply the operations.

Properties of Adding or Subtracting Complex Numbers

• Closure property: The complex numbers formed after adding or subtracting complex numbers are also complex numbers.
• Associative property: This property holds for the addition of complex numbers only. That is for any three complex numbers z₁, z₂, and z₃, we have

(z₁ + z₂) + z₃ = z₁ + (z₂ + z₃)

• Commutative property: This property holds for the addition of two complex numbers. For any two complex numbers z₁ and z₂, we have

z₁ + z₂ = z₂ + z₁ 

• Additive property: Here, 0 is the additive identity of complex numbers, since

z + 0 = 0 + z

• Additive inverse: For a complex number z, the additive inverse is -z, since z + (-z) = 0

People Also Read:

• Complex Numbers – Definition, Types, Formulas
• What are Complex Numbers?

Adding and Subtracting Complex Numbers Examples

Question 1. Find the sum of the two complex numbers z = 3 + 5i and w = 6 – 2i.

Solution:

Since the given complex numbers have real and imaginary parts, we can combine them to find the net sum of both the complex numbers.

z + w = (3 + 5i) + (6 – 2i) = (3 + 6) + i (5 – 2)

z + w = 9 + 3i

Thus, the sum of the complex numbers is equal to 9 + 3i.

Question 2. Subtract the complex numbers z = 2 – 3i and w = -4 + 2i.

Solution:

Since, we can combine the real and imaginary terms of the complex numbers and apply our operations, we can write

z – w = (2 – 3i) – (-4 + 2i) = (2 -(-4)) + i (-3 -2) 

z – w = 6 – 5i

Thus, the result is 6 – 5i.

Question 3. Given the complex numbers z₁ = 3 + 2i, z₂ = 5 – 3i and z₃ = 1 + 2i, find the value of z₁ + z₂ – z₃.

Solution:

Given the three complex numbers z₁ = 3 + 2i, z₂ = 5 – 3i and z₃ = 1 + 2i, we can apply associative property of complex numbers to find the result.

Thus, we can write,

z₁ + z₂ – z₃ = (z₁ + z₂) – z₃ = ((3 + 2i) + (5 – 3i)) – (1 + 2i)

z₁ + z₂ – z₃ = (8 – i) – (1 + 2i) = (8 – 1) + i(-1 – 2)

z₁ + z₂ – z₃ = 7 – 3i

So, the answer is 7 – 3i.

Question 4. Given the two complex numbers z and v, where z = 6 + 9i. If the sum of the two complex numbers is twice the value when v is subtracted from z, find the value of v.

Solution:

Given, the complex number z = 5 + 2i.

According to the question,

z + v = 2 (z – v)

z + v = 2z – 2v

3v = z

v = z/3

Putting the value of z = 6 + 9i, we get

v = (6 + 9i)/3 = 6/3 + i (9/3) = 2 + 3i

v = 2 + 3i

Adding and Subtracting Complex Numbers Worksheet

1. Add the following complex numbers: (3 + 4i) + (2 – 5i)

2. Subtract the following complex numbers: (5 – 2i) – (3 + 7i)

3. Find the sum of the complex numbers: (-2 + 6i) + (4 – 3i) + (1 + 2i)

4. Subtract the complex numbers: (2 – i) – (3 + 4i) – (5 – 2i)

5. If z₁ = 2 + 3i and z₂ = 4 – 5i find z₁ + z₂ and z₁ – z₂

FAQs on Adding and Subtracting Complex Numbers

How do you add or subtract complex numbers?

To add or subtract complex numbers, you simply combine or separate the real and imaginary parts. Similarly, to subtract them, you subtract the real parts and the imaginary parts separately.

How to add complex number fractions?

To add complex number fractions, you first find a common denominator, just like with real number fractions. Then, you add the fractions by adding their numerators and keeping the common denominator the same.

What is the standard form of adding complex numbers?

The standard form of adding complex numbers is to simply add their real parts and their imaginary parts separately. So, if you have two complex numbers a + bi and c + di, their sum is (a + c) + (b + d)i

What is the difference between z1 and z2 given that z₁ =- 6 + 8i and z₂ = 4 + 5i?

z₁ – z₂ ​= (- 6 + 8i) – (4 + 5i) = (-6 – 4) + (8i – 5i) = – 10 + 3i