Complex numbers are fundamental in chemical engineering calculations involving oscillations, wave functions, and signal processing. In Kenya's chemical industry, understanding complex numbers enables engineers to model dynamic systems such as reactor oscillations, control systems, and fluid flow behavior effectively. This chapter develops the mathematical tools for representing, manipulating, and applying complex numbers in engineering contexts.

3.1 Definition of complex numbers and their properties

Complex numbers extend the real number system to include solutions to equations like \(x^2 + 1 = 0\) which have no real roots. A complex number has a real part and an imaginary part, expressed as \(z = x + jy\), where \(j = \sqrt{-1}\). This section covers the algebraic and geometric representation of complex numbers, including conjugates, modulus, and argument.

3.1.1 Complex number definition and algebraic form

A complex number \(z\) is defined as

$$ z = x + jy $$

where \(x\) is the real part and \(y\) is the imaginary part. The imaginary unit \(j\) satisfies

$$ j^2 = -1 $$

This form is called the Cartesian or rectangular form.

Worked Examples

Example 1:
Identify the real and imaginary parts of \(z = 4 + j3\).

Given:
\(z = 4 + j3\)

Real part \(x = 4\)
Imaginary part \(y = 3\)

Answer: Real part = 4, Imaginary part = 3

Example 2:
Express \(z = -2 - j5\) in algebraic form and identify parts.

Given:
\(z = -2 - j5\)

Real part \(x = -2\)
Imaginary part \(y = -5\)

Answer: Real part = -2, Imaginary part = -5

Example 3:
Write the complex number with real part 7 and imaginary part \(-4\).

Given:
\(x = 7\), \(y = -4\)

$$ z = 7 - j4 $$

Answer: \(z = 7 - j4\)

3.1.2 Complex conjugate, modulus, and argument

The complex conjugate of \(z = x + jy\) is

$$ \overline{z} = x, jy $$

The modulus (magnitude) is

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

The argument \(\theta\) (angle with positive real axis) is

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

Worked Examples

Example 1:
Find the conjugate, modulus, and argument of \(z = 3 + j4\).

Given:
\(z = 3 + j4\)

Conjugate:$$ \overline{z} = 3 - j4 $$

Modulus:$$ |z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 $$

Argument:$$ \theta = \tan^{-1}(\frac{4}{3}) \approx 53.13^\circ $$

Answer:
Conjugate = \(3 - j4\), Modulus = 5, Argument = 53.13°

Example 2:
Calculate modulus and argument of \(z = -1 + j\sqrt{3}\).

Given:
\(z = -1 + j1.732\)

Modulus:$$ |z| = \sqrt{(-1)^2 + (1.732)^2} = \sqrt{1 + 3} = \sqrt{4} = 2 $$

Argument:
Since real part negative and imaginary positive, \(\theta\) in 2nd quadrant:

$$ \theta = 180^\circ - \tan^{-1}(\frac{1.732}{1}) = 180^\circ - 60^\circ = 120^\circ $$

Answer: Modulus = 2, Argument = 120°

Example 3:
Find conjugate and modulus of \(z = -4 - j3\).

Given:
\(z = -4 - j3\)

Conjugate:$$ \overline{z} = -4 + j3 $$

Modulus:$$ |z| = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 $$

Answer: Conjugate = \(-4 + j3\), Modulus = 5

3.1.3 Argand diagram representation of complex numbers

The Argand diagram is a graphical representation of complex numbers on the Cartesian plane where the horizontal axis is the real part and the vertical axis is the imaginary part. Each complex number corresponds to a point \((x, y)\).

Worked Examples

Example 1:
Plot the point for \(z = 2 + j3\) on the Argand diagram.

Given:
\(x = 2\), \(y = 3\)

Plot point at (2,3) with horizontal 2 units right and vertical 3 units up.

Answer: Point plotted at (2,3)

Example 2:
Determine the quadrant of \(z = -3 + j4\).

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

Since \(x < 0\) and \(y > 0\), point lies in quadrant II.

Answer: Quadrant II

Example 3:
Find coordinates and plot \(z = -5 - j2\).

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

Plot point at (-5,-2) in quadrant III.

Answer: Point plotted at (-5,-2)

3.1.4 Arithmetic operations on complex numbers

Operations include addition, subtraction, multiplication, and division. For \(z_1 = x_1 + jy_1\) and \(z_2 = x_2 + jy_2\):

• Addition:$$ z_1 + z_2 = (x_1 + x_2) + j(y_1 + y_2) $$

• Subtraction:$$ z_1 - z_2 = (x_1 - x_2) + j(y_1 - y_2) $$

• Multiplication:$$ z_1 z_2 = (x_1 x_2 - y_1 y_2) + j(x_1 y_2 + y_1 x_2) $$

• Division:$$ \frac{z_1}{z_2} = \frac{(x_1 x_2 + y_1 y_2) + j(y_1 x_2 - x_1 y_2)}{x_2^2 + y_2^2} $$

Worked Examples

Example 1:
Add \(z_1 = 3 + j2\) and \(z_2 = 1 + j4\).

Given:
\(z_1 = 3 + j2\), \(z_2 = 1 + j4\)

Addition:$$ z_1 + z_2 = (3 + 1) + j(2 + 4) = 4 + j6 $$

Answer: \(4 + j6\)

Example 2:
Subtract \(z_2 = 5 + j3\) from \(z_1 = 8 + j7\).

Given:
\(z_1 = 8 + j7\), \(z_2 = 5 + j3\)

Subtraction:$$ z_1 - z_2 = (8 - 5) + j(7 - 3) = 3 + j4 $$

Answer: \(3 + j4\)

Example 3:
Multiply \(z_1 = 2 + j3\) and \(z_2 = 4 - j1\).

Given:
\(z_1 = 2 + j3\), \(z_2 = 4 - j1\)

Multiplication:$$ (2)(4) - (3)(-1) + j[(2)(-1) + (3)(4)] = 8 + 3 + j(-2 + 12) = 11 + j10 $$

Answer: \(11 + j10\)

Example 4:
Divide \(z_1 = 5 + j6\) by \(z_2 = 1 - j2\).

Given:
\(z_1 = 5 + j6\), \(z_2 = 1 - j2\)

Calculate denominator:$$ 1^2 + (-2)^2 = 1 + 4 = 5 $$

Calculate numerator real part:$$ (5)(1) + (6)(-2) = 5 - 12 = -7 $$

Calculate numerator imaginary part:$$ (6)(1) - (5)(-2) = 6 + 10 = 16 $$

Division:$$ \frac{-7 + j16}{5} = -\frac{7}{5} + j\frac{16}{5} = -1.4 + j3.2 $$

Answer: \(-1.4 + j3.2\)

Chapter Summary

This chapter introduced the concept of complex numbers, defining them as numbers composed of a real part and an imaginary part. It explained how complex numbers can be expressed in terms of their conjugate, argument, and modulus, providing a foundation for their geometric interpretation. The use of the Argand diagram was demonstrated to visually represent complex numbers as points or vectors in the complex plane. Various arithmetic operations on complex numbers, including addition, subtraction, multiplication, and division, were covered to develop computational skills. The chapter also explored De Moivre's theorem, showing how it simplifies the computation of powers and roots of complex numbers. Finally, practical applications of complex numbers in engineering contexts were discussed, highlighting their importance in solving real-world problems involving oscillations, electrical circuits, and signal analysis.