Last updated on July 4th, 2025
Both addition and subtraction are considered as two basic operations in math. Here, we’ll see how the same operations can be used for complex numbers. A complex number is expressed in the form of a+ib, where a,b are real numbers. Let’s learn more about them and how to add or subtract them in this article.
In mathematics, a complex number is denoted as z. Since it is a combination of real and imaginary numbers, a complex number is represented as: z = a + ib. Here, the equation indicates an imaginary unit, while a and b are the real and imaginary parts, respectively. The value of i is (√-1).
While adding or subtracting complex numbers, the real and imaginary parts are combined. When the terms are combined, we can perform either addition or subtraction.
Before diving into addition and subtraction of complex numbers directly, take a look at the steps given below to have a better understanding:
We perform the addition of complex numbers by adding the real part and imaginary part of two numbers separately. For example, if z1 = a + ib and z2 =c +id are two complex numbers. The addition of these two numbers can be performed as:
z1 + z2 = (a + ib) + (c + id) = (a + c) + i (b + d)
Let us consider, z1 = (3 + 4i)
z2 = (2 + 5i)
Add (3 + 4i) + (2 + 5i)
When we add these two complex numbers, first we must add the real parts:
3 + 2 = 5
Next, we can add the imaginary parts:
4 + 5 = 9
Thus, z1 + z2 = (3 + 4i) + (2 + 5i) = 5 +9i
When we subtract complex numbers, we subtract the real parts and the imaginary parts individually. The formula for the subtraction of complex numbers is:
z1 - z2 = (a + ib) - (c + id) = (a - c) + i (b - d)
For example, subtract (6 + 5i) - (4 + 3i)
First, we must subtract the real parts:
6 - 4 = 2
Then, subtract the imaginary parts:
5 - 3 = 2
Thus, (6 + 5i) - (4 + 3i) = 2 + 2i
The key properties of the addition and subtraction of complex numbers are listed below:
Closure property: The closure property states that the result we obtain from the addition and subtraction of complex numbers is also a complex number. For example, if we subtract (7 + 5i) - (4 + 3i),
We get 3 + 2i, which is also a complex number.
Associative property: This property is subject only to the addition of three complex numbers. The subtraction of complex numbers does not show associative property, but the addition exhibits the property. If we add three complex numbers, then:
(z1+ z2) + z3 = z1+ (z2 + z3)
Commutative property: Commutative property states that only the addition of two or more complex numbers is commutative. For e.g., if z1 and z2 are complex numbers, then: z1 + z2 = z2 + z1
Additive identity: Additive identity is a number that, when added to a complex number, does not change its value. For any complex number z = a + bi, the additive identity is z + 0 = (a + bi) + 0 = a + bi
Additive inverse: -z is the additive inverse of a complex number z. Therefore, z + (-z) = 0.
Understanding the properties and concepts of complex numbers will help in various fields such as engineering, physics, mathematics, and computer science. Some practical, real-world applications of complex numbers are listed below:
While doing addition and subtraction of complex numbers, students often make mistakes in calculations. They often forget to consider the real part of the complex numbers. Here are some common errors and solutions to avoid them:
Add (2 + 3i) and (1 + 4i)
3 + 7i
To add two complex numbers, we can use the formula:
(a + ib) + (c + id) = (a + c) + i (b + d)
Where, a = 2
b = 3
c = 1
d = 4
Now we can substitute the values and identify the real and imaginary parts:
(2 + 3i) + (1 + 4i) = (2 + 1) + i (3 + 4)
= 3 + 7i
Thus, the sum of (2 + i3) and (1 + i4) is 3 + 7i.
Subtract (6 + 5i) from (3 + 2i)
-3 - 3i
To subtract two complex numbers, we can use the formula:
(a + ib) - (c + id) = (3 + 2i) - (6 + 5i)
Now we can substitute the values and identify the real and imaginary parts:
(a - c) + i (b - d) = (3 - 6) + i(2 -5)
Next, we can subtract the real parts:
(3 - 6) = -3
Subtract the imaginary parts:
(2 - 5) = -3
Thus, the difference when (6 + 5i) is subtracted from (3 + 2i) is -3 - 3i
Add (7 - 3i) and (-4 + 8i)
3 + 5i
To add two complex numbers, we can use the formula:
(a + ib) + (c + id) = (a + c) + i (b + d)
Where, a = 7
b = -3
c = -4
d = 8
Now we can substitute the values and identify the real and imaginary parts:
(a + c) + i (b + d) = (7 + (-4)) + i (-3 + 8)
Add the real parts:
7 + (-4) = 3
Add the imaginary parts:
-3 + 8 = 5
Thus, (7 - 3i) + (-4 + 8i) = 3 + 5i
Subtract (-6 + 5i) and (3- 4i)
-9 + 9i
To subtract two complex numbers, we can use the formula:
(a + ib) - (c + id) = (a - c) + i (b - d)
Where, a = - 6
b = 5
c = 3
d = -4
Now we can substitute the values:
(-6 + 5i) - (3 + (-4) i) = (-6 - 3) + i (5 - (-4)
Simplify the real parts:
-6 - 3= -9
Simplify the imaginary parts:
5 - (-4) = 5 + 4 = 9
Thus, the final result is -9 + 9i
(-6 + 5i) - (3- 4i) = -9 + 9i
Add the two complex numbers z = 4 - 8i and w = 5 - 6i
9 - 14i
To add two complex numbers, we can use the formula:
(a + ib) + (c + id) = (a + c) + i (b + d)
Where, a = 4
b = -8
c = 5
d = -6
Now, we can substitute the values:
(4 - 8i) + (5 - 6i) = (4 + 5) + i (-8 + (-6))
= 9 + i (-14)
= 9 - 14 i
Thus, (4 - 8i) + (5 - 6i) = 9 - 14i
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
: She loves to read number jokes and games.