Complex numbers are essential in Quantity Surveying for analysing phenomena such as vibrations in structures and electrical systems affecting construction projects. Representing complex numbers graphically using Argand diagrams allows Quantity Surveyors to visualise and manipulate these quantities effectively in two-dimensional space. This chapter focuses on the representation of complex numbers on the Argand plane, which forms the foundation for further operations such as addition, subtraction, and multiplication of complex numbers relevant to engineering and surveying applications.

3.1 Representation of complex number using Argand diagrams

3.1.1 Complex numbers and their geometric interpretation

A complex number has the form \(z = x + yi\), where \(x\) is the real part and \(y\) is the imaginary part, and \(i\) is the imaginary unit with the property \(i^2 = -1\). The Argand diagram represents \(z\) as a point or vector in the two-dimensional plane with the horizontal axis representing the real part and the vertical axis representing the imaginary part. The position of the point \(z\) is given by coordinates \((x, y)\).

The geometric interpretation allows Quantity Surveyors to visualise operations on complex numbers as movements or transformations in the plane. The magnitude (or modulus) of the complex number corresponds to the distance from the origin to the point, while the argument (or angle) corresponds to the angle between the positive real axis and the line joining the origin to the point.

The modulus \(|z|\) and argument \(\theta\) are given by the formulas:

$$|z| = \sqrt{x^2 + y^2}$$

$$\theta = \tan^{-1}(\frac{y}{x})$$

Worked Examples

Example 1: A Quantity Surveyor encounters a complex number \(z = 3 + 4i\) representing a force vector in a structural analysis. Find its modulus and argument.

Given: \(x=3\), \(y=4\)

$$|z| = \sqrt{3^2 + 4^2}$$

$$|z| = \sqrt{9 + 16}$$

$$|z| = \sqrt{25}$$

$$|z| = 5$$

$$\theta = \tan^{-1}(\frac{4}{3})$$

$$\theta = \tan^{-1}(1.3333)$$

$$\theta \approx 53.13^\circ$$

Answer: modulus = 5, argument = 53.13 degrees

Example 2: Find the modulus and argument of \(z = -2 + 2i\), which may represent a complex load in construction.

Given: \(x = -2\), \(y = 2\)

$$|z| = \sqrt{(-2)^2 + 2^2}$$

$$|z| = \sqrt{4 + 4}$$

$$|z| = \sqrt{8}$$

$$|z| = 2.828$$

$$\theta = \tan^{-1}(\frac{2}{-2})$$

$$\theta = \tan^{-1}(-1)$$

Since \(x < 0\) and \(y > 0\), \(\theta\) is in the second quadrant:

$$\theta = 180^\circ - 45^\circ = 135^\circ$$

Answer: modulus = 2.828, argument = 135 degrees

Example 3: A Quantity Surveyor needs to find the magnitude and angle of \(z = -3 - 4i\) representing a vibration force in a machine foundation.

Given: \(x = -3\), \(y = -4\)

$$|z| = \sqrt{(-3)^2 + (-4)^2}$$

$$|z| = \sqrt{9 + 16}$$

$$|z| = 5$$

$$\theta = \tan^{-1}(\frac{-4}{-3})$$

$$\theta = \tan^{-1}(1.3333)$$

Since \(x < 0\) and \(y < 0\), \(\theta\) is in the third quadrant:

$$\theta = 180^\circ + 53.13^\circ = 233.13^\circ$$

Answer: modulus = 5, argument = 233.13 degrees

3.1.2 Plotting complex numbers on the Argand diagram

Plotting complex numbers on the Argand diagram involves marking the point corresponding to the real and imaginary parts in a rectangular coordinate system. The real axis is horizontal, and the imaginary axis is vertical. This visual representation helps Quantity Surveyors interpret complex quantities in engineering contexts such as load analysis or signal processing.

Steps to plot a complex number \(z = x + yi\) on the Argand diagram:

1. Draw the horizontal real axis and vertical imaginary axis intersecting at the origin.

2. Locate the point on the real axis corresponding to \(x\).

3. Locate the point on the imaginary axis corresponding to \(y\).

4. From the origin, move \(x\) units along the real axis.

5. From this point, move \(y\) units parallel to the imaginary axis.

6. Mark the resulting point as \(z\).

7. Draw a vector from the origin to the point \(z\).

Worked Examples

Example 1: Plot \(z = 4 + 3i\) on the Argand diagram.

Given: \(x=4\), \(y=3\)

Steps:

1. Draw axes.

2. Locate 4 units on the real axis.

3. From 4 on real axis, move 3 units up on imaginary axis.

4. Mark point (4,3).

5. Draw vector from origin to (4,3).

Example 2: Plot \(z = -2 + 5i\).

Given: \(x = -2\), \(y = 5\)

Steps:

1. Draw axes.

2. Locate -2 on real axis (left of origin).

3. From -2 on real axis, move 5 units up on imaginary axis.

4. Mark point (-2,5).

5. Draw vector from origin to (-2,5).

Example 3: Plot \(z = -3 - 4i\).

Given: \(x = -3\), \(y = -4\)

Steps:

1. Draw axes.

2. Locate -3 on real axis.

3. From -3 on real axis, move 4 units down (since \(y\) is negative).

4. Mark point (-3,-4).

5. Draw vector from origin to (-3,-4).

3.1.3 Polar form representation of complex numbers

Complex numbers can also be represented in polar form as \(z = r(\cos \theta + i \sin \theta)\), where \(r\) is the modulus and \(\theta\) the argument. This form is particularly useful for multiplication, division, and finding powers and roots of complex numbers, which can be relevant in Quantity Surveying when dealing with periodic phenomena or alternating quantities.

The conversion from rectangular to polar form uses:

$$r = \sqrt{x^2 + y^2}$$

$$\theta = \tan^{-1}(\frac{y}{x})$$

The polar form is then:

$$z = r(\cos \theta + i \sin \theta)$$

Worked Examples

Example 1: Convert \(z = 1 + i\) to polar form.

Given: \(x=1\), \(y=1\)

$$r = \sqrt{1^2 + 1^2} = \sqrt{2} = 1.414$$

$$\theta = \tan^{-1}(\frac{1}{1}) = 45^\circ$$

$$z = 1.414 (\cos 45^\circ + i \sin 45^\circ)$$

Example 2: Convert \(z = -3 + 3i\) to polar form.

Given: \(x = -3\), \(y = 3\)

$$r = \sqrt{(-3)^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 4.243$$

Since \(x<0\) and \(y>0\), \(\theta = 180^\circ - \tan^{-1}(3/3) = 180^\circ - 45^\circ = 135^\circ\)

$$z = 4.243 (\cos 135^\circ + i \sin 135^\circ)$$

Example 3: Convert \(z = -4 - 4i\) to polar form.

Given: \(x = -4\), \(y = -4\)

$$r = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 5.657$$

Since \(x<0\), \(y<0\), \(\theta = 180^\circ + \tan^{-1}(4/4) = 180^\circ + 45^\circ = 225^\circ\)

$$z = 5.657 (\cos 225^\circ + i \sin 225^\circ)$$

3.1.4 Modulus and argument calculations in surveying contexts

In Quantity Surveying, determining the magnitude and direction of complex quantities like forces, displacements, or electrical loads is crucial. The modulus gives the size, while the argument gives the direction relative to a reference axis. Calculating these accurately aids in cost estimation, structural analysis, and project planning.

The modulus and argument are calculated as:

$$|z| = \sqrt{x^2 + y^2}$$

$$\theta = \tan^{-1}(\frac{y}{x})$$

Careful quadrant consideration is necessary for the argument to reflect the correct direction.

Worked Examples

Example 1: A load vector is represented by \(z = 6 + 8i\). Find its magnitude and direction.

Given: \(x=6\), \(y=8\)

$$|z| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$$

$$\theta = \tan^{-1}(\frac{8}{6}) = \tan^{-1}(1.3333) = 53.13^\circ$$

Answer: magnitude = 10, direction = 53.13 degrees

Example 2: A displacement vector \(z = -5 + 12i\) is measured. Find its magnitude and argument.

Given: \(x = -5\), \(y = 12\)

$$|z| = \sqrt{(-5)^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$$

Since \(x < 0\), \(y > 0\), argument is:

$$\theta = 180^\circ - \tan^{-1}(\frac{12}{5}) = 180^\circ - 67.38^\circ = 112.62^\circ$$

Answer: magnitude = 13, argument = 112.62 degrees

Example 3: Electrical load represented by \(z = -7 - 24i\). Find magnitude and direction.

Given: \(x = -7\), \(y = -24\)

$$|z| = \sqrt{(-7)^2 + (-24)^2} = \sqrt{49 + 576} = \sqrt{625} = 25$$

Since \(x < 0\), \(y < 0\):

$$\theta = 180^\circ + \tan^{-1}(\frac{24}{7}) = 180^\circ + 73.74^\circ = 253.74^\circ$$

Answer: magnitude = 25, argument = 253.74 degrees

Chapter Summary

This chapter introduced the representation of complex numbers on Argand diagrams, where each complex number corresponds to a point in the plane defined by its real and imaginary parts. It covered plotting complex numbers and interpreting their magnitude and argument geometrically. The chapter then explored various operations involving complex numbers, including addition, subtraction, multiplication, division, and conjugation, with an emphasis on both algebraic and geometric interpretations. Finally, the chapter focused on calculations using De Moivre's theorem, demonstrating how to raise complex numbers to integer powers and extract roots efficiently by working with their polar forms. Through these topics, students gained the skills to visualize, manipulate, and apply complex numbers in engineering contexts. The combination of geometric insight and algebraic techniques equips learners to solve practical problems involving complex quantities.