Biomedical engineers in Kenya routinely encounter signals, impedances, and system responses that involve quantities best represented as complex numbers. Mastery of complex numbers enables precise analysis of electrical circuits in bioinstrumentation, modeling of physiological systems, and interpretation of biomedical imaging data. This chapter introduces complex numbers, their components, and graphical representation, forming a foundation for advanced applications in biomedical engineering mathematics.

1.1 Introduction to Complex Numbers

Biomedical engineers use complex numbers to represent signals and system behaviors that cannot be fully described by real numbers alone. This section covers the fundamental definitions and visualization tools critical for working with complex quantities.

1.1.1 Definition of a Complex Number

A complex number combines a real part and an imaginary part into a single mathematical entity. It is expressed as:

$$ z = x + yi $$

where \(x\) and \(y\) are real numbers, and \(i\) is the imaginary unit defined by:

$$ i^2 = -1 $$

Worked Examples

Example 1: A bio-signal is represented by the complex number \(3 + 4i\). Identify the real and imaginary parts.

Given: \( z = 3 + 4i \)

Real part \(x = 3\)

Imaginary part \(y = 4\)

Answer: Real part = 3, Imaginary part = 4

Example 2: Verify that \(i^2 = -1\) by calculation.

Given: \( i \)

Calculate:

$$ i^2 = i \times i $$

Since \( i = \sqrt{-1} \), then:

$$ i^2 = -1 $$

Answer: \( i^2 = -1 \)

Example 3: Express the complex number \( -2 - 5i \) in standard form.

Given: \( z = -2 - 5i \)

Standard form is \( x + yi \) where \( x = -2 \) and \( y = -5 \).

Answer: \( z = -2 - 5i \)

Example 4: If \( z = 7 + 0i \), what type of number is \(z\)?

Given: \( z = 7 + 0i \)

Since imaginary part \( y = 0 \), \( z \) is a real number.

Answer: \( z \) is a real number equal to 7

Example 5: Calculate \( (2 + 3i) + (4 - 5i) \).

Given: \( z_1 = 2 + 3i \), \( z_2 = 4 - 5i \)

Add real parts: \(2 + 4 = 6\)

Add imaginary parts: \(3i + (-5i) = -2i\)

Sum:

$$ z = 6 - 2i $$

Answer: \( 6 - 2i \)

1.1.2 Real and Imaginary Parts

The real part of a complex number is the component without \(i\), and the imaginary part is the coefficient of \(i\). These parts can be extracted and manipulated separately in biomedical engineering calculations.

If:

$$ z = x + yi $$

then:

• Real part: \( \text{Re}(z) = x \)
• Imaginary part: \( \text{Im}(z) = y \)

Worked Examples

Example 1: Determine the real and imaginary parts of \( z = -6 + 8i \).

Given: \( z = -6 + 8i \)

Real part \( x = -6 \)

Imaginary part \( y = 8 \)

Answer: Real part = -6, Imaginary part = 8

Example 2: For the complex number \( z = 0 - 9i \), find the real and imaginary parts.

Given: \( z = 0 - 9i \)

Real part \( x = 0 \)

Imaginary part \( y = -9 \)

Answer: Real part = 0, Imaginary part = -9

Example 3: If a complex impedance in a biomedical circuit is \( Z = 5 + 12i \) ohms, state the resistance and reactance.

Given: \( Z = 5 + 12i \) ohms

Resistance \( R = \text{Re}(Z) = 5 \, \Omega \)

Reactance \( X = \text{Im}(Z) = 12 \, \Omega \)

Answer: Resistance = 5 \(\Omega\), Reactance = 12 \(\Omega\)

Example 4: Extract real and imaginary parts from \( z = -3.5 - 0.7i \).

Given: \( z = -3.5 - 0.7i \)

Real part \( x = -3.5 \)

Imaginary part \( y = -0.7 \)

Answer: Real part = -3.5, Imaginary part = -0.7

Example 5: For \( z = 10i \), find the real and imaginary parts.

Given: \( z = 0 + 10i \)

Real part \( x = 0 \)

Imaginary part \( y = 10 \)

Answer: Real part = 0, Imaginary part = 10

1.1.3 Complex Plane: Argand Diagram

The Argand diagram is a graphical representation of complex numbers on a plane, with the horizontal axis representing the real part and the vertical axis representing the imaginary part.

A complex number \( z = x + yi \) is represented as the point \( (x, y) \) in this plane.

The magnitude (modulus) \( r \) and the argument (angle) \( \theta \) are defined as:

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

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

Worked Examples

Example 1: Plot \( z = 3 + 4i \) on the Argand diagram and find its modulus and argument.

Given: \( z = 3 + 4i \), \( x = 3 \), \( y = 4 \)

Calculate modulus:

$$ r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 $$

Calculate argument:

$$ \theta = \tan^{-1} ( \frac{4}{3} ) = \tan^{-1}(1.3333) \approx 53.13^\circ $$

Answer: Modulus \( r = 5 \), Argument \( \theta = 53.13^\circ \)

Example 2: For \( z = -5 + 12i \), find the modulus and argument.

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

Calculate modulus:

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

Calculate argument (noting \(x<0\), quadrant II):

$$ \theta = \tan^{-1} ( \frac{12}{-5} ) = \tan^{-1}(-2.4) \approx -67.38^\circ $$

Since \(x<0\), add 180°:

$$ \theta = 180^\circ - 67.38^\circ = 112.62^\circ $$

Answer: Modulus \( r = 13 \), Argument \( \theta = 112.62^\circ \)

Example 3: Find modulus and argument of \( z = -7 - 7i \).

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

Calculate modulus:

$$ r = \sqrt{(-7)^2 + (-7)^2} = \sqrt{49 + 49} = \sqrt{98} \approx 9.8995 $$

Calculate argument (quadrant III):

$$ \theta = \tan^{-1} ( \frac{-7}{-7} ) = \tan^{-1}(1) = 45^\circ $$

Add 180° because both \(x,y<0\):

$$ \theta = 180^\circ + 45^\circ = 225^\circ $$

Answer: Modulus \( r \approx 9.8995 \), Argument \( \theta = 225^\circ \)

Example 4: Determine modulus and argument of \( z = 0 + 10i \).

Given: \( x = 0 \), \( y = 10 \)

Calculate modulus:

$$ r = \sqrt{0^2 + 10^2} = \sqrt{100} = 10 $$

Calculate argument:

Since \( x=0 \), \( \theta = 90^\circ \) (positive imaginary axis)

Answer: Modulus \( r = 10 \), Argument \( \theta = 90^\circ \)

Example 5: Find modulus and argument of \( z = 8 + 0i \).

Given: \( x = 8 \), \( y = 0 \)

Calculate modulus:

$$ r = \sqrt{8^2 + 0^2} = 8 $$

Calculate argument:

Since \( y=0 \), \( \theta = 0^\circ \) (positive real axis)

Answer: Modulus \( r = 8 \), Argument \( \theta = 0^\circ \)

Chapter Summary

This chapter began with an introduction to complex numbers, defining them as numbers composed of real and imaginary parts and representing them on the complex plane using the Argand diagram. It then covered arithmetic operations including addition, subtraction, multiplication, division, and the use of the conjugate to simplify division. The chapter proceeded to explore the polar form of complex numbers, demonstrating how to convert between Cartesian and polar forms, and introduced the exponential form through Euler's formula. Further, it examined complex number operations such as calculating powers using De Moivre's Theorem and finding roots of complex numbers. The properties and applications of complex exponential functions were also discussed, particularly in solving differential equations. The final section focused on solving equations with complex numbers, addressing quadratic equations with real and complex coefficients, analyzing the nature of their roots, and solving systems of linear equations using complex numbers, highlighting practical applications such as AC circuit analysis.