103 LearnersLast updated on July 4th, 2025
Addition and Subtraction of Complex Numbers
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.
What is Adding and Subtracting Complex Numbers?
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.
How to Add or Subtract Complex Numbers?
Before diving into addition and subtraction of complex numbers directly, take a look at the steps given below to have a better understanding:
- Before adding or subtracting complex numbers, we should group the real part and the imaginary part separately.
- All real numbers are complex numbers with an imaginary part of zero, but not all complex numbers are purely real. For example, 4 + i0 is a real number because the imaginary part is 0. It is both a real and a complex number.
- The formula for adding complex numbers is:
(a + ib) + (c + id) = (a + c) + i (b + d)
Given below is the formula for subtracting complex numbers:
(a + ib) - (c + id) = (a - c) + i (b - d)
Addition of Complex Numbers
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
Subtraction of Complex Numbers
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
What are the Properties of Adding or Subtracting Complex Numbers?
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.
Real-Life Applications of Addition and Subtraction of Complex Numbers
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:
- To determine the overall voltage and current in an alternating current circuit, electrical engineers add and subtract the voltage and current represented in complex numbers.
- To combine and adjust frequencies and amplitudes in sound processing, engineers can use the addition and subtraction of complex numbers. It aids in signal modification of radio waves, image processing, and adjusting frequencies.
- For wind direction, speed, or movement of the wind, the navigators, or sailors use the calculation of complex numbers. For example, they can set the real part as an east-west movement, and the imaginary part as a north-south movement.
- In engineering, vibrations and oscillations can be modeled using complex numbers, allowing for easier manipulation of phase and amplitude.
Common Mistakes and How to Avoid Them on Addition And Subtraction of Complex Numbers
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:
Mistake 1
Mistakenly Adding or Subtracting Real and Imaginary Parts Together
When adding or subtracting complex numbers, remember to add or subtract the real and imaginary parts separately. If students incorrectly combine all terms, the sum or difference will be incorrect. For example,
(4 + 4i) + (3 + 4i)
The real parts: 4 and 3 (4 + 3 = 7)
The imaginary parts: (4i + 4i = 8i)
(4 + 4i) + (3 + 4i) = 7 + 8i
Mistake 2
Forgetting to Include the Imaginary Unit i
Remember to write the imaginary unit i with the imaginary part of the complex number. Sometimes, students forget to include imaginary unit i in the final answer, leading to mistakes. For example, (8 + 6i) - (3 + 2i) = 5 + 4i. This is a correct complex number.
Mistake 3
Incorrectly Distributing Negative Signs
The distribution of negative signs can often confuse while subtracting two complex numbers. Because of this confusion, students can misplace the signs when writing the formula for subtracting complex numbers. The correct formula is z1 - z2 = (a + ib) - (c + id) = (a - c) + i (b - d).
Mistake 4
Forgetting the Value of i2
The correct value calculation of i2 involves complex numbers. The value of i2 is -1. Forgetting the value can lead to incorrect results. Sometimes, the negative sign of 1 may confuse the students.
Mistake 5
Ignoring the Formulas for Addition and Subtraction
Students must memorize the formulas before adding or subtracting complex numbers. The addition formula is:
(a + ib) + (c + id) = (a + c) + i (b + d)
The formula for subtraction is:
(a + ib) - (c + id) = (a - c) + i (b - d)
Solved Examples of Addition And Subtraction of Complex Numbers
Problem 1
Add (2 + 3i) and (1 + 4i)
3 + 7i
Explanation
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.
Problem 2
Subtract (6 + 5i) from (3 + 2i)
-3 - 3i
Explanation
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
Problem 3
Add (7 - 3i) and (-4 + 8i)
3 + 5i
Explanation
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
Problem 4
Subtract (-6 + 5i) and (3- 4i)
-9 + 9i
Explanation
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
Problem 5
Add the two complex numbers z = 4 - 8i and w = 5 - 6i
9 - 14i
Explanation
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
FAQs of Addition And Subtraction of Complex Numbers
1.Define a complex number.
2.What are the formulas for the addition and subtraction of complex numbers?
3.How to add two complex numbers?
4.List the properties of the addition and subtraction of complex numbers.
5.Can the subtraction of two complex numbers result in a real number?
6.How can children in India use numbers in everyday life to understand Addition and Subtraction of Complex Numbers?
7.What are some fun ways kids in India can practice Addition and Subtraction of Complex Numbers with numbers?
8.What role do numbers and Addition and Subtraction of Complex Numbers play in helping children in India develop problem-solving skills?
9.How can families in India create number-rich environments to improve Addition and Subtraction of Complex Numbers skills?
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Hiralee Lalitkumar Makwana
About the Author
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.
Fun Fact
: She loves to read number jokes and games.
