Equality of Complex Numbers Revision Notes for Leaving Cert Mathematics

Equality of Complex Numbers

What does it mean for complex numbers to be equal?

Two complex numbers are equal when their real parts match and their imaginary parts match exactly. This is the fundamental rule you need to remember when working with complex number equations.

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The Key Rule for Complex Number Equality

When you have two complex numbers in the form $a + bi$ and $c + di$ :

If $a + bi = c + di$ , then $a = c$ and $b = d$

This means:

The real parts must be equal: $a = c$
The imaginary parts must be equal: $b = d$

How to solve equality problems

Understanding the method is crucial for solving these problems effectively. The approach involves systematically comparing corresponding parts of the complex numbers.

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Step-by-Step Method:

Rearrange both sides to standard form ( $a + bi$ )
Separate the real and imaginary parts
Equate real parts with real parts
Equate imaginary parts with imaginary parts
Solve the resulting equations

Worked example 1

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Worked Example: Basic Equality Problem

Question: If $5 - i = x + (5 - 2y)i$ , find the values of $x$ and $y$ .

Solution: The equation is already in standard form on both sides.

Step 1: Identify and equate the real parts

Left side real part: $5$
Right side real part: $x$
Therefore: $5 = x$ , so $x = 5$

Step 2: Identify and equate the imaginary parts

Left side imaginary part: $-1$ (coefficient of $i$ )
Right side imaginary part: $(5 - 2y)$ (coefficient of $i$ )
Therefore: $-1 = 5 - 2y$

Step 3: Solve for $y$

$-1 = 5 - 2y$
$2y = 5 + 1 = 6$
$y = 3$

Answer: $x = 5$ and $y = 3$

Worked example 2

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Worked Example: System of Equations

Question: $a$ and $b$ are real numbers such that $a(2 + i) + 8 - bi = 5b - 3 - i$ . Find the values of $a$ and $b$ .

Solution: Step 1: Expand and rearrange the left side

$a(2 + i) + 8 - bi = 2a + ai + 8 - bi$
$= (2a + 8) + i(a - b)$

Step 2: Write both sides in standard form

Left side: $(2a + 8) + i(a - b)$
Right side: $(5b - 3) + i(-1)$

Step 3: Equate the real parts

$2a + 8 = 5b - 3$
$2a - 5b = -11$ ... (equation 1)

Step 4: Equate the imaginary parts

$a - b = -1$ ... (equation 2)

Step 5: Solve the system of equations From equation 2: $a = b - 1$

Substitute into equation 1:

$2(b - 1) - 5b = -11$
$2b - 2 - 5b = -11$
$-3b = -9$
$b = 3$

From equation 2: $a = 3 - 1 = 2$

Answer: $a = 2$ and $b = 3$

Worked example 3

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Worked Example: Direct Comparison

Question: Find the real numbers $p$ and $q$ if $(2p - 3) + (q + 1)i = 7 - 4i$ .

Solution: Both sides are already in standard form.

Step 1: Equate real parts

$2p - 3 = 7$
$2p = 10$
$p = 5$

Step 2: Equate imaginary parts

$q + 1 = -4$
$q = -5$

Answer: $p = 5$ and $q = -5$

Exam tips

These practical tips will help you avoid common mistakes and demonstrate clear mathematical reasoning in your solutions.

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Essential Exam Strategies:

Always rearrange both sides to standard form first
Double-check your identification of real and imaginary parts
Remember that the coefficient of i is the imaginary part
When there's no $i$ term visible, the imaginary part is $0$
Show all steps clearly in your working

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Key Points to Remember:

Two complex numbers are equal when both their real parts and imaginary parts match exactly
Always separate into real and imaginary parts before solving
The coefficient of i gives you the imaginary part
Set up two separate equations: one for real parts, one for imaginary parts
This method works for any complex number equality problem

Equality of Complex Numbers (Leaving Cert Mathematics): Revision Notes