Adding and Subtracting Complex Numbers Revision Notes for Leaving Cert Mathematics

Adding and Subtracting Complex Numbers

What are complex numbers?

A complex number is a special type of number that extends our regular number system. When we try to find the square root of negative numbers (like √-4), we can't get an answer using only real numbers. This is where complex numbers come into play.

Complex numbers consist of two parts:

A real part (ordinary numbers we're familiar with)
An imaginary part (involving the square root of negative numbers)

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Complex numbers were developed to solve mathematical problems that couldn't be solved with real numbers alone, particularly when dealing with square roots of negative numbers.

The imaginary unit

The foundation of complex numbers is the imaginary unit, represented by the letter i.

Definition: $i = \sqrt{-1}$

This means that $i^2 = -1$ .

Using this imaginary unit, we can express the square roots of negative numbers:

$\sqrt{-9} = \sqrt{9} \cdot \sqrt{-1} = 3i$
$\sqrt{-25} = \sqrt{25} \cdot \sqrt{-1} = 5i$
$\sqrt{-1} = i$

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Remember that $i^2 = -1$ is the fundamental relationship that makes all complex number operations possible. This is the key to understanding complex numbers.

Standard form of complex numbers

Every complex number can be written in the standard form:

$a + bi$

where:

a is the real part
b is the coefficient of the imaginary part
i is the imaginary unit

Examples of identifying parts

For the complex number $3 + 4i$ :

Real part = 3
Imaginary part = 4

For the complex number $-2 + 5i$ :

Real part = -2
Imaginary part = 5

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Important notes:

A real number like 5 can be written as $5 + 0i$
A pure imaginary number like $-6i$ can be written as $0 - 6i$

Adding complex numbers

To add two complex numbers, we add the real parts together and add the imaginary parts together separately.

Rule: $(a + bi) + (c + di) = (a + c) + (b + d)i$

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Worked Example 1: Adding Complex Numbers

Calculate: $(3 + 2i) + (4 - 3i)$

Step 1: Identify the real and imaginary parts

First number: real part = 3, imaginary part = 2
Second number: real part = 4, imaginary part = -3

Step 2: Add the real parts: $3 + 4 = 7$

Step 3: Add the imaginary parts: $2 + (-3) = -1$

Step 4: Write in standard form: $7 - i$

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Worked Example 2: Adding Complex Numbers

Calculate: $(-5 + 3i) + (-1 - 7i)$

Step 1: Add the real parts: $(-5) + (-1) = -6$

Step 2: Add the imaginary parts: $3 + (-7) = -4$

Step 3: Final answer: $-6 - 4i$

Subtracting complex numbers

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

Rule: $(a + bi) - (c + di) = (a - c) + (b - d)i$

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Worked Example 3: Subtracting Complex Numbers

Calculate: $(-6 + 3i) - (5 - 2i)$

Step 1: Rewrite as addition: $(-6 + 3i) + (-5 + 2i)$

Step 2: Subtract the real parts: $(-6) - 5 = -11$

Step 3: Subtract the imaginary parts: $3 - (-2) = 3 + 2 = 5$

Step 4: Final answer: $-11 + 5i$

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Worked Example 4: Finding Sum of Complex Numbers

If $z_1 = 1 + 2i$ and $z_2 = 5 - 2i$ , find $z_1 + z_2$ :

Step 1: $(1 + 2i) + (5 - 2i)$

Step 2: Add real parts: $1 + 5 = 6$

Step 3: Add imaginary parts: $2 + (-2) = 0$

Step 4: Final answer: $6 + 0i = 6$

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Worked Example 5: Complex Number Operations

If $z_1 = 1 + 2i$ and $z_2 = 5 - 2i$ , find $2z_1 - z_2$ :

Step 1: First find $2z_1 = 2(1 + 2i) = 2 + 4i$

Step 2: Now calculate: $(2 + 4i) - (5 - 2i)$

Step 3: Subtract real parts: $2 - 5 = -3$

Step 4: Subtract imaginary parts: $4 - (-2) = 6$

Step 5: Final answer: $-3 + 6i$

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Key Strategies for Success:

Always separate real and imaginary parts - treat them as separate calculations
Be careful with signs - particularly when subtracting negative numbers
Write your final answer in standard form $a + bi$
Check your work by ensuring you've combined like terms correctly

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Common Exam Tips:

When the question asks for the answer "in the form $a + bi$ ", make sure your final answer is clearly written this way
Remember that if the imaginary part equals zero, you just have a real number
If the real part equals zero, you have a pure imaginary number
Always show your working clearly, separating the real and imaginary calculations

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Remember - Key Points:

Complex numbers have the form $a + bi$ where $i = \sqrt{-1}$
To add complex numbers: add real parts together and imaginary parts together
To subtract complex numbers: subtract real parts and subtract imaginary parts
Always write your final answer in the standard form $a + bi$
Treat real and imaginary parts as completely separate calculations

Adding and Subtracting Complex Numbers (Leaving Cert Mathematics): Revision Notes