Subtraction of Two Complex Numbers
2026-02-28 00:59 Diff
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Last updated on September 1, 2025

The mathematical operation of finding the difference between two complex numbers is known as the subtraction of complex numbers. It helps simplify expressions involving real and imaginary parts and solve problems in various applications of complex numbers.

What is Subtraction of Two Complex Numbers?

Subtracting complex numbers involves finding the difference between their corresponding real and imaginary parts.

A complex number is expressed in the form a + bi, where a is the real part and b is the imaginary part.

To subtract two complex numbers, subtract the real parts and the imaginary parts separately.

How to Subtract Two Complex Numbers?

When subtracting two complex numbers, follow these steps:

Separate real and imaginary parts: Write each complex number in the form a + bi.

Subtract real parts: Subtract the real part of the second complex number from the real part of the first.

Subtract imaginary parts: Subtract the imaginary part of the second complex number from the imaginary part of the first.

Combine results: Combine the results to form a new complex number.

Methods to Subtract Two Complex Numbers

The following are methods to subtract complex numbers:

Method 1: Horizontal Method

To apply the horizontal method for subtracting complex numbers, use these steps:

Step 1: Write both complex numbers in a horizontal line with a minus sign between them.

Step 2: Subtract the real parts and the imaginary parts separately.

Step 3: Combine the results to form the new complex number.

Example: Subtract (3 + 2i) from (5 + 4i).

Step 1: (5 + 4i) - (3 + 2i)

Step 2: (5 - 3) + (4i - 2i)

Step 3: 2 + 2i

Method 2: Column Method

Using the column method, write the complex numbers one below the other, aligning like parts vertically. Then subtract the corresponding parts.

Example: Subtract (2 + 3i) from (6 + 5i).

Solution: Align the parts vertically: 6 + 5i ← Minuend - 2 + 3i ← Subtrahend --------- 4 + 2i Therefore, the result is 4 + 2i.

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Properties of Subtraction of Complex Numbers

Subtraction of complex numbers has the following properties:

Subtraction is not commutative Changing the order of the numbers changes the result, i.e., (a + bi) - (c + di) ≠ (c + di) - (a + bi)

Subtraction is not associative When more than two complex numbers are involved, changing the grouping changes the result. ((a + bi) − (c + di)) − (e + fi) ≠ (a + bi) − ((c + di) − (e + fi))

Subtraction is the addition of the opposite sign Subtracting a complex number is the same as adding its opposite. (a + bi) − (c + di) = (a + bi) + (−c − di)

Subtracting zero from a complex number leaves the number unchanged

Subtracting zero from any complex number results in the same complex number: (a + bi) - 0 = (a + bi)

Tips and Tricks for Subtraction of Complex Numbers

Here are some tips for dealing with the subtraction of complex numbers efficiently:

Tip 1: Keep track of both real and imaginary parts separately.

Tip 2: Write complex numbers in the standard form to simplify subtraction.

Tip 3: Use parentheses to avoid confusion with signs, especially with negative complex numbers.

Forgetting to separate parts

Students often forget to handle real and imaginary parts separately. Always ensure to subtract these parts individually before combining them.

Problem 1

Use the horizontal method, (4 + 5i) - (1 + 3i) = (4 - 1) + (5i - 3i) = 3 + 2i

Okay, lets begin

Subtract (6 - 2i) from (8 + 3i)

Well explained 👍

Problem 2

Use the horizontal method, (8 + 3i) - (6 - 2i) = (8 - 6) + (3i + 2i) = 2 + 5i

Okay, lets begin

Subtract (-3 + 4i) from (2 - i)

Well explained 👍

Problem 3

(2 - i) - (-3 + 4i) = (2 + 3) + (-i - 4i) = 5 - 5i

Okay, lets begin

Subtract (5i) from (7 + 3i)

Well explained 👍

Problem 4

(7 + 3i) - (5i) = 7 + (3i - 5i) = 7 - 2i

Okay, lets begin

Subtract (3 - i) from (0 + i)

Well explained 👍

No, real and imaginary parts should be handled separately; subtract real parts from real parts and imaginary from imaginary.

1.Is subtraction commutative for complex numbers?

No, changing the order of subtraction affects the result; it is not commutative.

2.What are complex numbers?

Complex numbers are numbers in the form a + bi, where a is the real part and b is the imaginary part.

3.What is the first step of subtracting complex numbers?

The first step is to identify and separate the real and imaginary parts of the complex numbers.

4.What methods are used for subtracting complex numbers?

The horizontal method and the column method are commonly used for subtracting complex numbers.

Common Mistakes and How to Avoid Them in Subtraction of Complex Numbers

Subtraction in complex numbers can lead to common mistakes, but being aware of these can help avoid them.

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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.