Complex Numbers and Transformations Revision Notes for Leaving Cert Mathematics

Complex Numbers and Transformations

Complex numbers can be transformed in various ways using mathematical operations. These transformations create predictable geometric changes when we represent complex numbers on the Argand diagram. Understanding these transformations helps us visualise how operations affect complex numbers graphically.

Multiplying complex numbers by real numbers (scaling)

When you multiply a complex number by a real number, you create a scaling transformation. This means the complex number gets larger or smaller while maintaining its direction from the origin.

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Key rule: If $z = a + bi$ and you multiply by real number $k$ , then:

$kz = k(a + bi) = ka + kbi$
Both the real part and imaginary part are multiplied by the same factor $k$

How scaling affects the modulus

The modulus (distance from origin) changes in a predictable way:

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If $|z|$ is the modulus of $z$ , then $|kz| = k|z|$
When $k = 2$ , the modulus doubles
When $k = 3$ , the modulus triples
When $k = \frac{1}{2}$ , the modulus halves

The diagram above shows how multiplying $z_1$ and $z_2$ by different real numbers creates vectors along the same direction but with different lengths.

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Worked Example: Scaling a Complex Number

If $z_1 = 3 + 2i$ :

$2z_1 = 2(3 + 2i) = 6 + 4i$
$3z_1 = 3(3 + 2i) = 9 + 6i$

Notice how all three points lie on the same straight line through the origin.

Adding the same complex number (translation)

When you add the same complex number to different complex numbers, you create a translation. This moves all points the same distance in the same direction.

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Key rule: Adding complex number $w = c + di$ to any complex number $z$ results in:

$z + w$ moves $z$ by distance $c$ horizontally and $d$ vertically

The diagram shows what happens when we add $4 + 2i$ to several different complex numbers. Each original point moves exactly 4 units right and 2 units up.

Translation properties:

All points move the same distance
All points move in the same direction
The shape and size of any figure remains unchanged
Only the position changes

Multiplying complex numbers by i (rotation)

Multiplying a complex number by i creates a rotation of 90° anticlockwise around the origin. This is one of the most important transformations in complex number theory.

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Key rotation rules:

$z \times i$ rotates $z$ through 90° anticlockwise
$z \times i^2$ rotates $z$ through 180°
$z \times i^3$ rotates $z$ through 270° anticlockwise

The circular diagram shows how successive multiplications by $i$ create quarter-turn rotations around the origin.

Working with powers of i

Remember these essential relationships:

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$i^1 = i$
$i^2 = -1$
$i^3 = -i$
$i^4 = 1$ (back to the start)

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Worked Example: Rotating with Powers of i

If $z = 4 + i$ :

$iz = i(4 + i) = 4i + i^2 = 4i - 1 = -1 + 4i$
$i^2z = -1(4 + i) = -4 - i$
$i^3z = -i(4 + i) = -4i - i^2 = -4i + 1 = 1 - 4i$

Worked example

Let's work through a complete example using an Argand diagram.

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Complete Worked Example: Complex Number Transformations

From the diagram, we can read the coordinates:

$z_1 = 2 + 2i$
$z_2 = 6 + 6i$
$z_3 = 7 + 3i$

Part (i): If $z_2 = a \cdot z_1$ , find $a$ .

Since $z_2 = 6 + 6i$ and $z_1 = 2 + 2i$ : $6 + 6i = a(2 + 2i)$ $6 + 6i = 2a + 2ai$

Comparing coefficients:

Real parts: $6 = 2a$ , so $a = 3$
Imaginary parts: $6 = 2a$ , so $a = 3$

Therefore, $a = 3$ .

Part (ii): If $z_3 = z_1 + z$ , find $z$ .

Since $z_3 = 7 + 3i$ and $z_1 = 2 + 2i$ : $7 + 3i = (2 + 2i) + z$ $z = (7 + 3i) - (2 + 2i)$ $z = 5 + i$

Exam tips

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Key Study Tips:

Always plot carefully: Use graph paper or a ruler to ensure accuracy when plotting complex numbers
Check your scaling: When multiplying by real numbers, verify that all points lie on straight lines through the origin
Rotation memory aid: Think "i = 90° anticlockwise" - this helps remember the rotation direction
Translation check: When adding complex numbers, verify that all points move the same distance in the same direction

Key formulas to memorise

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Essential Formulas:

Scaling: $|kz| = k|z|$ where $k$ is real
Rotation by i: $z \times i^n$ rotates by $90n°$ anticlockwise
Powers of i: $i^1 = i$ , $i^2 = -1$ , $i^3 = -i$ , $i^4 = 1$

Remember!

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Key Points to Remember:

Scaling transformations multiply both real and imaginary parts by the same factor and change the distance from the origin proportionally
Translation transformations add the same complex number to all points, moving everything the same distance in the same direction
Rotation by i turns complex numbers 90° anticlockwise around the origin each time
All transformations preserve the fundamental relationships between complex numbers while changing their positions
The Argand diagram is essential for visualising these transformations clearly

Complex Numbers and Transformations (Leaving Cert Mathematics): Revision Notes