Adding and Subtracting Complex Numbers
Adding and Subtracting Complex Numbers are mathematical operations on complex numbers. Before getting into the detail of the addition and subtraction of complex numbers, let us recall the meaning of complex numbers. A complex number is the combination of a real number and an imaginary number. It is of the form a + ib and is usually represented by z. The real and imaginary parts of a complex number are added individually while adding complex numbers. Similarly, for the subtraction of complex numbers, we subtract the real and imaginary parts of the complex numbers separately.
In this article, we will explore the concept of adding and subtracting complex numbers along with their rules and steps with the help of examples. We will also learn the concept of addition and subtraction of complex numbers in polar form.
What is Adding and Subtracting Complex Numbers?
The addition and subtraction of complex numbers are fundamental operations that are applied to complex numbers. Just like when we add or subtract polynomials, we combine the like terms. Similarly, for adding and subtracting complex numbers, we combine the real parts and imaginary parts of the complex numbers and then apply the operation. Let us see the formula for adding and subtracting complex numbers z_{1} = a + ib and z_{2} = c + id, where a, b, c, d are real numbers:
Adding Complex Numbers
While performing the operation of addition of complex numbers, we combine the real parts and imaginary parts of the complex numbers and add them. The formula for adding complex numbers is given by,
z_{1} + z_{2 }= a + ib + c + id
= (a + c) + (ib + id)
= (a + c) + i(b + d)
Hence we have (a + ib) + (c + id) = (a + c) + i(b + d)
Subtracting Complex Numbers
For subtracting complex numbers, we consider the real and imaginary parts of the complex numbers separately and subtract the real and imaginary parts of one complex number from the real and imaginary parts, respectively, of the other complex number. The formula for subtraction of complex numbers is given by,
z_{1}  z_{2 }= (a + ib)  (c + id)
= a + ib  c  id
= (a  c) + (ib  id)
= (a  c) + i(b  d)
Hence we have (a + ib)  (c + id) = (a  c) + i(b  d)
Steps and Rules for Adding and Subtracting Complex Numbers
Now, we know the formulas for the addition and subtraction of complex numbers. Next, we will understand the process for the same stepwise. Given below are the steps for adding and subtracting complex numbers:
 Step 1: Segregate the real and imaginary parts of the complex numbers.
 Step 2: Add (subtract) the real parts of the complex numbers.
 Step 3: Add (subtract) the imaginary parts of the complex numbers.
 Step 4: Give the final answer in a + ib format.
Properties of Adding and Subtracting Complex Numbers
Given below is the list of properties of addition and subtraction of complex numbers:
 The Closure Property: The sum and difference of complex numbers is also a complex number. Hence, it holds the closure property.
 Commutative Property: The addition of complex numbers is commutative but the subtraction of complex numbers is not commutative.
 Associative Property: Adding complex numbers is associative but the subtraction of complex numbers is not associative.
 Additive Identity: 0 is the additive identity of the complex numbers, i.e., for a complex number z, we have z + 0 = 0 + z = z.
 Additive Inverse: For a complex number z, the additive inverse in complex numbers is z, i.e., z + (z) = 0
Important Notes on Adding and Subtracting Complex Numbers
 Adding and subtracting complex numbers is just like adding and subtracting two binomials. i.e., we just need to combine the like terms.
 All real numbers are complex numbers but all complex numbers need not be real numbers.
 Subtracting complex numbers does not hold commutative law.
 For adding and subtracting complex numbers in polar form, we first change the complex numbers to rectangular form and then perform the operation. Then, we convert the final answer into polar form.
Related Topics
Examples on Adding and Subtracting Complex Numbers

Example 1: Add the two complex numbers z = 3  4i and w = 5 + 7i.
Solution: For adding complex numbers z and w, we will use the formula (a + ib) + (c + id) = (a + c) + i(b + d). Here a = 3, b = 4, c = 5, d = 7
z + w = (3  4i) + (5 + 7i)
= (3  5) + i (4 + 7)
= 2 + i3
Answer: (3  4i) + (5 + 7i) = 2 + 3i

Example 2: Subtract the complex numbers 12 + 6i and 7 + 5i.
Solution: For subtracting complex numbers, we will use the formula (a + ib)  (c + id) = (a  c) + i(b  d). Here a = 12, b = 6, c = 7, d = 5
(12 + 6i)  (7 + 5i) = (12  7) + i(6  5)
= 19 + i
Answer: (12 + 6i)  (7 + 5i) = 19 + i
FAQs on Adding and Subtracting Complex Numbers
What is Adding and Subtracting Complex Numbers in Math?
Adding and subtracting complex numbers are fundamental operations that are applied to complex numbers. Just like when we add or subtract polynomials, we combine the like terms. Similarly, for adding and subtracting complex numbers, we combine the real parts and imaginary parts of the complex numbers and then apply the operation.
What is Adding Complex Numbers?
While performing the operation of addition of complex numbers, we combine the real parts and imaginary parts of the complex numbers and add them.
What is Subtracting Complex Numbers?
For subtracting complex numbers, we consider the real and imaginary parts of the complex numbers separately and subtract the real and imaginary parts of one complex number from the real and imaginary parts, respectively, of the other complex number.
How Do You Add and Subtract Complex Numbers?
The real and imaginary parts of a complex number are added individually while adding complex numbers. Similarly, for the subtraction of complex numbers, we subtract the real and imaginary parts of the complex numbers separately.
What are the Properties of Adding and Subtracting Complex Numbers?
Some of the important properties of adding and subtracting complex numbers are:
 The sum and difference of complex numbers is also a complex number.
 Adding complex numbers is associative but the subtraction of complex numbers is not associative.
 The addition of complex numbers is commutative but the subtraction of complex numbers is not commutative.
How to Find the Sum and Difference of Two Complex Numbers?
To find the sum of two complex numbers, we combine the real parts and the imaginary parts of the complex numbers and then add them and give the final answer in a + bi format. We follow the same process to find the difference of two complex numbers. The only difference is that we subtract the real parts and imaginary parts here instead of adding them.
What are the Formulas for Addition and Subtraction of Complex Numbers?
For subtracting complex numbers, we use the formula (a + ib)  (c + id) = (a  c) + i(b  d) and for adding complex numbers, we use the formula (a + ib) + (c + id) = (a + c) + i(b + d).
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